diff --git "a/papers/0909/0909.0910.tex" "b/papers/0909/0909.0910.tex" new file mode 100644--- /dev/null +++ "b/papers/0909/0909.0910.tex" @@ -0,0 +1,8567 @@ + + +\documentclass{amsbook} + +\usepackage{graphics} + + +\newcommand{\HH}{{\mathcal H}} +\newcommand{\LL}{{\mathcal L}} +\newcommand{\Sc}{{\mathcal S}} +\newcommand{\vth}{\vartheta} +\newcommand{\CC}{{\mathbb C}} +\newcommand{\RR}{{\mathbb R}} +\newcommand{\ZZ}{{\mathbb Z}} +\newcommand{\PP}{{\mathbb P}} +\newcommand{\tDl}{\tilde{\Delta}} +\newcommand{\tla}{\tilde{\lambda}} +\newcommand{\vph}{\varphi} +\newcommand{\bbR}{{\rm I\!R}} +\newcommand{\cS}{\mathcal S} +\newcommand{\vq}{\vec{q}} +\newcommand{\vp}{\vec{p}} +\newcommand{\vf}{\vec{f}} +\newcommand{\vF}{\vec{F}} +\newcommand{\e}{\epsilon} +\newcommand{\U}{{\mathcal U}} +\newcommand{\tu}{\tilde{u}} +\newcommand{\tq}{\tilde{q}} +\newcommand{\tQ}{\tilde{Q}} +\newcommand{\vQ}{\vec{Q}} +\newcommand{\A}{{\mathcal A}} +\newcommand{\B}{{\mathcal B}} +\newcommand{\C}{{\mathcal C}} +\renewcommand{\k}{\kappa} +\newcommand{\ga}{\gamma} +\newcommand{\Ga}{\Gamma} +\newcommand{\ve}{{\bf e}} +\newcommand{\he}{\hat{e}} +\newcommand{\hk}{\hat{k}} +\newcommand{\vv}{\vec{v}} +\newcommand{\hcS}{\hat{\cS}} +\newcommand{\tcS}{\tilde{\cS}} +\newcommand{\hS}{\hat{S}} +\newcommand{\tS}{\tilde{S}} +\newcommand{\hD}{\hat{D}} +\newcommand{\hJ}{\hat{J}} +\newcommand{\tD}{\tilde{D}} +\newcommand{\tC}{\tilde{C}} +\newcommand{\dl}{\delta} +\newcommand{\Dl}{\Delta} +\renewcommand{\th}{\theta} +\newcommand{\ra}{\rightarrow} +\newcommand{\al}{\alpha} +\newcommand{\be}{\beta} +\newcommand{\sg}{\sigma} +\newcommand{\Sg}{\Sigma} +\newcommand{\bM}{\bar{M}} +\newcommand{\pa}{\partial} +\newcommand{\z}{\zeta} +\newcommand{\Z}{\ZZ^2/\{0\}} +\newcommand{\hQ}{\hat{Q}} +\newcommand{\hv}{\hat{v}} +\newcommand{\hw}{\hat{w}} +\newcommand{\hx}{\hat{x}} +\newcommand{\hy}{\hat{y}} +\newcommand{\hz}{\hat{z}} +\newcommand{\bv}{\bar{v}} +\newcommand{\bw}{\bar{w}} +\newcommand{\La}{\Lambda} +\newcommand{\tLa}{\tilde{\Lambda}} +\newcommand{\la}{\lambda} +\newcommand{\bq}{\bar{q}} +\newcommand{\bp}{\bar{p}} +\newcommand{\bQ}{\bar{Q}} +\newcommand{\bE}{\bar{E}} +\newcommand{\nid}{\noindent} +\newcommand{\F}{{\mathcal F}} +\newcommand{\N}{{\mathcal N}} +\newcommand{\rc}{S_\omega} +\newcommand{\hrc}{\hat{S}_\omega} +\newcommand{\bW}{\bar{W}} +\newcommand{\hN}{\hat{N}} +\newcommand{\hF}{\hat{F}} +\newcommand{\tF}{\tilde{F}} +\newcommand{\om}{\omega} +\newcommand{\Om}{\Omega} +\newcommand{\na}{\nabla} +\newcommand{\lag}{\langle} +\newcommand{\rag}{\rangle} +\newcommand{\tx}{\tilde{x}} +\newcommand{\ty}{\tilde{y}} +\newcommand{\tz}{\tilde{z}} +\newcommand{\vtQ}{\vec{\tilde{Q}}} +\newcommand{\ttau}{\tilde{\tau}} +\newcommand{\htau}{\hat{\tau}} +\newcommand{\hrho}{\hat{\rho}} +\newcommand{\hvth}{\hat{\vartheta}} +\newcommand{\tb}{\tilde{b}} +\newcommand{\W}{{\mathcal W}} +\renewcommand{\O}{{\mathcal O}} + +\newcommand{\bd}{\bar{d}} +\newcommand{\bb}{\bar{b}} +\newcommand{\ba}{\bar{a}} +\newcommand{\bc}{\bar{c}} +\newcommand{\Q}{\widehat{Q}} + +\newcommand{\q}{\widehat{q}} + +\newcommand{\R}{\widehat{R}} + +\renewcommand{\r}{\widehat{r}} + +\renewcommand{\a}{\widehat{a}} + +\newcommand{\im}{\mathop{\rm Im}\nolimits} + +\newcommand{\sign}{\mathop{\rm sign}\nolimits} + +\newcommand{\BD}{{B\"acklund-Darboux transformations}} + + +\newcommand{\Der}{\mathop{\rm Der}\nolimits} + +\def\maprightu#1{\smash{ + \mathop{\longrightarrow}\limits^{#1}}} +\def\maprightd#1{\smash{ + \mathop{\longrightarrow}\limits_{#1}}} +\def\mapdownl#1{ + \llap{$\vcenter{\hbox{$\scriptstyle#1$}}$}\Big\downarrow} +\def\mapdownr#1{\Big\downarrow + \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} +\def\mapupl#1{ + \llap{$\vcenter{\hbox{$\scriptstyle#1$}}$}\Big\uparrow} +\def\mapupr#1{\Big\uparrow + \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} + + + +\includeonly{preface,setup,integrable,NLSappl,feight,MelV,invman,horbit, +hshoe,Sdbc,lax,leuler,arnold,misce,chaos,index} + +\newtheorem{theorem}{Theorem}[chapter] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{proposition}[theorem]{Proposition} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{example}[theorem]{Example} +\newtheorem{xca}[theorem]{Exercise} + +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} + +\numberwithin{section}{chapter} +\numberwithin{equation}{chapter} + + +\newcommand{\abs}[1]{\lvert#1\rvert} + +\newcommand{\blankbox}[2]{\parbox{\columnwidth}{\centering +\setlength{\fboxsep}{0pt}\fbox{\raisebox{0pt}[#2]{\hspace{#1}}}}} + +\begin{document} +\frontmatter +\title{Chaos in Partial Differential Equations} + +\author{Y. Charles Li}\address{Department of Mathematics, University of Missouri, +Columbia, MO 65211} + + + + + + + + + + + + +\maketitle + +\setcounter{page}{6} +\tableofcontents + +\clearpage{}\chapter*{Preface} + +The area: Chaos in Partial Differential Equations, is at its fast developing +stage. Notable results have been obtained in recent years. The present book +aims at an overall survey on the existing results. On the other hand, we +shall try to make the presentations introductory, so that beginners can +benefit more from the book. + +It is well-known that the theory of chaos in finite-dimensional dynamical +systems has been well-developed. That includes both discrete maps and systems +of ordinary differential equations. Such a theory has produced important +mathematical theorems and led to important applications in physics, +chemistry, biology, and engineering etc.. For a long period of time, +there was no theory on chaos in partial differential equations. On the other +hand, the demand for such a theory is +much stronger than for finite-dimensional systems. Mathematically, +studies on infinite-dimensional systems pose much more challenging problems. +For example, as phase spaces, Banach spaces possess much more structures +than Euclidean spaces. In terms of applications, most of important natural +phenomena are described by partial differential equations -- nonlinear wave +equations, Maxwell equations, Yang-Mills equations, and Navier-Stokes +equations, to name a few. Recently, the author and collaborators have +established a systematic theory on chaos in nonlinear wave equations. + +Nonlinear wave equations are the most important class of equations in +natural sciences. They describe a wide spectrum of phenomena -- motion of +plasma, nonlinear optics (laser), water waves, vortex motion, to name a +few. Among these nonlinear wave equations, there is a class of equations +called soliton equations. This class of equations describes a variety of +phenomena. In particular, the same soliton equation describes several +different phenomena. Mathematical theories on soliton equations have been well +developed. Their Cauchy problems are completely solved through inverse +scattering transforms. Soliton equations are integrable Hamiltonian +partial differential equations which are the natural counterparts of +finite-dimensional integrable Hamiltonian systems. We have established a +standard program for proving the existence of chaos in perturbed soliton +equations, with the machineries: 1. Darboux transformations for soliton +equations, 2. isospectral theory for soliton equations under periodic boundary +condition, 3. persistence of invariant manifolds and Fenichel fibers, +4. Melnikov analysis, 5. Smale horseshoes and symbolic dynamics, 6. shadowing +lemma and symbolic dynamics. + +The most important implication of the theory on chaos in partial +differential equations in theoretical physics will be on the study of +turbulence. For that goal, we chose the 2D Navier-Stokes +equations under periodic boundary conditions to begin a dynamical system +study on 2D turbulence. Since they possess Lax pair +and Darboux transformation, the 2D Euler equations are the +starting point for an analytical study. The high Reynolds number 2D +Navier-Stokes equations are viewed as a singular perturbation of the 2D Euler +equations through the perturbation parameter $\e = 1/Re$ which is the +inverse of the Reynolds number. + +Our focus will be on nonlinear wave equations. New results on shadowing lemma +and novel results related to Euler equations of inviscid fluids will also be +presented. The chapters on figure-eight structures and Melnikov vectors +are written in great details. The readers can learn these machineries +without resorting to other references. In other chapters, details of proofs +are often omitted. Chapters 3 to 7 illustrate how to prove the existence of +chaos in perturbed soliton equations. Chapter 9 contains the most recent +results on Lax pair structures of Euler equations of inviscid fluids. In +chapter 12, we give brief comments on other related topics. + +The monograph will be of interest to researchers in mathematics, physics, +engineering, chemistry, biology, and science in general. Researchers who +are interested in chaos in high dimensions, will find the book of particularly +valuable. The book is also accessible to graduate students, and can be +taken as a textbook for advanced graduate courses. + +I started writing this book in 1997 when I was at MIT. This project continued +at Institute for Advanced Study during the year 1998-1999, and at University +of Missouri - Columbia since 1999. In the Fall of 2001, I started to rewrite +from the old manuscript. Most of the work was done in the summer of 2002. +The work was partially supported by an AMS centennial fellowship in 1998, +and a Guggenheim fellowship in 1999. + +Finally, I would like to thank my wife Sherry and my son Brandon for their +strong support and appreciation.\clearpage{} + +\mainmatter +\clearpage{}\chapter{General Setup and Concepts} + +We are mainly concerned with the Cauchy problems of +partial differential equations, and view them as defining flows +in certain Banach spaces. Unlike the Euclidean space +$\RR^n$, such Banach spaces admit a variety of norms +which make the structures in infinite dimensional dynamical +systems more abundant. The main difficulty in studying infinite +dimensional dynamical systems often comes from the fact that the +evolution operators for the partial differential equations are +usually at best $C^0$ in time, in contrast to finite +dimensional dynamical systems where the evolution operators are +$C^1$ smooth in time. The well-known concepts for finite +dimensional dynamical systems can be generalized to infinite +dimensional dynamical systems, and this is the main task of this +chapter. + + +\section{Cauchy Problems of Partial Differential + Equations} + +The types of evolution equations studied in this book can be +casted into the general form, +\begin{equation} + \partial_t Q = G(Q, \partial_xQ, \dots, \partial^{\ell}_x Q)\ , +\label{geq} +\end{equation} +where $t \in \RR^1$ (time), $x = (x_1, \dots, x_n) \in +\RR^n$, $Q = (Q_1, \dots, Q_m)$ and $G= (G_1, \dots, G_m)$ +are either real or complex valued functions, and +$\ell$, $m$ and $n$ are integers. The equation (\ref{geq}) is +studied under certain boundary conditions, for example, + +\begin{itemize} +\item periodic boundary conditions, e.g. $Q$ is periodic in each + component of $x$ with period $2\pi $, +\item decay boundary conditions, e.g. $Q \ra 0$ as $x \ra \infty$. +\end{itemize} +Thus we have Cauchy problems for the equation (\ref{geq}), and we +would like to pose the Cauchy problems in some Banach spaces +$\mathcal{H}$, for example, + +\begin{itemize} +\item $\mathcal{H}$ can be a Sobolev space $H^k$, +\item $\mathcal{H}$ can be a Solobev space $H^k_{e,p}$ of even + periodic functions. +\end{itemize} +We require that the problem is well-posed in $\mathcal{H}$, for +example, + +\begin{itemize} +\item for any $Q_0 \in \mathcal{H}$, there exists a unique +solution $Q = Q(t,Q_0) \in C^0[(-\infty, \infty); \mathcal{H}]$ +or $C^0[[0; \infty), \mathcal{H}]$ to the equation (\ref{geq}) +such that $Q(0,Q_0)=Q_0$, +\item for any fixed $t_0 \in (-\infty,\infty)$ or $[0,\infty)$, +$Q(t_0,Q_0)$ is a $C^r$ function of $Q_0$, for $Q_0 \in \mathcal{H}$ +and some integer $r \geq 0$. +\end{itemize} + +{\bf Example:} +Consider the integrable cubic nonlinear Schr\"{o}dinger (NLS) equation, +\begin{equation} + iq_t = q_{xx} + 2 \left[\mid q \mid^2 - \omega^2 \right] q\,, +\label{inls} +\end{equation} +where $i=\sqrt{-1}$, $t\in \RR^1$, $x \in \RR^1$, $q$ is a +complex-valued +function of $(t,x)$, and $\omega$ is a real constant. +We pose the periodic boundary condition, +\begin{displaymath} + q(t, x +1) = q(t,x)\,. +\end{displaymath} +The Cauchy problem for equation (\ref{inls}) is posed in the +Sobolev space $H^1$ of periodic functions, +\begin{eqnarray*} + \mathcal{H}& \equiv& \left\{ Q = (q, \bar{q}) \ \bigg| \ q(x + + 1) = q(x), \ q \in H^1_{[0,1]}: \hbox{the} \right.\\ +&&\qquad \hbox{Sobolev space $H^1$ over the period interval}\ [0,1] +\bigg\}\, , +\end{eqnarray*} +and is well-posed \cite{Caz89} \cite{Bou93} \cite{Bou94}. + +Fact 1: For any $Q_0 \in \mathcal{H}$, there exists a unique solution $Q += Q(t,Q_0) \in C^0 [ ( -\infty, \infty), \mathcal{H} ]$ to +the equation (\ref{inls}) such that $Q (0,Q_0) = Q_0$. + +Fact 2: For any fixed $t_0 \in ( -\infty, \infty)$, $Q (t_0, Q_0)$ is a +$C^2$ function of $Q_0$, for $Q_0 \in \mathcal{H}$. + + +\section{Phase Spaces and Flows} + + +For finite dimensional dynamical systems, the phase spaces are +often $\RR^n$ or $\CC^n$. For infinite dimensional +dynamical systems, we take the Banach space $\mathcal{H}$ +discussed in the previous section as the counterpart. + +\begin{definition} +We call the Banach space $\mathcal{H}$ in which the Cauchy +problem for (\ref{geq}) is well-posed, a \emph{phase +space}. Define an operator $F^t$ labeled by $t$ as +\begin{displaymath} + Q(t,Q_0) = F^t (Q_0)\,; +\end{displaymath} +then $F^t:\mathcal{H} \to \mathcal{H}$ is called the +\emph{evolution operator} (or flow) for the system (\ref{geq}). +\end{definition} + +A point $p \in \mathcal{H}$ is called a \emph{fixed point} if $F^t (p) = +p$ for any $t$. Notice that here the fixed point $p$ is in fact a +function of $x$, which is the so-called stationary solution of +(\ref{geq}). Let $q \in \mathcal{H}$ be a point; then $\ell_q \equiv \{ F^t +(q), \hbox{for all}\ t \}$ is called the orbit with initial point +$q$. An orbit $\ell_q$ is called a \emph{periodic orbit} if there +exists a $T \in(-\infty, \infty)$ such that $F^T(q) = q$. An +orbit $\ell_q$ is called a \emph{homoclinic orbit} if there +exists a point $q_{\ast} \in \mathcal{H}$ such that $F^t(q) \to q_{\ast}$, +as $\mid t\mid \to \infty$, and $q_{\ast}$ is called the asymptotic +point of the homoclinic orbit. An orbit $\ell_q$ is called a +\emph{heteroclinic orbit} if there exist two different points +$q_\pm \in \mathcal{H}$ such that $F^t(q) \to q_{\pm}$, as $t \to \pm +\infty$, and $q_{\pm}$ are called the asymptotic points of the +heteroclinic orbit. An orbit $\ell_q$ is said to be homoclinic to +a submanifold $W$ of $\mathcal{H}$ if $\inf_{Q\in W} \parallel +F^t(q) -Q \parallel \to 0$, as $\mid t \mid \to \infty$. + +{\bf Example 1:} +Consider the same Cauchy problem for the system (\ref{inls}). +The fixed points of (\ref{inls}) satisfy the second order +ordinary differential equation +\begin{equation} +q_{xx} + 2 \left[\mid q \mid^2 - \omega^2 \right] q = 0\,. +\label{ftq} +\end{equation} +In particular, there exists a circle of fixed points $q = \omega +e^{i\gamma}$, where $\gamma \in [0,2 \pi]$. For simple periodic +solutions, we have +\begin{equation} + q = ae^{i\theta (t)}, \quad \theta(t) = - \left[2 (a^2 - + \omega^2) t - \gamma \right]\,; +\label{spsl} +\end{equation} +where $a> 0 $, and $\gamma \in [0, 2\pi]$. For +orbits homoclinic to the circles (\ref{spsl}), we have +\begin{eqnarray} + q &=& \frac{1}{\Lambda} \bigg [ \cos 2 p - \sin p \ \mbox{sech}\ \tau \cos + 2 \pi x - i \sin 2 p \tanh \tau \bigg ] a e^{i \theta + (t)}\,,\label{horb} \\ \nonumber \\ +&&\quad \Lambda = 1 + \sin p \ \mbox{sech}\ \tau \cos 2 \pi x\,,\nonumber +\end{eqnarray} +where $\tau = 4 \pi \sqrt{a^2 - \pi^2} \ t + \rho$, $p = \arctan +\bigg [ \frac{\sqrt{a^2 - \pi^2}}{\pi} \bigg ]$, $\rho \in (-\infty, \infty)$ +is the B\"{a}cklund parameter. Setting $a= \omega$ in +(\ref{horb}), we have heteroclinic orbits asymptotic to points on +the circle of fixed points. The expression +(\ref{horb}) is generated from (\ref{spsl}) through a B\"acklund-Darboux +transformation \cite{LM94}. + +{\bf Example 2:} +Consider the sine-Gordon equation, +\begin{displaymath} + u_{tt} - u_{xx} + \sin u = 0\ , +\end{displaymath} +under the decay boundary condition that $u$ belongs to the +Schwartz class in $x$. The well-known ``breather'' solution, +\begin{equation} + u(t,x) = 4 \arctan \left[ \frac{\tan \nu \cos [(\cos + \nu)t]}{\cosh [(\sin \nu) x ]} \right]\,, +\label{br} +\end{equation} +where $\nu$ is a parameter, is a periodic orbit. The expression +(\ref{br}) is generated from trivial solutions through a +B\"{a}cklund-Darboux transformation \cite{EFM90}. + +\section{Invariant Submanifolds} + +Invariant submanifolds are the main objects in studying phase +spaces. In phase spaces for partial differential equations, +invariant submanifolds are often submanifolds with +boundaries. Therefore, the following concepts on invariance are +important. + +\begin{definition}[Overflowing and Inflowing Invariance] +A submanifold $W$ with boundary $\partial W$ is +\begin{itemize} +\item overflowing invariant if for any $t>0$, $\bar{W} \subset + F^t \circ W$, where $\bar{W} = W \cup \partial W$, +\item inflowing invariant if any $t>0$, $F^t \circ \bar{W} + \subset W$, +\item invariant if for any $t>0$, $F^t \circ \bar{W} = \bar{W}$. +\end{itemize} +\end{definition} + +\begin{definition}[Local Invariance] +A submanifold $W$ with boundary $\partial W$ is locally invariant +if for any point $q \in W$, if $\bigcup\limits_{t \in [0,\infty)} F^t +(q) \not \subset W$, then there exists $T \in (0,\infty)$ such +that $\bigcup\limits_{t \in [0,T)} F^t (q) \subset W$, and $F^T(q) \in +\partial W$; and if $\bigcup\limits_{t \in (-\infty,0]} F^t(q) +\not \subset W$, then there exists $T \in (-\infty,0)$ such that +$\bigcup\limits_{t\in (T,0]} F^t(q) \subset W$, and $F^T(q) \in \partial +W$. +\end{definition} + +Intuitively speaking, a submanifold with boundary is locally +invariant if any orbit starting from a point inside the +submanifold can only leave the submanifold through its boundary +in both forward and backward time. + +{\bf Example:} +Consider the linear equation, +\begin{equation} + iq_t= (1 + i) q_{xx} + iq\,, +\label{liq} +\end{equation} +where $i = \sqrt{-1}$, $t\in \RR^1$, $x \in \RR^1$, and $q$ is a +complex-valued function of $(t,x)$, under periodic boundary +condition, +\begin{displaymath} + q (x + 1) = q(x)\,. +\end{displaymath} +Let $q = e^{\Omega_j t + ik_jx}$; then +\begin{displaymath} + \Omega_j = (1-k^2_j) + i\, k^2_j\,, +\end{displaymath} +where $k_j = 2 j \pi$, $(j \in \ZZ )$. $\Omega_0 = 1$, and when +$\mid j\mid >0$, $R_e \{ \Omega_j \} <0$. We take the $H^1$ space +of periodic functions of period 1 to be the phase space. Then the +submanifold +\begin{displaymath} + W_0 = \left\{ q \in H^1 \ \bigg| \ q = c_0, \,\, c_0 \, \hbox{is + complex and}\, \parallel q \parallel <1 \right\} +\end{displaymath} +is an outflowing invariant submanifold, the submanifold +\begin{displaymath} + W_1 = \left\{ q \in H^1 \ \bigg| \ q = c_1 e^{ik_1x}, \ \ + c_1 \, \hbox{is complex}, \, \hbox{and}\, + \parallel q \parallel <1 \right\} +\end{displaymath} +is an inflowing invariant submanifold, and the submanifold +\begin{displaymath} + W = \left\{ q \in H^1 \ \bigg| \ q = c_0 + c_1 e^{ik_1x}, \ \ c_0 \, + \hbox{and} \, c_1 \, \hbox{are complex}, \, \hbox{and}\, + \parallel q \parallel <1 \right\} +\end{displaymath} +is a locally invariant submanifold. The \emph{unstable subspace} is +given by +\begin{displaymath} +W^{(u)} = \left\{ q \in H^1 \ \bigg| \ q = c_0, \ \ c_0 \, \hbox{is + complex} \right\}\,, +\end{displaymath} +and the \emph{stable subspace} is given by +\begin{displaymath} + W^{(s)} = \left\{ q \in H^1 \ \bigg| \ q= \sum_{j \in Z/\{0\}} c_j + \, e^{ik_jx}, \ \ c_j\hbox{'s are complex} \right\}\,. +\end{displaymath} +Actually, a good way to view the partial differential equation +(\ref{liq}) as defining an infinite dimensional dynamical system +is through Fourier transform, let +\begin{displaymath} + q(t,x) = \sum_{j \in Z} c_j (t) e^{ik_jx}\,; +\end{displaymath} +then $c_j (t)$ satisfy +\begin{displaymath} + \dot{c}_j = \left[ (1-k^2_j) + ik^2_j \right] c_j\,, \quad j + \in \ZZ\,; +\end{displaymath} +which is a system of infinitely many ordinary differential +equations. + +\section{Poincar\'{e} Sections and Poincar\'{e} Maps} + +In the infinite dimensional phase space $\mathcal{H}$, Poincar\'{e} +sections can +be defined in a similar fashion as in a finite dimensional phase +space. Let $l_q$ be a periodic or homoclinic orbit in $\mathcal{H}$ under +a flow $F^t$, and $q_{\ast}$ be a point on $l_q$, then the +Poincar\'{e} section $\Sigma$ can be defined to be any +codimension 1 subspace which has a transversal intersection with +$l_q$ at $q_{\ast}$. Then the flow $F^t$ will induce a Poincar\'e map +$P$ in the neighborhood of $q_{\ast}$ in $\Sigma_0$. Phase +blocks, e.g. Smale horseshoes, can be defined using the +norm. + + + + + +\clearpage{} +\clearpage{}\chapter{Soliton Equations as Integrable Hamiltonian PDEs} + +\section{A Brief Summary} + +Soliton equations are integrable Hamiltonian partial +differential equations. For example, the Korteweg-de Vries (KdV) equation +\[ +u_t = -6uu_x -u_{xxx}\ , +\] +where $u$ is a real-valued function of two variables $t$ and $x$, can +be rewritten in the Hamiltonian form +\[ +u_t = \pa_x {\dl H \over \dl u} \ , +\] +where +\[ +H= \int \left [ {1 \over 2} u_x^2 - u^3 \right ] dx \ , +\] +under either periodic or decay boundary conditions. It is integrable in +the classical Liouville sense, i.e., there exist enough functionally +independent constants of motion. These constants of motion can be generated +through isospectral theory or B\"acklund transformations \cite{AI79}. The +level sets of these constants of motion are elliptic tori \cite{PT87} +\cite{MM75} \cite{MT76} \cite{FM76}. + +There exist soliton equations which possess level sets which are normally +hyperbolic, for example, the focusing cubic nonlinear Schr\"odinger equation +\cite{LM94}, +\[ +iq_t = q_{xx} + 2 |q|^2q\ , +\] +where $i =\sqrt{-1}$ and $q$ is a complex-valued function of two variables +$t$ and $x$; the sine-Gordon equation \cite{MO95}, +\[ +u_{tt} = u_{xx} +\sin u \ , +\] +where $u$ is a real-valued function of two variables $t$ and $x$, etc. + +Hyperbolic foliations are very important since they are the sources of chaos +when the integrable systems are under perturbations. We will investigate +the hyperbolic foliations of three typical types of soliton equations: +(i). (1+1)-dimensional soliton equations represented by the focusing +cubic nonlinear Schr{\"{o}}dinger equation, (ii). soliton lattices +represented by the focusing cubic nonlinear Schr{\"{o}}dinger lattice, +(iii). (1+2)-dimensional soliton equations represented by the +Davey-Stewartson II equation. +\begin{remark} +For those soliton equations which have only elliptic level sets, the +corresponding representatives can be chosen to be the KdV equation for +(1+1)-dimensional soliton equations, the Toda lattice for soliton +lattices, and the KP equation for (1+2)-dimensional soliton equations. +\end{remark} + +Soliton equations are canonical equations which model a variety of physical +phenomena, for example, nonlinear wave motions, nonlinear optics, plasmas, +vortex dynamics, etc. \cite{AS81} \cite{AC91}. Other typical examples of such +integrable Hamiltonian partial differential equations are, e.g., the +defocusing cubic nonlinear Schr\"odinger equation, +\[ +iq_t = q_{xx} - 2 |q|^2q\ , +\] +where $i =\sqrt{-1}$ and $q$ is a complex-valued function of two variables +$t$ and $x$; the modified KdV equation, +\[ +u_t = \pm 6u^2u_x -u_{xxx}\ , +\] +where $u$ is a real-valued function of two variables $t$ and $x$; the +sinh-Gordon equation, +\[ +u_{tt} = u_{xx} +\sinh u \ , +\] +where $u$ is a real-valued function of two variables $t$ and $x$; the +three-wave interaction equations, +\[ +{\pa u_i \over \pa t} + a_i {\pa u_i \over \pa x} = b_i \bar{u}_j +\bar{u}_k \ , +\] +where $i,j,k=1,2,3$ are cyclically permuted, $a_i$ and $b_i$ are real +constants, $u_i$ are complex-valued functions of $t$ and $x$; the +Boussinesq equation, +\[ +u_{tt}-u_{xx}+(u^2)_{xx} \pm u_{xxxx} = 0 \ , +\] +where $u$ is a real-valued function of two variables $t$ and $x$; +the Toda lattice, +\[ +\pa^2 u_n / \pa t^2 = \exp \left \{ -(u_n-u_{n-1})\right \}- +\exp \left \{ -(u_{n+1}-u_n)\right \} \ , +\] +where $u_n$'s are real variables; the focusing cubic nonlinear Schr{\"{o}}dinger lattice, +\[ +i{\pa q_n \over \pa t} = (q_{n+1}-2q_n+q_{n-1})+|q_n|^2(q_{n+1}+q_{n-1})\ , +\] +where $q_n$'s are complex variables; the +Kadomtsev-Petviashvili (KP) equation, +\[ +(u_t +6uu_x +u_{xxx})_x = \pm 3 u_{yy}\ , +\] +where $u$ is a real-valued function of three variables $t$, $x$ and $y$; +the Davey-Stewartson II equation, +\[ +\left \{ \begin{array}{l} +i \pa_t q = [\pa_x^2 - \pa_y^2]q + [2|q|^2 + u_y]q\ , \cr \cr +[\pa_x^2 + \pa_y^2]u = -4 \pa_y |q|^2 \ , \cr +\end{array} \right. +\] +where $i =\sqrt{-1}$, $q$ is a complex-valued function of three +variables $t$, $x$ and $y$; and $u$ is a real-valued function of three +variables $t$, $x$ and $y$. For more complete list of soliton equations, +see e.g. \cite{AS81} \cite{AC91}. + +The cubic nonlinear Schr\"odinger equation is one of our main focuses +in this book, which can be written in the Hamiltonian form, +\[ +i q_t = {\dl H \over \dl \bar{q}} \ , +\] +where +\[ +H= \int [-|q_x|^2 \pm |q|^4] dx \ , +\] +under periodic boundary conditions. Its phase space is defined as +\begin{eqnarray*} +\HH^k &\equiv& \bigg \{ \vq = \left ( \begin{array}{c} q \cr r \cr \end{array} +\right )\ \bigg | \ r=-\bq, \ q(x+1)=q(x), \\ +& & q \in H^k_{[0,1]}:\ +\mbox{the Sobolev space}\ H^k \ \mbox{over the period interval} \ +[0,1] \bigg \}\ . +\end{eqnarray*} +\begin{remark} +It is interesting to notice that the cubic nonlinear Schr\"odinger +equation can also be written in Hamiltonian form in spatial variable, i.e., +\[ +q_{xx}=iq_t \pm 2 |q|^2 q \ , +\] +can be written in Hamiltonian form. Let $p=q_x$; then +\[ +{\pa \over \pa x}\left ( \begin{array}{c} q \\ \bar{q} \\ \bar{p} +\\ p \\ \end{array} \right ) = J \left ( \begin{array}{c} +{\dl H \over \dl q} \\ \\ +{\dl H \over \dl \bq} \\ \\ {\dl H \over \dl \bp} \\ \\ {\dl H \over \dl p} +\\ \end{array} \right ) \ , +\] +where +\[ +J= \left ( \begin{array}{cccc} 0&0&1&0\\ 0&0&0&1 \\ -1&0&0&0 \\ 0&-1&0&0 +\\ \end{array} \right ) \ , +\] +\[ +H=\int [|p|^2 \mp |q|^4 - {i \over 2} (q_t \bq -\bq_t q)] dt \ , +\] +under decay or periodic boundary conditions. We do not know whether or not +other soliton equations have this property. +\end{remark} + +\section{A Physical Application of the Nonlinear Schr\"odinger Equation} + +The cubic nonlinear Schr\"{o}dinger (NLS) equation has many +different applications, i.e. it describes many different physical phenomena, +and that is why it is called a canonical equation. Here, as an example, +we show how the NLS equation describes the motion of a vortex +filament -- the beautiful Hasimoto derivation \cite{Has72}. Vortex +filaments in an inviscid fluid are known to preserve their identities. +The motion of a very thin isolated vortex filament $\vec{X}=\vec{X}(s,t)$ +of radius $\e$ in an incompressible inviscid unbounded fluid by its +own induction is described asymptotically by +\begin{equation} +\pa \vec{X}/\pa t =G \k \vec{b}\ , \label{indu} +\end{equation} +where $s$ is the length measured along the filament, $t$ is the time, +$\k$ is the curvature, $\vec{b}$ is the unit vector in the direction +of the binormal and $G$ is the coefficient of local induction, +\[ +G={\Ga \over 4\pi}[\ln (1/\e) +O(1)]\ , +\] +which is proportional to the circulation $\Ga$ of the filament and +may be regarded as a constant if we neglect the second order term. +Then a suitable choice of the units of time and length reduces +(\ref{indu}) to the nondimensional form, +\begin{equation} +\pa \vec{X}/\pa t =\k \vec{b}\ . \label{nindu} +\end{equation} +Equation (\ref{nindu}) should be supplemented by the equations of +differential geometry (the Frenet-Seret formulae) +\begin{equation} +\pa \vec{X}/\pa s =\vec{t}\ , \ \ \pa \vec{t}/\pa s =\k \vec{n}\ , \ \ +\pa \vec{n}/\pa s =\tau \vec{b}-\k \vec{t}\ , \ \ +\pa \vec{b}/\pa s =-\tau \vec{n}\ , +\label{FSf} +\end{equation} +where $\tau$ is the torsion and $\vec{t}$, $\vec{n}$ and $\vec{b}$ +are the tangent, the principal normal and the binormal unit vectors. +The last two equations imply that +\begin{equation} +\pa (\vec{n}+i \vec{b})/\pa s =-i\tau (\vec{n}+i\vec{b})-\k \vec{t}\ , +\label{apa1} +\end{equation} +which suggests the introduction of new variables +\begin{equation} +\vec{N} = (\vec{n}+i \vec{b})\exp \bigg \{ i \int_0^s \tau ds \bigg \}\ , +\label{apa2} +\end{equation} +and +\begin{equation} +q=\k \exp \bigg \{ i \int_0^s \tau ds \bigg \}\ . +\label{apa3} +\end{equation} +Then from (\ref{FSf}) and (\ref{apa1}), we have +\begin{equation} +\pa \vec{N}/\pa s =-q \vec{t}\ , \ \ +\pa \vec{t}/\pa s =\mbox{Re} \{ q \overline{\vec{N}} \} = {1 \over 2} +(\bq \vec{N} + q \overline{\vec{N}})\ . +\label{apa4} +\end{equation} +We are going to use the relation ${\pa^2 \vec{N} \over \pa s \pa t} = +{\pa^2 \vec{N} \over \pa t \pa s}$ to derive an equation for $q$. +For this we need to know $\pa \vec{t}/\pa t$ and $\pa \vec{N}/\pa t$ +besides equations (\ref{apa4}). From (\ref{nindu}) and (\ref{FSf}), +we have +\begin{eqnarray*} +& & \pa \vec{t}/\pa t = {\pa^2 \vec{X} \over \pa s \pa t}= \pa +(\k \vec{b})/\pa s =(\pa \k /\pa s) \vec{b} - \k \tau \vec{n} \\ +& & = \k \ \mbox{Re} \{ ({1 \over \k} \pa \k / \pa s +i \tau ) +(\vec{b}+i \vec{n}) \}\ , +\end{eqnarray*} +i.e. +\begin{equation} +\pa \vec{t}/\pa t =\ \mbox{Re} \{ i(\pa q / \pa s)\overline{\vec{N}} \} +={1 \over 2} i [ (\pa q / \pa s)\overline{\vec{N}}- +(\pa q / \pa s)^-\vec{N}]\ . +\label{apa5} +\end{equation} +We can write the equation for $\pa \vec{N}/\pa t$ in the following form: +\begin{equation} +\pa \vec{N}/\pa t =\al \vec{N}+\be \overline{\vec{N}}+\ga \vec{t}\ , +\label{apa6} +\end{equation} +where $\al$, $\be$ and $\ga$ are complex coefficients to be determined. +\begin{eqnarray*} +\al +\bar{\al} &=& {1 \over 2} [\pa \vec{N}/\pa t \cdot \overline{\vec{N}} ++\pa \overline{\vec{N}}/\pa t \cdot \vec{N}] \\ +&=& {1 \over 2} \pa (\vec{N}\cdot \overline{\vec{N}})/\pa t = 0\ , +\end{eqnarray*} +i.e. $\al =i R$ where $R$ is an unknown real function. +\begin{eqnarray*} +& & \be ={1 \over 2}\pa \vec{N}/\pa t \cdot \vec{N} = {1 \over 4} +\pa (\vec{N}\cdot \vec{N})/\pa t = 0\ , \\ +& & \ga = -\vec{N} \cdot \pa \vec{t}/\pa t = -i \pa q/\pa s \ . +\end{eqnarray*} +Thus +\begin{equation} +\pa \vec{N}/\pa t =i[R\vec{N}-(\pa q/\pa s)\vec{t}\ ]\ . +\label{apa7} +\end{equation} +From (\ref{apa4}), (\ref{apa7}) and (\ref{apa5}), we have +\begin{eqnarray*} +{\pa^2 \vec{N} \over \pa s \pa t} &=& -(\pa q/\pa t) \vec{t} - q +\pa \vec{t}/\pa t \\ +&=& -(\pa q/\pa t) \vec{t} - {1 \over 2} iq [ (\pa q / \pa s) +\overline{\vec{N}}-(\pa q / \pa s)^-\vec{N}] \ , \\ +{\pa^2 \vec{N} \over \pa t \pa s} &=& i[(\pa R/ \pa s)\vec{N}-Rq +\vec{t}-(\pa^2 q / \pa s^2)\vec{t} \\ +& & -{1 \over 2}(\pa q/ \pa s)(\bq \vec{N}+q \overline{\vec{N}})]\ . +\end{eqnarray*} +Thus, we have +\begin{equation} +\pa q/\pa t =i[\pa^2 q/\pa s^2 + R q]\ , +\label{apa8} +\end{equation} +and +\begin{equation} +{1 \over 2}q \pa \bq/ \pa s = \pa R/ \pa s -{1 \over 2}(\pa q/ \pa s)\bq \ . +\label{apa9} +\end{equation} +The comparison of expressions for ${\pa^2 \vec{t} \over \pa s \pa t}$ +from (\ref{apa4}) and (\ref{apa5}) leads only to (\ref{apa8}). Solving +(\ref{apa9}), we have +\begin{equation} +R={1 \over 2}(|q|^2 +A)\ , +\label{apa10} +\end{equation} +where $A$ is a real-valued function of $t$ only. Thus we have the +cubic nonlinear Schr\"{o}dinger equation for $q$: +\[ +-i\pa q/\pa t = \pa^2 q/\pa s^2 +{1 \over 2}(|q|^2 +A) q \ . +\] +The term $Aq$ can be transformed away by defining the new variable +\[ +\tilde{q} = q \exp [ -{1 \over 2} i \int_0^t A(t) dt ] \ . +\] + + + +\clearpage{} +\clearpage{}\chapter{Figure-Eight Structures} + +For finite-dimensional Hamiltonian systems, figure-eight structures are often +given by singular level sets. These singular level sets are also called +separatrices. Expressions for such figure-eight structures can be obtained by +setting the Hamiltonian and/or other constants of motion to special values. +For partial differential equations, such an approach is not feasible. +For soliton equations, expressions for figure-eight structures can be obtained +via B\"acklund-Darboux transformations \cite{LM94} \cite{Li92} \cite{Li00a}. + +\section{1D Cubic Nonlinear Schr\"odinger (NLS) Equation \label{1DCNSE}} + +We take the focusing nonlinear Schr\"odinger equation (NLS) +as our first example to show how to construct figure-eight structures. +If one starts from the conservational laws of the NLS, it turns out that +it is very elusive to get the separatrices. On the contrary, starting +from the B\"acklund-Darboux transformation to be presented, one can find +the separatrices rather easily. We consider the NLS +\begin{equation} +iq_t = q_{xx} + 2 |q|^{2} q \ , +\label{NLS} +\end{equation} +under periodic boundary condition $q(x + 2 \pi ) = q(x)$. +The NLS is an integrable system by virtue of the Lax pair \cite{ZS72}, +\begin{eqnarray} +\varphi_x &=& U \varphi \ , \label{Lax1}\\ +\varphi_t &=& V \varphi \ , \label{Lax2} +\end{eqnarray} +where +\[ +U = i \lambda \sigma_3 +\, + \, i \left( \begin{array}{cc} + 0 & q\\ + -r & 0 + \end{array} \right)\ , +\] +\[ +V = \, 2\, i\, \lambda^2 \sigma_3 +\,+iqr\sigma_3 + \, + \left( \begin{array}{cc} + 0 & 2i\lambda q + q_x\\ + \\ + -2i\lambda r+r_x & 0 + \end{array} \right)\ , +\] +where $\sigma_3$ denotes the third Pauli matrix +$\sigma_3 = \mbox{diag}(1,-1)$, $r=-\bq$, and $\la$ is the spectral parameter. +If $q$ satisfies the NLS, then the compatibility of the over determined +system (\ref{Lax1}, \ref{Lax2}) is guaranteed. +Let $M=M(x)$ be the fundamental matrix solution to the ODE (\ref{Lax1}), +$M(0)$ is the $2\times 2$ identity matrix. We introduce the so-called +transfer matrix $T = T(\lambda, \vec{q})$ where $\vec{q} = (q, -\bq)$, +$T= M(2 \pi)$. +\begin{lemma} Let $Y(x)$ be any solution to the ODE (\ref{Lax1}), then +\[ +Y(2n\pi )=T^n \ Y(0)\ . +\] +\end{lemma} +Proof: Since $M(x)$ is the fundamental matrix, +\[ +Y(x)=M(x)\ Y(0)\ . +\] +Thus, +\[ +Y(2 \pi)=T\ Y(0)\ . +\] +Assume that +\[ +Y(2 l\pi)=T^l \ Y(0)\ . +\] +Notice that $Y(x+2 l\pi)$ also solves the ODE (\ref{Lax1}); then +\[ +Y(x+2 l \pi)=M(x)\ Y(2 l\pi)\ ; +\] +thus, +\[ +Y(2 (l+1) \pi)=T\ Y(2 l\pi )=T^{l+1}\ Y(0)\ . +\] +The lemma is proved. Q.E.D. + +\begin{definition} +We define the Floquet discriminant $\Dl$ as, +\[ +\Delta(\lambda, \vec{q}) =\ \mbox{trace}\ \{ T(\lambda, \vec{q}) \}\ . +\] +We define the periodic and anti-periodic points +$\la^{(p)}$ by the condition +\[ +|\Delta(\lambda^{(p)}, \vq)| = 2\ . +\] +We define the critical points $\la^{(c)}$ by the condition +\[ +{\pa \Delta(\lambda, \vq) \over \pa \lambda} \bigg|_{\la =\la^{(c)}} = 0\ . +\] +A multiple point, denoted $\lambda^{(m)}$, is a critical +point for which +\[ +|\Delta(\lambda^{(m)}, \vq)| = 2. +\] +The algebraic multiplicity of $\lambda^{(m)}$ is defined as the order +of the zero of $\Delta(\lambda) \pm 2$. Ususally it is 2, but it can exceed +2; when it does equal 2, we call the multiple point a double point, and +denote it by $\lambda^{(d)}$. +The geometric multiplicity of $\lambda^{(m)}$ is defined as the +maximum number of linearly independent solutions to the ODE (\ref{Lax1}), +and is either 1 or 2. +\end{definition} +Let $q(x,t)$ be a solution to the NLS (\ref{NLS}) for which the linear +system (\ref{Lax1}) has a complex double point $\nu$ of geometric +multiplicity 2. We denote two linearly independent solutions of the +Lax pair (\ref{Lax1},\ref{Lax2}) at $\lambda = \nu$ by +$(\phi^+, \phi^-)$. Thus, a general solution of the linear systems +at $(q,\nu)$ is given by +\begin{equation} +\phi (x,t) = c_+ \phi^+\ +\ c_- \phi^- \ . +\label{4.2} +\end{equation} +We use $\phi$ to define a Gauge matrix \cite{SZ87} $G$ by +\begin{equation} + G = G (\lambda ; \nu ; \phi ) = N +\left( \begin{array}{cl} + \lambda-\nu & \quad 0\\ +0 & \lambda - \bar{\nu} + \end{array}\right) +N^{-1}\ , +\label{4.3} +\end{equation} +where +\begin{equation} +N = \left ( \begin{array}{lr} + \phi_1 & -{\bar{\phi}}_2 \\ +\phi_2 &\ \ {\bar{\phi}}_1 + \end{array} \right )\ . +\label{4.4} +\end{equation} +Then we define $Q$ and $\Psi$ by +\begin{equation} + Q(x,t) = q(x,t) \ + \ +2 (\nu-{\bar{\nu}})\ +\frac{\phi_1{\bar{\phi}}_2}{\phi_1 {\bar{\phi}}_1+ +\phi_2{\bar{\phi}}_2} +\label{4.5} +\end{equation} +and +\begin{equation} +\Psi (x,t; \lambda) \ = \ G(\lambda; \nu ; \phi ) \ \psi (x,t; \lambda) +\label{4.6} +\end{equation} +where $\psi$ solves the Lax pair (\ref{Lax1},\ref{Lax2}) at +$(q,\nu)$. +Formulas (\ref{4.5}) and (\ref{4.6}) are the B\"acklund-Darboux +transformations for the +potential and eigenfunctions, respectively. We have the +following \cite{SZ87} \cite{LM94}, +\begin{theorem} +Let $q(x,t)$ be a solution to the NLS equation (\ref{NLS}), +for which the linear system (\ref{Lax1}) has +a complex double point $\nu$ of geometric multiplicity 2, +with eigenbasis $(\phi^+, \phi^-)$ for the Lax pair +(\ref{Lax1},\ref{Lax2}), and define +$Q(x,t)$ and $\Psi (x,t;\lambda)$ by +(\ref{4.5}) and (\ref{4.6}). Then +\begin{enumerate} +\item $Q(x,t)$ is an solution of NLS, with spatial period $2\pi$, +\item $Q$ and $q$ have the same Floquet spectrum, +\item $Q(x,t)$ is homoclinic to $q(x,t)$ in the +sense that $Q(x,t) \longrightarrow q_{\theta_\pm}(x,t)$, +expontentially as $\exp (-\sigma_\nu|t|)$, as $t +\longrightarrow \pm \infty$, where $q_{\theta_\pm}$ is a +``torus translate'' of $q, \sigma_\nu$ is the nonvanishing growth +rate associated to the complex double point $\nu$, and explicit +formulas exist for this growth rate and for the translation +parameters $\theta_\pm$, +\item $\Psi (x,t;\lambda)$ solves the +Lax pair (\ref{Lax1},\ref{Lax2}) at $(Q, \lambda)$. +\end{enumerate} +\label{theorem 4} +\end{theorem} +This theorem is quite general, constructing homoclinic solutions +from a wide class of starting solutions $q(x,t)$. It's proof is +one of direct verification \cite{Li92}. + +We emphasize several qualitative features of these homoclinic orbits: +(i) $Q(x,t)$ is homoclinic to a torus which itself +possesses rather complicated spatial and temporal structure, and is +not just a fixed point. (ii) Nevertheless, the homoclinic orbit +typically has still more complicated spatial structure than its +``target torus''. (iii) When there are several complex double points, each +with nonvanishing growth rate, one can iterate the B\"acklund-Darboux +transformations to generate more complicated homoclinic orbits. +(iv) The number of complex double points with nonvanishing growth +rates counts the dimension of the unstable manifold of the critical +torus in that two unstable directions are coordinatized by the +complex ratio $c_+/c_-$. Under even symmetry only one real +dimension satisfies the constraint of evenness, as will be clearly +illustrated in the following example. (v) These B\"acklund-Darboux formulas +provide global expressions for the stable and unstable manifolds of the +critical tori, which represent figure-eight structures. + +{\bf Example:} As a concrete example, we take $q(x,t)$ to be the special +solution +\begin{equation} +q_c = c \exp \left \{ -i [2c^2 t + \gamma ] \right \}\ . +\label{4.7} +\end{equation} +Solutions of the Lax pair (\ref{Lax1},\ref{Lax2}) can +be computed explicitly: +\begin{equation} +\phi^{(\pm)}(x,t;\lambda)= e^{\pm i\kappa(x + 2 \lambda t)} +\left( \begin{array}{cl} + c e^{-i(2 c^2 t +\gamma)/2} \\ +(\pm\kappa - \lambda)e^{i(2 c^2 t+\gamma)/2} + \end{array}\right) \ , +\label{4.8} +\end{equation} +where +\[ +\kappa = \kappa (\lambda) = \sqrt{c^2 + +\lambda^2}\ . +\] +With these solutions one can construct the fundamental matrix +\begin{equation} +M(x;\lambda ; q_c) = +\Bigg [ \begin{array}{cl} +\cos \kappa x+ i \frac \lambda \kappa \sin \kappa x &\ +i \frac {q_c}{\kappa} \sin \kappa x \\ +i \frac {\overline{q_c}}{\kappa }\sin \kappa x & +\cos \kappa x - i \frac \lambda \kappa \sin \kappa x +\end{array}\Bigg ]\ , +\label{4.9} +\end{equation} +from which the Floquet discriminant can be computed: +\begin{equation} +\Delta (\lambda ; q_c) = 2 \cos (2 \kappa \pi)\ . +\label{4.10} +\end{equation} +From $\Delta$, spectral quantities can be +computed: +\begin{enumerate} + \item simple periodic points: $\lambda^\pm + = \pm i \ c\ , $ + \item double points: + $\kappa(\lambda^{(d)}_j) + = j/2\ , \ \ j \in \ZZ \ ,\ + j \neq 0 \ ,$ + \item critical points: + $\lambda^{(c)}_j = \lambda^{(d)}_j\ ,\quad j \in + \ZZ\ ,\ \ j \neq 0 \ ,$ + \item simple periodic points: $\lambda^{(c)}_0 = 0\ $. +\end{enumerate} +For this spectral data, there are 2N purely +imaginary +double points, +\begin{equation} + (\lambda^{(d)}_j)^2 = j^2/4 - c^2, \ \ +j =1, 2, \cdots , N; +\label{4.11} +\end{equation} +where +\[ +\bigg [N^2/4 - c^2 \bigg ] < 0 < +\bigg [(N+1)^2/4 - c^2 \bigg ] \ . +\] +From this spectral data, the +homoclinic orbits can be explicitly computed through B\"acklund-Darboux +transformation. +Notice that to have temporal growth (and decay) in the eigenfunctions +(\ref{4.8}), one needs $\la$ to be complex. Notice also that the +B\"acklund-Darboux transformation is built with quadratic products in +$\phi$, thus choosing $\nu = \la_j^{(d)}$ will guarantee periodicity of +$Q$ in $x$. +When $N=1$, the B\"acklund-Darboux +transformation at one purely imaginary double point $\la_1^{(d)}$ yields $Q = +Q(x, t; c, \gamma; c_+/c_-)$ \cite{LM94}: +\begin{eqnarray} +Q &=& \bigg [ \cos 2p - \sin p \ \ \mbox{sech} \tau \ \cos (x +\vth)- +i \sin 2p \tanh \tau \bigg ] \nonumber \\ +& & \bigg [1 + \sin p \ \ \mbox{sech} \tau +\ \cos (x +\vth)\bigg ]^{-1} ce^{-i(2 c^2 t+\gamma)} \label{4.13} \\ +& & \ra e^{\mp 2ip}ce^{-i(2 c^2 t + \gamma)}\quad +\mbox{ as } \ \rho \ra \mp \infty ,\nonumber +\end{eqnarray} +where $c_+/c_- \equiv \exp(\rho + i\beta)$ and $p$ is defined by $1/2 + +i\sqrt{c^2 - 1/4} = c \exp(ip)$, $\tau \equiv \sigma t -\rho$, and +$\vth = p - (\beta +\pi/2)$. + +Several points about this homoclinic orbit need to be made: +\begin{enumerate} +\item The orbit depends +only upon the ratio $c_+/c_-$, and not upon $c_+$ and $c_-$ +individually. + +\item $Q$ is homoclinic to the plane wave +orbit; however, a phase +shift of $-4p$ occurs when one compares the asymptotic behavior of +the orbit as $t \rightarrow \ - \ \infty$ with its behavior as +$t\rightarrow \ +\ \infty$. + +\item For small p, the formula +for $Q$ becomes more transparent: +\[ Q \simeq\bigg [ (\cos 2p - i\, +\sin 2p\, \tanh \tau) - 2 \sin\, +p \ \mbox{sech}\ \tau \cos +(x +\vth)\bigg ] ce^{-i(2 c^2 t + \gamma)}. \] + + +\item An evenness constraint on $Q$ in $x$ can be enforced by +restricting the phase $\phi$ to be one of two values +\[ +\phi = 0, \pi . \hskip 1truein \mbox{(evenness)} +\] +In this manner, the even symmetry disconnects the level set. +Each component constitutes one loop of the figure eight. While the target +q is independent of +$x$, each of these loops has $x$ dependence through the $\cos(x)$. +One loop has exactly this dependence and can be interpreted as a +spatial +excitation located near $x=0$, while the second loop has the +dependence $\cos (x - \pi )$, which we interpret as spatial +structure located near $ x = \pi $. In this example, the disconnected +nature of the level set is clearly related to distinct spatial +structures on the individual loops. See Figure \ref{1fig8} for +an illustration. +\item Direct calculation shows that the transformation matrix +$M(1;\la_1^{(d)};Q)$ is similar to a Jordan form when $t \in (-\infty,\infty)$, +\[ +M(1;\la_1^{(d)};Q) \sim \left ( \begin{array}{lr} -1 & 1 \cr 0 & -1 \cr +\end{array} \right )\ , +\] +and when $t \ra \pm \infty$, $M(1;\la_1^{(d)};Q) \longrightarrow -I$ (the +negative of the 2x2 identity matrix). Thus, when $t$ is finite, the +algebraic multiplicity ($=2$) of $\la =\la_1^{(d)}$ with the potential $Q$ +is greater than the geometric multiplicity ($=1$). +\end{enumerate} +\begin{figure} +\includegraphics{fig1.eps} +\caption{An illustration of the figure-eight structure.} +\label{1fig8} +\end{figure} + +In this example the dimension of the loops need not +be one, but is determined by the number of purely imaginary double +points which in turn is controlled by the amplitude $c$ of the plane +wave target and by the spatial period. (The dimension of the +loops increases linearly with the spatial period.) When there are +several complex double points, B\"acklund-Darboux transformations must +be iterated to produce complete +representations. Thus, B\"acklund-Darboux transformations give +global representations of the figure-eight structures. + +\subsection{Linear Instability} + +The above figure-eight structure corresponds to the following linear +instability of Benjamin-Feir type. Consider the uniform solution to the +NLS (\ref{NLS}), +\[ +q_c = c e^{i \th(t)}\ , \ \ \ \ \th(t)=-[2c^2t +\ga ] \ . +\] +Let +\[ +q= [c + \tq ] e^{i \th(t)}\ , +\] +and linearize equation (\ref{NLS}) at $q_c$, we have +\[ +i \tq_t = \tq_{xx} + 2 c^2 [ \tq + \bar{\tq}]\ . +\] +Assume that $\tq$ takes the form, +\[ +\tq = \bigg [ A_j e^{\Om_j t} + B_j e^{\bar{\Om}_j t} \bigg ] \cos k_j x \ , +\] +where $k_j = 2 j \pi$, ($j=0,1,2, \cdot \cdot \cdot$), $A_j$ and $B_j$ +are complex constants. Then, +\[ +\Om_j^{(\pm)} = \pm k_j \sqrt{4c^2 -k_j^2}\ . +\] +Thus, we have instabilities when $c > 1/2$. + +\subsection{Quadratic Products of Eigenfunctions} + +Quadratic products of eigenfunctions play a crucial role in +characterizing the hyperbolic structures of soliton equations. +Its importance lies in the following aspects: (i). Certain +quadratic products of eigenfunctions solve the linearized soliton +equation. (ii). Thus, they are the perfect candidates for building +a basis to the invariant linear subbundles. (iii). Also, they +signify the instability of the soliton equation. (iv). Most +importantly, quadratic products of eigenfunctions can serve as +Melnikov vectors, e.g., for Davey-Stewartson equation \cite{Li00a}. + +Consider the linearized NLS equation at any solution $q(t,x)$ +written in the vector form: +\begin{eqnarray} +i\pa_t (\dl q) &=& (\dl q)_{xx} +2 [ q^2\overline{\dl q} + +2 |q|^2 \dl q ]\ , +\nonumber \\ +\label{lNLS} \\ +i\pa_t (\overline{\dl q}) &=& -(\overline{\dl q})_{xx} -2 +[ \bq^2\dl q + 2 |q|^2 \overline{\dl q} ]\ ,\nonumber +\end{eqnarray} +we have the following lemma \cite{LM94}. +\begin{lemma} +Let $\vph^{(j)} = \vph^{(j)}(t,x;\la,q)$ ($j=1,2$) be any two +eigenfunctions solving the Lax pair +(\ref{Lax1},\ref{Lax2}) at an arbitrary $\la$. Then +\[ +\left ( \begin{array}{c} \dl q \cr \cr \overline{\dl q} \cr +\end{array} \right )\ , \ \ +\left ( \begin{array}{c} \vph_1^{(1)} \vph_1^{(2)} \cr \cr +\vph_2^{(1)} \vph_2^{(2)}\cr \end{array} \right )\ , \ \ +\mbox{and} \ +S\left ( \begin{array}{c} \vph_1^{(1)} \vph_1^{(2)} \cr \cr +\vph_2^{(1)} \vph_2^{(2)}\cr \end{array} \right )^{-}\ ,\ +\ \mbox{where}\ +S=\left ( \begin{array}{lr} 0&1 \cr 1&0 \cr \end{array} \right ) +\] +solve the same equation (\ref{lNLS}); thus +\[ +\Phi = \left ( \begin{array}{c} \vph_1^{(1)} \vph_1^{(2)} \cr \cr +\vph_2^{(1)} \vph_2^{(2)}\cr \end{array} \right ) + +S\left ( \begin{array}{c} \vph_1^{(1)} \vph_1^{(2)} \cr \cr +\vph_2^{(1)} \vph_2^{(2)}\cr \end{array} \right )^{-} +\] +solves the equation (\ref{lNLS}) and satisfies the reality +condition $\Phi_2 = \bar{\Phi}_1$. +\end{lemma} + +Proof: Direct calculation leads to the conclusion. Q.E.D. + +The periodicity condition $\Phi(x+2\pi ) = \Phi(x)$ can be easily +accomplished. For example, we can take $\vph^{(j)}$ ($j=1,2$) +to be two linearly independent Bloch functions $\vph^{(j)} = +e^{\sg_j x} \psi^{(j)}$ ($j=1,2$), where $\sg_2=-\sg_1$ and +$\psi^{(j)}$ are periodic functions $\psi^{(j)}(x+2\pi )= +\psi^{(j)}(x)$. Often we choose $\la$ to be a double point +of geometric multiplicity $2$, so that $\vph^{(j)}$ are already +periodic or antiperiodic functions. + +\section{Discrete Cubic Nonlinear Schr\"odinger Equation \label{DCNSE}} + +Consider the discrete focusing cubic nonlinear Schr\"odinger equation +(DNLS) +\begin{equation} +i \dot{q}_n = {1 \over h^2}[q_{n+1}-2 q_n +q_{n-1}] + |q_n|^2(q_{n+1}+ +q_{n-1}) +-2 \om^2 q_n \ , \label{DNLS} +\end{equation} +under periodic and even boundary conditions, +\[ +q_{n+N}=q_n\ , \ \ \ \ q_{-n}=q_n\ , +\] +where $i = \sqrt{-1}$, $q_n$'s are complex variables, $n \in \ZZ$, $\om$ +is a positive parameter, $h=1/N$, and $N$ is a positive integer +$N \geq 3$. The DNLS is integrable by virtue of the Lax pair +\cite{AL76}: +\begin{eqnarray} +\varphi_{n+1}&=&L^{(z)}_n\varphi_n\ , \label{dLax1} \\ +\dot{\varphi}_n&=&B^{(z)}_n\varphi_n\ , \label{dLax2} +\end{eqnarray} +\noindent +where +\begin{eqnarray*} +L^{(z)}_n&=&\left( \begin{array}{cc} + z& ihq_n \cr + ih\bq_n & 1/z \cr + \end{array} \right), +\\ +\\ +B^{(z)}_n&=&{i\over h^2}\left( \begin{array}{cc} + b_n^{(1)} & -izhq_n+(1/z)ihq_{n-1} \\ + -izh\bq_{n-1}+(1/z)ih\bq_n& b_n^{(4)} + \end{array} \right), \\ +& & b_n^{(1)} = 1-z^2+2i\la h-h^2q_n\bq_{n-1}+\om^2 h^2 ,\\ +& & b_n^{(4)} = 1/z^2-1+2i\la h+h^2\bq_nq_{n-1}-\om^2 h^2, +\end{eqnarray*} +\noindent +and where $z = \exp(i\la h)$. Compatibility of the over-determined +system (\ref{dLax1},\ref{dLax2}) gives the ``Lax representation'' +\[ +\dot{L}_n=B_{n+1}L_n-L_nB_n +\] +of the DNLS (\ref{DNLS}). Let $M(n)$ be the fundamental matrix solution +to (\ref{dLax1}), the {\em Floquet discriminant} is defined as +\[ +\Dl = \ \mbox{trace}\ \{ M(N) \} \ . +\] +Let $\psi^+$ and $\psi^-$ be any two solutions to (\ref{dLax1}), and +let $W_n(\psi^+,\psi^-)$ be the Wronskian +\[ +W_n(\psi^+,\psi^-) = \psi_n^{(+,1)}\psi_n^{(-,2)}-\psi_n^{(+,2)} +\psi_n^{(-,1)} \ . +\] +One has +\[ +W_{n+1}(\psi^+,\psi^-)=\rho_n W_n(\psi^+,\psi^-)\ , +\] +where $\rho_n = 1 +h^2|q_n|^2$, and +\[ +W_N(\psi^+,\psi^-)=D^2 \ W_0(\psi^+,\psi^-)\ , +\] +where $D^2 = \prod^{N-1}_{n=0}\rho_n$. +Periodic and antiperiodic points $z^{(p)}$ are defined by +\[ +\Delta(z^{(p)})=\pm 2D\ . +\] +A critical point $z^{(c)}$ is defined by the condition +$$ + {d\Delta \over dz}\bigg |_{z=z^{(c)}} = 0. +$$ +\noindent +A multiple point $z^{(m)}$ is a critical point which is also a periodic +or antiperiodic point. The {\em algebraic multiplicity} of +$z^{(m)}$ is defined as the order of the zero of $\Delta(z)\pm 2D$. Usually +it is $2$, but it can exceed $2$; when it does equal $2$, we call the +multiple point a {\em double point}, and denote it by $z^{(d)}$. The +{\em geometric multiplicity} of $z^{(m)}$ is defined as the dimension of +the periodic (or antiperiodic) eigenspace of (\ref{dLax1}) at $z^{(m)}$, +and is either $1$ or $2$. + +Fix a solution $q_n(t)$ of the DNLS (\ref{DNLS}), for which (\ref{dLax1}) +has a double point $z^{(d)}$ of geometric multiplicity 2, which is not +on the unit circle. +We denote two linearly independent solutions (Bloch functions) of the +discrete Lax pair (\ref{dLax1},\ref{dLax2}) at $z=z^{(d)}$ by $(\phi_n^+,\phi_n^-)$. Thus, a general solution of the discrete Lax pair +(\ref{dLax1};\ref{dLax2}) at $(q_n(t),z^{(d)})$ is given by +\[ +\phi_n = c^+ \phi_n^+ + c^- \phi_n^-, +\] +\nid +where $c^+$ and $c^-$ are complex parameters. We use $\phi_n$ +to define a transformation matrix $\Ga_n$ by +\[ +\Ga_n=\left(\begin{array}{cc} z+(1/z)a_n & b_n \cr c_n &-1/z+z d_n \cr + \end{array} \right), +\] +\nid +where, +\begin{eqnarray*} +a_n &=& {z^{(d)} \over (\bar{z}^{(d)})^2\Dl_n}\bigg [|\phi_{n2}|^2+|z^{(d)}|^2|\phi_{n1}|^2 + \bigg ],\\ +d_n &=& -{1 \over z^{(d)}\Dl_n}\bigg [|\phi_{n2}|^2+|z^{(d)}|^2|\phi_{n1}|^2 + \bigg ],\\ +b_n &=& {|z^{(d)}|^4-1 \over (\bar{z}^{(d)})^2\Dl_n}\phi_{n1}\bar{\phi}_{n2}, \\ +c_n &=& {|z^{(d)}|^4-1 \over z^{(d)}\bar{z}^{(d)}\Dl_n}\bar{\phi}_{n1}\phi_{n2}, \\ +\Dl_n &=& -{1 \over \bar{z}^{(d)}}\bigg [|\phi_{n1}|^2+|z^{(d)}|^2|\phi_{n2}|^2 + \bigg ]. +\end{eqnarray*} +\nid +From these formulae, we see that +\[ +\bar{a}_n=-d_n,\ \ \bar{b}_n=c_n. +\] +\nid +Then we define $Q_n$ and $\Psi_n$ by +\begin{equation} +Q_n\equiv {i\over h}b_{n+1}-a_{n+1}q_n +\label{BD1} +\end{equation} +\nid +and +\begin{equation} +\Psi_n(t;z)\equiv \Ga_n(z;z^{(d)};\phi_n)\psi_n(t;z) +\label{BD2} +\end{equation} +\nid +where $\psi_n$ solves the discrete Lax pair (\ref{dLax1},\ref{dLax2}) +at $(q_n(t),z)$. Formulas (\ref{BD1}) and (\ref{BD2}) are the +B\"acklund-Darboux transformations for the potential and eigenfunctions, +respectively. We have the following theorem \cite{Li92}. +\begin{theorem} +Let $q_n(t)$ denote a solution of the DNLS (\ref{DNLS}), for which +(\ref{dLax1}) has a double point $z^{(d)}$ of geometric multiplicity 2, +which is not on the unit circle. We denote two linearly independent +solutions of the discrete Lax pair (\ref{dLax1},\ref{dLax2}) +at $(q_n, z^{(d)})$ by $(\phi_n^+,\phi_n^-)$. We define $Q_n(t)$ and +$\Psi_n(t;z)$ by (\ref{BD1}) and (\ref{BD2}). Then +\begin{enumerate} +\item $Q_n(t)$ is also a solution of the DNLS (\ref{DNLS}). (The eveness +of $Q_n$ can be obtained by choosing the complex B\"acklund parameter +$c^+/c^-$ to lie on a certain curve, as shown in the example below.) +\item $\Psi_n(t;z)$ solves the discrete Lax pair (\ref{dLax1},\ref{dLax2}) +at $(Q_n(t),z)$. +\item $\Dl(z;Q_n)=\Dl(z;q_n)$, for all $z\in C$. +\item $Q_n(t)$ is homoclinic to $q_n(t)$ in the sense that $Q_n(t) \ra +e^{i\th_{\pm}}\ q_n(t)$, exponentially as $\exp (-\sg |t|)$ as $t \ra +\pm \infty$. Here $\th_{\pm}$ are the phase shifts, $\sg$ is a +nonvanishing growth rate associated to the double point $z^{(d)}$, and +explicit formulas can be developed for this growth rate and +for the phase shifts $\th_{\pm}$. +\end{enumerate} +\label{Backlund} +\end{theorem} + +\nid +{\bf Example:} We start with the uniform solution of (\ref{DNLS}) +\begin{equation} +q_n=q_c \ ,\ \forall n; \ \ \ \ q_c=a\exp \bigg \{-i[2(a^2-\om^2)t - \ga] +\bigg \}\ . +\label{ucsl} +\end{equation} +We choose the amplitude $a$ in the range +\begin{eqnarray} +& & N\tan{\pi \over N}< a 3 \ , +\nonumber \\ \label{constr} \\ +& & 3\tan{\pi \over 3}< a < \infty\ ,\ \ \ \mbox{when}\ N=3 \ ;\nonumber +\end{eqnarray} +so that there is only one set of quadruplets of double points +which are not on the unit circle, and denote one of them by $z=z_1^{(d)}= +z_1^{(c)}$ which corresponds to $\be = \pi / N$. The homoclinic orbit +$Q_n$ is given by +\begin{equation} +Q_n = q_c (\hat{E}_{n+1})^{-1} \bigg [ \hat{A}_{n+1} - 2 \cos \be +\sqrt{\rho \cos^2 \be -1}\hat{B}_{n+1} \bigg ]\ , +\label{hetorb} +\end{equation} +where +\[ +\hat{E}_n = ha\cos \be +\sqrt{\rho \cos^2 \be - 1} \ \mbox{sech}\ [ +2\mu t +2p] \cos [(2n-1)\be +\vth ]\ , +\] +\[ +\hat{A}_{n+1} = ha\cos \be +\sqrt{\rho \cos^2 \be - 1} \ \mbox{sech}\ [ +2\mu t +2p] \cos [(2n+3)\be +\vth ]\ , +\] +\[ +\hat{B}_{n+1} = \cos \varphi + i \sin \varphi \tanh [ 2\mu t +2p] ++ \ \mbox{sech}\ [ 2\mu t +2p] \cos [ 2(n+1)\be +\vth ]\ , +\] +\[ +\be = \pi / N\ , \ \ \rho = 1+h^2 a^2\ , \ \ \mu = 2h^{-2} \sqrt{\rho} +\sin \be \sqrt{\rho \cos^2 \be -1}\ , +\] +\[ +h=1/N,\ \ c_+/c_- = i e^{2p} e^{i\vth}\ , \ \ \vth \in [0,2\pi]\ , \ \ +p \in (-\infty, \infty)\ , +\] +\[ +z_1^{(d)}=\sqrt{\rho}\cos \be +\sqrt{\rho \cos^2 \be -1}\ , \ \ +\th(t)=(a^2-\om^2)t - \ga/2\ , +\] +\[ +\sqrt{\rho \cos^2 \be -1} + i \sqrt{\rho} \sin \be = ha e^{i \varphi}\ , +\] +where $\varphi=\sin^{-1} [\sqrt{\rho}(ha)^{-1} \sin \be ], +\ \ \varphi \in (0, \pi/2)$. + +Next we study the ``evenness'' condition: $Q_{-n} = Q_n$. It turns out +that the choices $\vth = - \be\ , \ - \be +\pi$ in the formula of +$Q_n$ lead to the evenness of $Q_n$ in $n$. In terms of figure eight +structure of $Q_n$, $\vth = - \be$ corresponds to one ear of the +figure eight, and $\vth = - \be +\pi$ corresponds to the other +ear. The even formula for $Q_n$ is given by, +\begin{equation} +Q_n = q_c \bigg [ \Ga / \La_n -1 \bigg ]\ , \label{ehetorb} +\end{equation} +where +\[ +\Ga = 1-\cos 2 \varphi - i \sin 2 \varphi \tanh [ 2 \mu t + 2p]\ , +\] +\[ +\La_n = 1 \pm \cos \varphi [\cos \be ]^{-1}\ \mbox{sech}[2 \mu t + 2p] +\cos [2n\be]\ , +\] +where (`+' corresponds to $\vth = -\be$). + +The heteroclinic orbit (\ref{ehetorb}) represents the figure eight +structure. If we denote by $S$ the circle, we have the topological +identification: +\[ +\mbox{(figure 8)}\ \otimes S = \bigcup_{p \in (-\infty,\infty),\ +\ga \in [0,2\pi]} Q_n(p, \ga, a, \om, \pm, N)\ . +\] + +\section{Davey-Stewartson II (DSII) Equations} + +Consider the Davey-Stewartson II equations (DSII), +\begin{equation} + \left \{ \begin{array}{l} +i \partial_t q = [ \partial^2_x - \partial^2_y]q+ [ 2( +|q|^2 - \omega^2) + u_y ] q \, , \cr \cr +[\partial^2_x + \partial^2_y] u = -4 \partial_y |q|^2 \, , \cr + \end{array} \right. +\label{fDS2} +\end{equation} +where $q$ and $u$ are respectively complex-valued and +real-valued functions of three variables $(t,x,y)$, and $\om$ is a +positive constant. We pose periodic boundary conditions, +\begin{eqnarray*} +& & q(t,x + L_1,y) = q(t,x,y) = q(t,x,y+ L_2) \, , \\ +& & u(t,x + L_1,y) = u(t,x,y) = u(t,x,y+ L_2) \, , +\end{eqnarray*} +and the even constraint, +\begin{eqnarray*} +& & q(t,-x,y) = q (t,x,y) = q(t,x,-y) \, ,\\ +& & u(t,-x,y) = u(t,x,y) = u(t,x,-y) \, . +\end{eqnarray*} +Its Lax pair is defined as: +\begin{eqnarray} + L \psi &=& \lambda \psi\,, \label{LP1} \\ + \partial_t \psi &=& A \psi\,,\label{LP2} +\end{eqnarray} +where $\psi = \left( \psi_1, \psi_2\right)$, and +\[ + L = \left( +\begin{array}{lr} +D^{-} & q\\ \\ +\bq & D^{+} +\end{array} +\right)\,, +\] +\[ +A = i \left[ +2 \left( +\begin{array}{cc} +- \partial^2_x & q \partial_x\\ +\bq \partial_x & \partial^2_x +\end{array} +\right) \, + \, +\left( +\begin{array}{cc} +r_1 & (D^+ q)\\ +-(D^{-} \bq ) & r_2 +\end{array} +\right) +\right]\, , +\] +\begin{equation} + D^+ = \alpha \partial_y + \partial_x\,, \qquad D^{-} = \alpha + \partial_y - \partial_x\, , \qquad \al^2 = -1\ . +\label{DD} +\end{equation} +$r_1$ and $r_2$ have the expressions, +\begin{equation} + r_1 = \frac{1}{2} [-w+iv] \ , \ \ \ r_2= \frac{1}{2} [w+iv] \, , +\label{res1} +\end{equation} +where $u$ and $v$ are real-valued functions satisfying +\begin{eqnarray} +& &[\partial^2_x + \partial^2_y] w += 2 [\partial^2_x - \partial^2_y] |q|^2 \, , \label{res2} \\ +& &[\partial^2_x + \partial^2_y] v += i4 \alpha \partial_x \partial_y |q|^2 \, , \label{res3} +\end{eqnarray} +and $w=2(|q|^2 - \om^2 )+u_y$. +Notice that DSII (\ref{fDS2}) is +invariant under the transformation $\sg$: +\begin{equation} + \sigma \circ (q,\bq, r_1, r_2; \al) = (q,\bq, -r_2, -r_1; -\al)\,. +\label{IT} +\end{equation} +Applying the transformation $\sigma$ (\ref{IT}) to the Lax pair +(\ref{LP1}, \ref{LP2}), we have a congruent Lax pair for which +the compatibility condition gives the same DSII. +The congruent Lax pair is given as: +\begin{eqnarray} + \hat{L} \hat{\psi} &=& \lambda \hat{\psi} + \,,\label{CLP1} \\ + \partial_t \hat{\psi} &=& \hat{A} + \hat{\psi} + \,,\label{CLP2} +\end{eqnarray} +where $\hat{\psi} = (\hat{\psi}_1, \hat{\psi}_2)$, and +\[ + \hat{L} = + \left( + \begin{array}{cc} +- D^+ & q\\ \\ +\bq & -D^- + \end{array} + \right)\,, +\] +\[ +\hat{A} = i \left[ + 2 \left( + \begin{array}{cc} +- \partial^2_x & q \partial_x\\ +\bq \partial_x & \partial^2_x + \end{array} + \right) + + \left( + \begin{array}{cc} +-r_2 & -(D^- q)\\ +(D^+\bq ) & -r_1 + \end{array} + \right) +\right]\,. +\] +The compatibility condition of the Lax pair (\ref{LP1}, \ref{LP2}), +\begin{displaymath} + \partial_t L = [A, L]\ , +\end{displaymath} +where $[A, L] = AL - LA$, and the compatibility condition of the +congruent Lax pair (\ref{CLP1}, \ref{CLP2}), +\begin{displaymath} + \partial_t \hat{L} = [\hat{A}, \hat{L}] + \end{displaymath} +give the same DSII (\ref{fDS2}). Let $(q,u)$ be a solution to the DSII +(\ref{fDS2}), and let $\lambda_0$ be any value of $\lambda$. Let +$\psi = (\psi_1, \psi_2)$ be a solution to the Lax +pair (\ref{LP1}, \ref{LP2}) at $(q, \bar q, r_1, r_2; +\lambda_0)$. Define the matrix operator: +\begin{displaymath} + \Gamma = +\left[ + \begin{array}{cc} + \wedge + a & b\\ + c & \wedge + d + \end{array} +\right]\,, +\end{displaymath} +where $\wedge = \alpha \partial_y - \lambda$, and $a$, $b$, $c$, +$d$ are functions defined as: +\begin{eqnarray*} + a &=& \frac{1}{\Delta} \left[ \psi_2 \wedge_2 \bar{\psi}_2 + + \bar{\psi}_1 \wedge_1 \psi_1 \right]\,,\\[2ex] + b &=& \frac{1}{\Delta} \left[ \bar{\psi}_2 \wedge_1 \psi_1 - + \psi_1 \wedge_2 \bar{\psi}_2 \right]\,,\\[2ex] + c &=& \frac{1}{\Delta} \left[ \bar{\psi}_1 \wedge_1 \psi_2 + - \psi_2 \wedge_2 \bar{\psi}_1 \right]\,,\\[2ex] + d &=& \frac{1}{\Delta} \left[ \bar{\psi}_2 \wedge_1 \psi_2 + + \psi_1 \wedge_2 \bar{\psi}_1 \right]\,, +\end{eqnarray*} +in which $\wedge_1 = \alpha \partial_y - \lambda_0$, $\wedge_2 = +\alpha \partial_y + \bar{\lambda}_0$, and +\begin{displaymath} + \Delta = - \left[ | \psi_1 |^2 + |\psi_2|^2 \right]\,. +\end{displaymath} +Define a transformation as follows: +\begin{displaymath} + \left\{ + \begin{array}{ccc} +(q,r_1,r_2) &\rightarrow& (Q,R_1,R_2)\,, \\ +\phi &\rightarrow& \Phi\,; + \end{array} +\right. +\end{displaymath} +\begin{eqnarray} + Q &=& q - 2b\,,\nonumber \\[2ex] + R_1 &=& r_1 + 2(D^+a)\,, \label{DSBT}\\[2ex] + R_2 &=& r_2 - 2 (D^- d)\,,\nonumber\\[2ex] + \Phi &=& \Gamma \phi\,;\nonumber +\end{eqnarray} +where $\phi$ is any solution to the Lax pair (\ref{LP1}, +\ref{LP2}) at $(q, \bar{q}, r_1, r_2; \lambda)$, $D^+$ +and $D^-$ are defined in (\ref{DD}), we have the following theorem +\cite{Li00a}. +\begin{theorem} +The transformation (\ref{DSBT}) is a B\"acklund-Darboux +transformation. That is, the function $Q$ defined +through the transformation (\ref{DSBT}) is also a solution to the +DSII (\ref{fDS2}). The function $\Phi$ defined through the transformation +(\ref{DSBT}) solves the Lax pair (\ref{LP1}, \ref{LP2}) at $(Q, +\bar{Q}, R_1, R_2; \lambda)$. +\label{DSTH} +\end{theorem} + +\subsection{An Example \label{dsex}} + +Instead of using $L_1$ and $L_2$ to describe the periods of the +periodic boundary condition, one can introduce $\k_1$ and $\k_2$ +as $L_1 = \frac{2\pi }{\k_1}$ and $L_2 = \frac{2\pi }{\k_2}$. Consider +the spatially independent solution, +\begin{equation} +q_c = \eta \exp \{ -2i [ \eta^2 - \om^2 ] t + i \ga \} \ . +\label{us} +\end{equation} +The dispersion relation for the linearized DSII at $q_c$ is +\[ +\Om = \pm \frac{|\xi_1^2 - \xi_2^2|}{\sqrt{\xi_1^2 +\xi_2^2}} +\sqrt{4 \eta^2 - (\xi_1^2 +\xi_2^2)}\ , \ \ \mbox{for} \ +\dl q \sim q_c \exp \{ i (\xi_1 x +\xi_2 y) +\Om t \} \ , +\] +where $\xi_1 = k_1 \k_1$, $\xi_2 = k_2 \k_2$, and $k_1$ and $k_2$ +are integers. We restrict $\k_1$ and $\k_2$ as follows to have only +two unstable modes ($\pm \k_1, 0$) and ($0, \pm \k_2$), +\[ +\k_2 < \k_1 < 2 \k_2\ , \ \ +\k_1^2 < 4 \eta^2 < \min \{ \k_1^2 + \k_2^2, 4 \k_2^2 \} \ , +\] +or +\[ +\k_1 < \k_2 < 2 \k_1\ , \ \ +\k_2^2 < 4 \eta^2 < \min \{ \k_1^2 + \k_2^2, 4 \k_1^2 \} \ . +\] +The Bloch eigenfunction of the Lax pair (\ref{LP1}) and (\ref{LP2}) +is given as, +\begin{equation} + \psi = c(t) \left[ + \begin{array}[]{c} +-q_c \\ \chi + \end{array} \right] +\exp \left\{ i(\xi_1 x + \xi_2y) \right\} \, , +\label{slLax} +\end{equation} +where +\begin{eqnarray*} +& & c(t) = c_0 \exp \left\{ \left[ 2\xi_1(i \alpha \xi_2 - \lambda ) + + ir_2 \right] t \right\} \, , \\ +& & r_2 - r_1 = 2 ( \left| q_c \right|^2 - \omega^2 ) \, , \\ +& & \chi = (i \alpha \xi_2- \lambda )-i\xi_1 \, , \\ +& & (i \alpha \xi_2 - \lambda)^2 + \xi^2_1 = \eta^2 \, . +\end{eqnarray*} +For the iteration of the B\"acklund-Darboux transformations, one +needs two sets of eigenfunctions. First, we choose +$\xi_1 = \pm \frac{1}{2} \k_1$, $\xi_2=0$, $\lambda_0 = \sqrt{\eta^2 - +\frac{1}{4}\k_1^2}$ (for a fixed branch), +\begin{eqnarray} + \psi^{\pm} = c^{\pm} \left[ + \begin{array}{c} + -q_c \\ \\ \chi^{\pm} + \end{array} \right] + \exp \left\{ \pm i \frac{1}{2}\k_1x \right\} \, , +\label{efunc1} +\end{eqnarray} +where +\begin{eqnarray*} +& & c^{\pm} = c^{\pm}_0 \exp \left\{ \left[ \mp \k_1 \lambda_0 + ir_2 + \right] t \right\} \, , \\ +& & \chi^{\pm} = - \lambda_0 \mp i \frac{1}{2} \k_1 = + \eta e^{\mp i (\frac{\pi}{2} +\vth_1)} \, . +\end{eqnarray*} +We apply the B\"acklund-Darboux transformations with $\psi = +\psi^+ + \psi^-$, which generates +the unstable foliation associated with the $(\k_1,0)$ and $(-\k_1,0)$ +linearly unstable modes. Then, we choose $\xi_2 = \pm +\frac{1}{2}\k_2$, $\lambda =0$, $\xi^0_1 = \sqrt{\eta^2-\frac{1}{4}\k_2^2}$ +(for a fixed branch), +\begin{equation} + \phi_{\pm} = c_{\pm} \left[ + \begin{array}{c} + -q_c \\ \\ \chi_{\pm} + \end{array} \right] + \exp \left\{ i (\xi^0_1 x \pm \frac{1}{2}\k_2 y) \right\} \, , +\label{efunc2} +\end{equation} +where +\begin{eqnarray*} +& & c_{\pm} = c^0_{\pm} \exp \left\{ \left[ \pm i \alpha \k_2\xi^0_1 + + ir_2 \right] t \right\} \, , \\ +& & \chi_{\pm} = \pm i \alpha \frac{1}{2}\k_2 - i\xi^0_1 +=\pm \eta e^{\mp i \vth_2}\, . +\end{eqnarray*} +We start from these eigenfunctions $\phi_{\pm}$ to generate +$\Gamma \phi_{\pm}$ through \BD, and then iterate the \BD ~with +$\Gamma \phi_+ + \Gamma \phi_-$ to generate the unstable +foliation associated with all the linearly unstable modes $(\pm +\k_1,0)$ and $(0, \pm \k_2)$. It turns out that the following +representations are convenient, +\begin{eqnarray} +\psi^\pm &=& \sqrt{c_0^+c_0^-}e^{ir_2 t}\left ( \begin{array}{c} +v_1^\pm \cr v_2^\pm \cr \end{array} \right ) \ , +\label{rwf1} \\ +\phi_\pm &=& \sqrt{c^0_+c^0_-}e^{i \xi_1^0 x + ir_2 t}\left ( \begin{array}{c} +w_1^\pm \cr w_2^\pm \cr \end{array} \right ) \ , +\label{rwf2} +\end{eqnarray} +where +\[ +v_1^\pm = -q_c e^{\mp \frac{\tau}{2} \pm i \tx} \ , \ \ +v_2^\pm = \eta e^{\mp \frac{\tau}{2} \pm i \tz} \ , +\] +\[ +w_1^\pm = -q_c e^{\pm \frac{\htau}{2} \pm i \hy} \ , \ \ +w_2^\pm = \pm \eta e^{\pm \frac{\htau}{2} \pm i \hz}\ , +\] +and +\[ +c_0^+/c_0^- = e^{\rho + i \vth }\ , \ \ +\tau = 2\k_1 \la_0 t - \rho \ , \ \ +\tx = \frac{1}{2} \k_1 x + \frac{\vth}{2}\ , +\tz = \tx - \frac{\pi}{2} - \vth_1 \ , +\] +\[ +c^0_+/c^0_- = e^{\hrho + i \hvth }\ , \ \ +\htau = 2i\al \k_2 \xi_1^0 t + \hrho \ , \ \ +\hy = \frac{1}{2} \k_2 y + \frac{\hvth}{2}\ , +\hz = \hy - \vth_2 \ . +\] +The following representations are also very useful, +\begin{eqnarray} +\psi &=& \psi^+ + \psi^- = 2 \sqrt{c_0^+c_0^-}e^{ir_2 t} +\left ( \begin{array}{c} +v_1 \cr v_2 \cr \end{array} \right ) \ , +\label{rwf3} \\ +\phi &=& \phi^+ + \phi^- = 2 \sqrt{c^0_+c^0_-}e^{i \xi_1^0 x + +ir_2 t}\left ( \begin{array}{c} +w_1 \cr w_2 \cr \end{array} \right ) \ , +\label{rwf4} +\end{eqnarray} +where +\[ +v_1 = -q_c [ \cosh \frac{\tau}{2} \cos \tx - i \sinh \frac{\tau}{2} \sin \tx ] +\ , \ \ v_2 = \eta [ \cosh \frac{\tau}{2} \cos \tz - i \sinh \frac{\tau}{2} +\sin \tz ] \ , +\] +\[ +w_1 = -q_c [ \cosh \frac{\htau}{2} \cos \hy + i \sinh \frac{\htau}{2} \sin +\hy ] +\ , \ \ w_2 = \eta [ \sinh \frac{\htau}{2} \cos \hz + i \cosh \frac{\htau}{2} +\sin \hz ] \ . +\] +Applying the \BD ~(\ref{DSBT}) with $\psi$ given in (\ref{rwf3}), we +have the representations, +\begin{eqnarray} +a &=& -\lambda_0 \ \mbox{sech}\ \tau \sin (\tx + \tz) \sin (\tx -\tz) +\label{aexp} \\ +& & \times \bigg [ 1 + \ \mbox{sech}\ \tau \cos (\tx + \tz) \cos (\tx -\tz) +\bigg ]^{-1}\, , \nonumber \\ +b &=& -q_c \tb = - \frac{\lambda_0 q_c}{\eta} \bigg [ \cos (\tx - \tz) - +i \tanh \tau \sin (\tx - \tz) \label{bexp}\\ +& & +\ \mbox{sech}\ \tau \cos (\tx + \tz)\bigg ] +\bigg [ 1+ \ \mbox{sech}\ \tau \cos (\tx + \tz) \cos (\tx - \tz) +\bigg ]^{-1} \, , \nonumber \\ +& & c= \overline{b} \, , \ \ \ \ d= - \overline{a} =-a \, . \label{cdexp} +\end{eqnarray} +The evenness of $b$ in $x$ is enforced by the requirement that +$\vth - \vth_1 = \pm \frac{\pi}{2}$, and +\begin{eqnarray} +a^{\pm} &=& \mp \lambda_0 \ \mbox{sech} \ \tau \cos \vth_1 \sin (\k_1 x) +\label{eaexp} \\ +& & \times \bigg [ 1 \mp \ \mbox{sech}\ \tau \sin \vth_1 \cos (\k_1 x) +\bigg ]^{-1}\, , \nonumber \\ +b^{\pm} &=& -q_c \tb^{\pm} = - \frac{\lambda_0 q_c}{\eta} \bigg [ +-\sin \vth_1 - +i \tanh \tau \cos \vth_1 \label{ebexp}\\ +& & \pm \ \mbox{sech}\ \tau \cos (\k_1 x) \bigg ] +\bigg [ 1 \mp \ \mbox{sech}\ \tau \sin \vth_1 \cos (\k_1 x) +\bigg ]^{-1} \, , \nonumber \\ +& & c= \overline{b} \, , \ \ \ \ d= - \overline{a} =-a \, . \label{ecdexp} +\end{eqnarray} +Notice also that $a^{\pm}$ is an odd function in $x$. Under +the above \BD, the +eigenfunctions $\phi_{\pm}$ (\ref{efunc2}) are transformed into +\begin{equation} + \varphi^{\pm} = \Gamma \phi_{\pm} \, , +\label{tphi} +\end{equation} +where +\begin{eqnarray*} + \Gamma = \left[ + \begin{array}{cc} +\Lambda + a & b \\ \\ +\overline{b} & \Lambda -a + \end{array} \right] \, , +\end{eqnarray*} +and $\Lambda = \alpha \partial_y - \lambda$ with $\lambda$ +evaluated at $0$. Let $\varphi = \varphi^+ + \varphi^-$ +(the arbitrary constants $c^0_{\pm}$ are already included in +$\varphi^{\pm}$), $\varphi$ has the representation, +\begin{equation} +\varphi = 2 \sqrt{c_+^0 c_-^0} e^{i \xi_1^0 x + i r_2 t}\left[ + \begin{array}{c} + -q_c W_1 \\ \\ \eta W_2 + \end{array} \right] \ , +\label{ief} +\end{equation} +where +\[ +W_1 = (\al \pa_y w_1) + a w_1 +\eta \tb w_2\ , \ \ +W_2 = (\al \pa_y w_2) - a w_2 +\eta \overline{\tb} w_1\ . +\] +We generate the coefficients in the \BD ~ +(\ref{DSBT}) with $\varphi$ (the iteration of the \BD), +\begin{eqnarray} +a^{(I)} &=& - \bigg [ W_2 (\al \pa_y \overline{W_2}) + +\overline{W_1} (\al \pa_y W_1) \bigg ]\bigg [ |W_1|^2 ++|W_2|^2 \bigg ]^{-1}\ , \label{ria}\\ +b^{(I)} &=& \frac{q_c}{\eta}\bigg [ \overline{W_2} (\al \pa_y W_1) - +W_1 (\al \pa_y \overline{W_2}) \bigg ]\bigg [ |W_1|^2 ++|W_2|^2 \bigg ]^{-1}\ , \label{rib} \\ +& & c^{(I)} = \overline{b^{(I)}} \, , \ \ \ \ \, d^{(I)} =-\overline{a^{(I)}} +\, , \label{ricd} +\end{eqnarray} +where +\begin{eqnarray*} +& & W_2 (\al \pa_y \overline{W_2}) + \overline{W_1} (\al \pa_y W_1) \\ +&=& \frac{1}{2} \al \k_2 \bigg \{ \cosh \htau \bigg [ -\al \k_2 a ++ i a \eta (\tb + \overline{\tb}) \cos \vth_2 \bigg ] \\ +&+& \bigg [ \frac{1}{4}\k_2^2 - a^2 -\eta^2 |\tb|^2 \bigg ] +\cos (\hy +\hz ) \sin \vth_2 + \sinh \htau \bigg [ a \eta +(\tb - \overline{\tb}) \sin \vth_2 \bigg ] \bigg \} \ , \\ \\ +& & |W_1|^2 +|W_2|^2 \\ +&=& \cosh \htau \bigg [ a^2 + \frac{1}{4}\k_2^2 + \eta^2 |\tb|^2 ++i\al \k_2 \eta \frac{1}{2} (\tb + \overline{\tb})\cos \vth_2 \bigg ] \\ +&+& \bigg [ \frac{1}{4}\k_2^2 - a^2 -\eta^2 |\tb|^2 \bigg ] +\sin (\hy +\hz ) \sin \vth_2 + \sinh \htau \bigg [ \al \k_2 \eta +\frac{1}{2} (\tb - \overline{\tb}) \sin \vth_2 \bigg ]\ , \\ \\ +& & \overline{W_2} (\al \pa_y W_1) - W_1 (\al \pa_y \overline{W_2}) \\ +&=& \frac{1}{2} \al \k_2 \bigg \{ \cosh \htau \bigg [ -\al \k_2 \eta \tb ++ i (-a^2 + \frac{1}{4}\k_2^2 + \eta^2 \tb^2 ) \cos \vth_2 \bigg ] \\ +&+& \sinh \htau (a^2 - \frac{1}{4}\k_2^2 + \eta^2 \tb^2 )\sin \vth_2 +\bigg \} \ . +\end{eqnarray*} +The new solution to the focusing Davey-Stewartson~II equation + (\ref{fDS2}) is given by +\begin{equation} + Q= q_c -2b -2b^{(I)} \, . +\label{newsl} +\end{equation} +The evenness of $b^{(I)}$ in $y$ is enforced by the requirement +that $\hvth - \vth_2 = \pm \frac{\pi}{2}$. In fact, we have +\begin{lemma} +Under the requirements that $\vth - \vth_1 = \pm \frac{\pi}{2}$, and +$\hvth - \vth_2 = \pm \frac{\pi}{2}$, +\begin{equation} + b(-x) = b(x) \, , \ \ \ b^{(I)}(-x,y)=b^{(I)}(x,y)=b^{(I)}(x,-y) \, , +\end{equation} +and $Q = q_c -2b-2b^{(I)}$ is even in both $x$ and $y$. +\label{evenla} +\end{lemma} + +Proof: It is a direct verification by noticing that under the +requirements, $a$ is an odd function in $x$. Q.E.D. + +The asymptotic behavior of $Q$ can be computed directly. In +fact, we have the asymptotic phase shift lemma. + +\begin{lemma}[Asymptotic Phase Shift Lemma] +For $\la_0 > 0$, $\xi_1^0 > 0$, and $i\al =1$, as $t \ra \pm \infty$, +\begin{equation} + Q = q_c -2b-2b^{(I)} \ra q_c e^{i\pi } e^{\mp i 2 (\vth_1 -\vth_2 )}\ . +\label{ayp} +\end{equation} +In comparison, the asymptotic phase shift of the +first application of the \BD ~ is given by +\[ +q_c - 2b \ra q_c e^{\mp i 2 \vth_1}\ . +\] +\end{lemma} + +\section{Other Soliton Equations} + +In general, one can classify soliton equations into two categories. +Category I consists of those equations possessing instabilities, +under periodic boundary condition. In their phase space, figure-eight +structures (i.e. separatrices) exist. Category II consists of +those equations possessing no instability, under periodic boundary +condition. In their phase space, no figure-eight structure +(i.e. separatrix) exists. Typical Category I soliton equations are, +for example, focusing nonlinear Schr\"odinger equation, sine-Gordon +equation \cite{Li03k}, modified KdV equation. Typical Category II +soliton equations are, +for example, KdV equation, defocusing nonlinear Schr\"odinger equation, +sinh-Gordon equation, Toda lattice. In principle, figure-eight structures +for Category I soliton equations can be constructed through B\"acklund-Darboux +transformations, as illustrated in previous sections. It should be remarked +that B\"acklund-Darboux transformations still exist for Category II soliton +equations, but do not produce any figure-eight structure. A good reference +on B\"acklund-Darboux transformations is \cite{MS91}. + + + + + + + + + + + + + + + + + + + + + + +\clearpage{} +\clearpage{}\chapter{Melnikov Vectors} + +\section{1D Cubic Nonlinear Schr\"odinger Equation \label{MVNLS}} + +We select the NLS (\ref{NLS}) as our first example to show how +to establish Melnikov vectors. We continue from Section \ref{1DCNSE}. +\begin{definition} +Define the sequence of functionals $F_j$ as follows, +\begin{equation} +F_j(\vq) = \Delta(\lambda^c_j (\vec{q}),\vec{q}), +\label{3.1} +\end{equation} +where $\la^c_j$'s are the critical points, $\vec{q}=(q, -\bq )$. +\end{definition} +We have the lemma \cite{LM94}: +\begin{lemma} +If $\lambda_j^c(\vec{q}) $ is a simple critical point of $\Delta +(\lambda)$ [i.e., $\Delta''(\lambda^c_j) \neq 0$], $F_j $ is +analytic in a neighborhood of $\vec{q}$, +with first derivative given by +\begin{equation} +{\delta F_{j}\over {\delta q}} = \,{\delta\Delta\over{\delta +q}}\bigg|_{\lambda = \lambda^c_{j}} \, + +{\partial\Delta\over {\partial\lambda}} +\bigg|_{\lambda = \lambda^c_{j}} \, {\delta +\lambda^c_{j}\over {\delta q}} += {\delta\Delta\over {\delta q}} +\bigg|_{\lambda = \lambda^c_{j}}\ , \label{gradf} +\end{equation} +where +\begin{equation} +\frac{\delta}{\delta {\vec{q}}} \ \Delta(\lambda;\vq) = i +\frac{\sqrt{\Delta^2 - 4}}{W [ \psi ^{+}, \psi ^{-}]} +\left[ \begin{array}{c} +\psi^{+}_{2} (x;\lambda) \psi ^{-} _{2} (x; \lambda) \\ +\\ +\psi^{+}_{1} (x; \lambda) \psi ^{-} _{1} (x; \lambda) +\end{array}\right] \ , +\label{3.2} +\end{equation} +and the Bloch eigenfunctions $\psi^\pm $ have the property that +\begin{eqnarray} +\psi^{\pm} (x+2\pi; \lambda) = \rho^{\pm 1} \psi^{\pm} (x; \lambda)\ , +\label{3.3} +\end{eqnarray} +for some $\rho$, also the Wronskian is given by +\[ +W [ \psi ^{+}, \psi ^{-}] = \psi ^{+}_1 \psi ^{-}_2 - +\psi ^{+}_2 \psi ^{-}_1 \ . +\] +In addition, $\Delta'$ is given by +\begin{eqnarray} +\frac{d\Delta}{d \lambda}= -i \frac{\sqrt{\Delta ^{2} - 4}}{W +[\psi +^{+},\psi ^{-}]} \int ^{2\pi}_{0} [\psi ^{+} _{1} \psi ^{-} _{2} + \psi _{2} +^{+} +\psi ^{-} _{1} ] \ dx \ . +\label{3.4} +\end{eqnarray} +\label{lemma 6} +\end{lemma} +Proof: To prove this lemma, one calculates using variation of +parameters: +$$ +\delta M (x; \lambda ) = M (x) \, \int _{0} ^{x} M ^{-1} (x') \delta +\hat{Q}(x') M (x') dx', +$$ +\noindent where +\begin{eqnarray*} +\delta \hat{Q} \equiv i\left( \begin{array}{c}0 \quad +\delta q\\ +\\ \delta \bq \quad 0 +\end{array}\right) \ . +\end{eqnarray*} + Thus; one obtains the formula +$$ +\delta \Delta (\lambda;\vq) = \ \mbox{trace}\ \bigg [M (2\pi) \int^{2\pi}_{0} +M^{-1} (x') \ \delta \hat{Q}(x') \ M (x') dx'\bigg ], +$$ +\noindent which gives + +\begin{eqnarray} +\frac{\delta \Delta (\lambda)}{\delta q(x)} &=& i \ \mbox{trace}\ +\bigg[ M^{-1} (x)\Bigl( +\begin{array}{cc} + 0 & 1 \\ 0 & 0 +\end{array} +\Bigr)M (x) M (2\pi)\bigg ]\ , \nonumber \\ +\label{derg} \\ +\frac{\delta \Delta (\lambda)}{\delta \bq(x)} &=& i \ \mbox{trace}\ +\bigg [ M^{-1} (x)\Bigl( +\begin{array}{cc} + 0 & 0 \\ 1 & 0 +\end{array} +\Bigr)M (x) M (2\pi)\bigg ] \ . \nonumber +\end{eqnarray} +Next, we use the Bloch eigenfunctions $\{ \psi^{\pm} \} $ +to form the matrix +\[ +N(x;\lambda) = \left( \begin{array}{cc} + \psi^+_1 & \psi^-_1 \\ + \psi^+_2 & \psi^-_2 +\end{array} \right) \ . +\] +Clearly, +\[ +N(x;\lambda) = M(x;\lambda) N(0;\lambda)\ ; +\] +or equivalently, +\begin{equation} +M(x;\lambda) = N(x;\lambda) [N(0;\lambda)]^{-1}\ . +\label{3.6} +\end{equation} +Since $\psi^\pm$ are Bloch eigenfunctions, one also has +$$ +N (x + 2\pi; \lambda) = N (x; \lambda) \left( \begin{array}{cc} + \rho & 0 \\ + 0 & \rho^{-1} +\end{array} \right)\ , +$$ +\noindent which implies +$$ +N (2\pi; \lambda ) = M (2\pi; \lambda) N (0; \lambda) = N (0; \lambda) +\left( \begin{array}{cc} + \rho & 0 \\ + 0 & \rho^{-1} +\end{array} \right)\ , +$$ +that is, +\begin{equation} +M (2\pi; \lambda ) = N (0; \lambda) +\left( \begin{array}{cc} + \rho & 0 \\ + 0 & \rho^{-1} +\end{array} \right) [N (0; \lambda)]^{-1}\ . +\label{3.7} +\end{equation} +For any $2 \times 2$ matrix $\sigma$, equations (\ref{3.6}) and (\ref{3.7}) +imply +\[ +\mbox{trace}\ \{[M(x)]^{-1} \ \sigma \ M(x) \ M(2\pi) \} = \mbox{trace}\ +\{[N(x)]^{-1} \ \sigma \ N(x) \ diag(\rho, \rho^{-1}) \}, +\] +which, through an explicit evaluation of (\ref{derg}), proves formula +(\ref{3.2}). +Formula (\ref{3.4}) is established similarly. These formulas, together +with the fact that $\lambda^c(\vec{q})$ is differentiable because it +is a simple zero of $\Delta'$, provide the representation of +$\frac{\dl F_j}{\dl \vq}$. Q.E.D. +\begin{remark} +Formula (\ref{gradf}) for the $\frac{\dl F_j}{\dl \vq}$ is actually valid even +if $\Dl''(\la^c_j(\vq);\vq)=0$. Consider a function $\vq_*$ at which +\[ +\Dl''(\la^c_j(\vq_*);\vq_*)=0, +\] +\nid +and thus at which $\la^c_j(\vq)$ fails to be analytic. For $\vq$ near +$\vq_*$, one has +\[ +\Dl'(\la^c_j(\vq);\vq)=0; +\] +\[ +{\dl \over \dl \vq}\Dl'(\la^c_j(\vq);\vq)=\Dl''{\dl \over \dl \vq}\la^c_j ++{\dl \over \dl \vq}\Dl'=0; +\] +\nid +that is, +\[ +{\dl \over \dl \vq}\la^c_j=-{1 \over \Dl''}{\dl \over \dl \vq}\Dl' \ . +\] +\nid +Thus, +\[ +{\dl \over \dl \vq} F_j = {\dl \over \dl \vq}\Dl+\Dl'{\dl \over \dl \vq} +\la^c_j = {\dl \over \dl \vq}\Dl-{\Dl' \over \Dl''} +{\dl \over \dl \vq}\Dl' \mid_{\la=\la^c_j(\vq)}. +\] +Since ${\Dl' \over \Dl''}\ra 0$, +as $\vq \ra \vq_*$, one still has formula (\ref{gradf}) at $\vq = \vq_*$: +\[ +{\dl \over \dl \vq} F_j={\dl \over \dl \vq}\Dl\mid_{\la=\la^c_j(\vq)}. +\] +\end{remark} +The NLS (\ref{NLS}) is a Hamiltonian system: +\begin{equation} +iq_t = \frac {\dl H} {\dl \bq } \ , +\label{hNLS} +\end{equation} +where +\[ +H = \int_{0}^{2\pi } \{ - |q_x|^2 + |q|^4 \} \ dx \ . +\] +\begin{corollary} +For any fixed $\la \in \CC$, $\Dl (\la, \vq)$ is a constant of motion +of the NLS (\ref{NLS}). In fact, +\[ +\{ \Dl (\la, \vq), H(\vq) \} = 0 \ , \ \ \ +\{ \Dl (\la, \vq), \Dl (\la', \vq) \} = 0 \ , \ \ \forall \la, +\la' \in \CC \ , +\] +where for any two functionals $E$ and $F$, their Poisson bracket is +defined as +\[ +\{ E, F \} = \int_{0}^{2\pi } \left [ \frac{\dl E}{\dl q} +\frac{\dl F}{\dl \bq} - \frac{\dl E}{\dl \bq} +\frac{\dl F}{\dl q} \right ] dx \ . +\] +\end{corollary} +Proof: The corollary follows from a direction calculation +from the spatial part (\ref{Lax1}) of the Lax pair and the +representation (\ref{3.2}). Q.E.D. + +For each fixed $\vq$, $\Dl$ is an entire function of +$\la$; therefore, can be determined by its values at +a countable number of values of $\la$. The invariance of $\Dl$ +characterizes the isospectral nature of the NLS equation. + +\begin{corollary} +The functionals $F_j$ are constants of motion of the NLS (\ref{NLS}). +Their gradients provide Melnikov vectors: +\begin{equation} +\mbox{grad} \ F_j(\vq) = i +\frac{\sqrt{\Delta^2 - 4}}{W [ \psi ^{+}, \psi ^{-}]} +\left[ \begin{array}{c} +\psi^{+}_{2} (x;\lambda^c_j) \psi ^{-} _{2} (x; \lambda^c_j) \\ +\\ +\psi^{+}_{1} (x; \lambda^c_j) \psi ^{-} _{1} (x; \lambda^c_j) +\end{array}\right] \ . +\label{3.8} +\end{equation} +\end{corollary} +The distribution of the critical points $\la^c_j$ are described by the +following counting lemma \cite{LM94}, +\begin{lemma}[Counting Lemma for Critical Points] +For $q \in H^1$, set $N = N(\| q\|_1) \in \ZZ^+$ by +$$ +N(\|q\|_1) = 2 \bigg[ \|q\|^2_0 \cosh +\bigg(\|q\|_0\bigg)\ +\ 3 \|q\|_1 \sinh +\bigg(\|q\|_0\bigg )\bigg], +$$ +\noindent where $[x] =$ first integer greater than $x$. +Consider +$$ +\Delta'(\lambda; \vq) = \frac d{d\lambda}\ \Delta (\lambda;\vq). +$$ Then +\begin{enumerate} +\item $\Delta'(\lambda;\vq)$ has exactly $2N+1$ zeros (counted +according to multiplicity) in the interior of the +disc $D = \{ \lambda\in \CC: \ |\lambda | +< (2N+1)\frac \pi{2}\};$ +\item $\forall k\, \in \ZZ,|k| > N, +\Delta'(\lambda,\vq)$ +has exactly one zero in each disc\newline +$\{\lambda \in \CC \colon\ |\lambda - k \pi | < +\frac\pi{4}\}$. +\item $\Delta' (\lambda; \vq)$ has no other zeros. +\item For $|\lambda | > (2N + 1)\frac\pi{2},$ +the zeros of $\Delta',\{\lambda^c_j,|j| > N\}$, are all +real, simple, and satisfy the asymptotics +\end{enumerate} +$$ +\lambda^c_j = j\pi + o(1) \ \ \mbox{ as }|j| \to +\infty . +$$ +\label{countle1} +\end{lemma} + +\subsection{Melnikov Integrals} + +When studying perturbed integrable systems, the figure-eight structures +often lead to chaotic dynamics through homoclinic bifurcations. An extremely +powerful tool for detecting homoclinic orbits is the so-called Melnikov +integral method \cite{Mel63}, which uses ``Melnikov integrals'' to +provide estimates of the distance between the center-unstable manifold +and the center-stable manifold of a normally hyperbolic invariant manifold. +The Melnikov +integrals are often integrals in time of the inner products of certain +Melnikov vectors with the perturbations in the perturbed integrable +systems. This implies that the Melnikov vectors play a key role in the +Melnikov integral method. First, we consider the case of one +unstable mode associated with a complex double point $\nu$, for which +the homoclinic orbit is given by B\"acklund-Darboux formula (\ref{4.5}), +\begin{eqnarray*} +Q(x,t) \equiv q(x,t) \ + \ +2 (\nu-\bar{\nu}) \ +\frac{\phi_1{\bar{\phi}}_2}{\phi_1 {\bar{\phi}}_1+ +\phi_2{\bar{\phi}}_2}\ , +\end{eqnarray*} +where $q$ lies in a normally hyperbolic invariant manifold and +$\phi$ denotes a general solution to the Lax pair (\ref{Lax1}, \ref{Lax2}) +at $(q, \nu)$, $\phi = c_+ \phi^+ + c_- \phi^-$, and $\phi^\pm$ are +Bloch eigenfunctions. Next, we consider the perturbed NLS, +\[ +iq_t =q_{xx} +2|q|^2q +i\e f(q,\bq)\ , +\] +where $\e$ is the perturbation parameter. +The {\em Melnikov integral} can be defined using the constant of motion +$F_j $, where $\lambda^c_j = \nu$ \cite{LM94}: +\begin{equation} +M_j \equiv \int_{-\infty}^{+\infty} \int_{0}^{2\pi} +\{ \frac {\dl F_j}{\dl q} f + \frac {\dl F_j}{\dl \bq} \bar{f} \}_{q=Q} +\ dx dt \ , +\label{6.1} +\end{equation} +where the integrand is evaluated along the unperturbed homoclinic +orbit $q = Q$, and the Melnikov vector $\frac{\dl F_j}{\dl \vq}$ +has been given in the last section, which can be expressed rather +explicitly using the B\"acklund-Darboux transformation \cite{LM94} . +We begin with the expression (\ref{3.8}), +\begin{eqnarray} +\frac{\delta F_j}{\delta \vec{q}} = i \frac{\sqrt{\Delta^2 - +4}}{W[\Phi^{+},\Phi^{-}]} \left( \begin{array}{c} + \Phi_2^{+} \Phi_2^{-} \\ + \Phi_1^{+} \Phi_1^{-} +\end{array} +\right) +\label{6.2} +\end{eqnarray} +where $\Phi^{\pm}$ are Bloch eigenfunctions at $(Q, \nu)$, which +can be obtained from B\"acklund-Darboux formula (\ref{4.5}): +\begin{eqnarray*} +\Phi^{\pm}(x,t; \nu) \ \equiv \ G(\nu; \nu ; +\phi) \ \phi^{\pm}(x,t; \nu)\ , +\end{eqnarray*} +with the transformation matrix $G$ given by +\begin{eqnarray*} + G = G (\lambda ; \nu ; \phi)= N +\left( \begin{array}{cl} + \lambda-\nu & \quad 0\\ +0 & \lambda - \bar{\nu} + \end{array}\right) +N^{-1}, +\end{eqnarray*} +\begin{eqnarray*} +N \equiv +\bigg [ \begin{array}{cl} + \phi_1 & -\bar{\phi}_2\\ +\phi_2 &\ \ \bar{\phi}_1 + \end{array} \bigg ]\ . +\end{eqnarray*} +These B\"acklund-Darboux formulas are rather easy to manipulate to obtain +explicit information. For example, the transformation matrix +$G(\lambda, \nu)$ has a simple limit as $\lambda \rightarrow \nu$: +\begin{eqnarray} +\lim_{\lambda \rightarrow \nu} G(\lambda, \nu) = \frac{\nu - +\bar{\nu}}{|\phi|^2} \left( \begin{array}{cc} + \phi_2 \bar{\phi}_2 & -\phi_1 \bar{\phi}_2 \\ + -\phi_2 \bar{\phi}_1 & \phi_1 \bar{\phi}_1 +\end{array} +\right) +\label{6.3} +\end{eqnarray} +where $|\phi|^2$ is defined by +\[ +|\vec{\phi}|^2 \equiv \phi_1 \bar{\phi}_1 +\phi_2 \bar{\phi}_2. +\] +With formula (\ref{6.3}) one quickly calculates +\[ +\Phi^\pm = \pm c_{\mp} \ \ W[\phi^{+},\phi^{-}] \ \ +\frac{\nu - {\bar{\nu}}}{| \phi |^2} +\left( \begin{array}{c} + {\bar {\phi}}_2 \\ + -{\bar {\phi}}_1 +\end{array} +\right) +\] +from which one sees that $\Phi^{+}$ and $\Phi^{-}$ +are linearly dependent at $(Q,\nu)$, +\[ +\Phi^{+} = -\frac{c_-}{c_+} \ \Phi^{-}. +\] +\begin{remark} +For $Q$ on the figure-eight, the two Bloch eigenfunctions +$\Phi^\pm$ are linearly dependent. Thus, the geometric +multiplicity of $\nu$ is only one, even though its +algebraic multiplicity is two or higher. +\end{remark} +Using L'Hospital's rule, one gets +\begin{eqnarray} +\frac{\sqrt{\Delta^2 - 4}}{W[\Phi^{+},\Phi^{-}]} = \frac{\sqrt{\Delta(\nu) +\Delta''(\nu)}}{(\nu - \bar{\nu}) \ \ W[\phi^{+},\phi^{-}]}\ . +\label{6.4} +\end{eqnarray} +With formulas (\ref{6.2}, \ref{6.3}, \ref{6.4}), one obtains the explicit +representation of the $\mbox{grad}\ F_j$ \cite{LM94}: +\begin{eqnarray} +\frac{\delta F_j}{\delta \vec{q}} \ = \ C_\nu \ \frac{c_+ c_- +W[\psi^{(+)},\psi^{(-)}] }{|\phi|^4} +\left( \begin{array}{c} + {\bar{\phi}}^2_2 \\ \\ + -{\bar{\phi}}^2_1 +\end{array} +\right)\ , +\label{6.5} +\end{eqnarray} +where the constant $C_\nu$ is given by +\[ +C_\nu \equiv i (\nu - \bar{\nu}) \ \ \sqrt{\Delta(\nu) \Delta''(\nu)}\ . +\] +With these ingredients, one obtains the following beautiful +representation of +the Melnikov function associated to the general complex double point +$\nu$ \cite{LM94}: +\begin{equation} +M_j = C_\nu \ c_+ c_- \int_{-\infty}^{+\infty} +\int_0^{2\pi} \ W[\phi^{+}, \phi^{-}] \ +\left[\frac{(\bar{\phi}_2^2) f(Q,\bar{Q}) + (\bar{\phi}_1^2) +\overline{f(Q,\bar{Q})}}{|\phi|^4} \right] \ \ dx dt. +\label{6.6} +\end{equation} +In the case of several complex double points, each associated with +an instability, one can iterate the B\"acklund-Darboux tranformations and use +those functionals $F_j$ which are associated with each +complex double point to obtain representations {\em Melnikov Vectors}. +In general, the relation between $\frac{\delta F_j}{\delta \vec{q}}$ and +double points can be summarized in the following lemma \cite{LM94}, +\begin{lemma} +Except for the trivial case $q = 0$, +\begin{eqnarray*} +(a). & & \ \ \frac{\delta F_j}{\delta q} = 0 \Leftrightarrow +\frac{\delta F_j}{\delta \bar{q}} = 0 +\Leftrightarrow M(2\pi,\lambda^c_j; \ \vec{q}_*) = \pm I. \\ +(b). & & \ \ \frac{\delta F_j}{\delta \vq}\mid_{\vq_*} = 0 \Rightarrow +\Delta'(\lambda_j^c(\vec{q}_*); \ \vec{q}_*) = 0, +\Rightarrow |F_j(\vec{q}_*)| = 2, \\ +& &\ \ \Rightarrow \lambda_j^c(\vec{q}_*) \ \ \mbox{is a multiple point}, +\end{eqnarray*} +where $I$ is the $2\times 2$ identity matrix. +\label{lemma 7} +\end{lemma} +The B\"acklund-Darboux transformation theorem indicates that +the figure-eight structure is attached to a complex double point. +The above lemma shows that at the origin of the figure-eight, +the gradient of $F_j$ vanishes. Together they indicate that the +the gradient of $F_j$ along the figure-eight is a perfect Melnikov +vector. + +{\bf Example:} When $\frac{1}{2} < c < 1$ in (\ref{4.7}), and choosing +$\vth = \pi $ in (\ref{4.13}), one can get the Melnikov vector field along +the homoclinic orbit (\ref{4.13}), +\begin{eqnarray} +& & \frac{\dl F_1}{\dl q} = 2 \pi\ \sin^2 p +\hbox{ sech} \tau \;\frac{[(- \sin p + i \cos p +\tanh \tau) \cos x+\hbox{ sech} \tau ]} {[1 - \sin p \hbox{ sech } +\tau \cos x ]^2} +\, c\; e^{i \theta}, \label{mvnls1} \\ +& & \frac{\dl F_1}{\dl \bq} = +\overline{\frac{\dl F_1}{\dl q}} \ . \nonumber +\end{eqnarray} + +\section{Discrete Cubic Nonlinear Schr\"odinger Equation} + +The discrete cubic nonlinear Schr\"odinger equation (\ref{DNLS}) +can be written in the Hamiltonian form \cite{AL76} \cite{Li92}: +\begin{equation} +i \dot{q}_n = \rho_n \pa H/ \pa \bq_n \ , \label{HDNLS} +\end{equation} +\\ +where $\rho_n = 1 +h^2|q_n|^2$, and +\[ +H={1 \over h^2}\sum^{N-1}_{n=0} +\bigg \{\bq_n(q_{n+1}+q_{n-1})-{2 \over h^2}(1+\om^2 h^2)\ln \rho_n +\bigg \} \ . +\] +$\sum^{N-1}_{n=0}\bigg \{\bq_n(q_{n+1}+q_{n-1})\bigg \}$ +itself is also a constant of motion. This invariant, together with $H$, +implies that $\sum^{N-1}_{n=0}\ln \rho_n$ is a constant of motion too. +Therefore, +\begin{equation} +D^2\equiv \prod^{N-1}_{n=0}\rho_n +\label{constD} +\end{equation} +is a constant of motion. We continue from Section \ref{DCNSE}. Using $D$, +one can define a normalized Floquet +discriminant $\tilde{\Dl}$ as +\[ +\tilde{\Dl} = \Dl / D\ . +\] +\begin{definition} +The sequence of invariants $\tF_j$ is defined as: +\begin{equation} +\tF_j(\vq) = \tilde{\Dl}(z^{(c)}_j(\vq);\vq)\ , +\label{coF} +\end{equation} +where $\vq = (q, -\bq)$, $q=(q_0,q_1,\cdot \cdot \cdot,q_{N-1})$. +\end{definition} +These invariants $\tF_j$'s are perfect candidate for building Melnikov +functions. The Melnikov vectors are given by the gradients of these +invariants. +\begin{lemma} +Let $z^{(c)}_j(\vq)$ be a simple critical point; then +\begin{equation} +{\dl \tF_j \over \dl \vq_n}(\vq) = {\dl \tilde{\Dl} \over \dl \vq_n} +(z^{(c)}_j(\vq);\vq)\ . +\label{derf} +\end{equation} +\begin{equation} +{\dl \tilde{\Dl} \over \dl \vq_n}(z;\vq) = {i h (\z -\z^{-1}) \over +2 W_{n+1}} \left (\begin{array}{c} \psi^{(+,2)}_{n+1}\psi^{(-,2)}_{n}+ +\psi^{(+,2)}_{n}\psi^{(-,2)}_{n+1} \cr \cr \psi^{(+,1)}_{n+1}\psi^{(-,1)}_{n}+ +\psi^{(+,1)}_{n}\psi^{(-,1)}_{n+1} \cr \end{array} \right )\ , +\label{fidr} +\end{equation} +where $\psi_n^\pm = (\psi_n^{(\pm,1)}, \psi_n^{(\pm,2)})^T$ are two +Bloch functions of the Lax pair (\ref{Lax1},\ref{Lax2}), such that +\[ +\psi^\pm_n = D^{n/N} \z^{\pm n/N} \tilde{\psi}^\pm_n\ , +\] +where $\tilde{\psi}^\pm_n$ are periodic in $n$ with period $N$, +$W_n = \ \mbox{det}\ ( \psi^+_n,\psi^-_n )$. +\end{lemma} +For $z^{(c)}_j = z^{(d)}$, the Melnikov vector field located on +the heteroclinic orbit (\ref{BD1}) is given by +\begin{equation} +{\dl \tF_j \over \dl \vQ_n} = K {W_n \over E_n A_{n+1}} +\left ( \begin{array}{c} [z^{(d)}]^{-2} \ \overline{\phi^{(1)}_n} +\ \overline{\phi^{(1)}_{n+1}} \cr \cr [\overline{z^{(d)}}]^{-2} +\ \overline{\phi^{(2)}_n} \ \overline{\phi^{(2)}_{n+1}} \cr \end{array} +\right )\ , \label{fibd} +\end{equation} +where $\vQ_n =(Q_n, -\bar{Q_n})$, +\[ +\phi_n = (\phi^{(1)}_n, \phi^{(2)}_n)^T= c_+ \psi_n^+ + c_- \psi_n^- \ , +\] +\[ +W_n = \left | \begin{array}{lr} \psi_n^+ & \psi_n^- \cr \end{array} +\right |\ , +\] +\[ +E_n = |\phi^{(1)}_n|^2 + |z^{(d)}|^2|\phi^{(2)}_n|^2\ , +\] +\[ +A_n = |\phi^{(2)}_n|^2 + |z^{(d)}|^2|\phi^{(1)}_n|^2\ , +\] +\[ +K=-{ihc_+c_- \over 2}|z^{(d)}|^4 (|z^{(d)}|^4 - 1)[\overline{z^{(d)}}]^{-1} +\sqrt{ \tDl(z^{(d)};\vq) \tDl''(z^{(d)};\vq)}\ , +\] +where +\[ +\tDl''(z^{(d)};\vq) = {\pa^2 \tDl(z^{(d)};\vq) \over \pa z^2}\ . +\] + +{\bf Example:} The Melnikov vector evaluated on the heteroclinic orbit +(\ref{hetorb}) is given by +\begin{equation} +{\dl \tF_1 \over \dl \vQ_n} = \hat{K} \bigg [ \hat{E}_n \hat{A}_{n+1} +\bigg ]^{-1} \ \mbox{sech}\ [2\mu t +2p]\left ( \begin{array}{c} +\hat{X}^{(1)}_n \cr - \hat{X}^{(2)}_n \cr \end{array} \right ) \ , +\label{melv} +\end{equation} +where +\begin{eqnarray*} +\hat{X}^{(1)}_n &=& \bigg [ \cos \be \ \mbox{sech}\ [ 2\mu t +2p] ++\cos [(2n+1)\be +\vth +\varphi] \\ +& & - i \tanh [ 2\mu t +2p] +\sin [(2n+1)\be +\vth +\varphi] \bigg ] e^{i2\th(t)}\ , +\end{eqnarray*} +\begin{eqnarray*} +\hat{X}^{(2)}_n &=& \bigg [ \cos \be \ \mbox{sech}\ [ 2\mu t +2p] ++\cos [(2n+1)\be +\vth -\varphi] \\ +& & - i \tanh [ 2\mu t +2p] +\sin [(2n+1)\be +\vth -\varphi] \bigg ] e^{-i2\th(t)}\ , +\end{eqnarray*} +\[ +\hat{K} = -2Nh^2a(1-z^4) [8 \rho^{3/2}z^2]^{-1}\sqrt{\rho \cos^2 \be -1}\ . +\] +Under the even constraint, the Melnikov vector evaluated on the +heteroclinic orbit (\ref{ehetorb}) is given by +\begin{equation} +{ \dl \tF_1 \over \dl \vQ_n}\bigg |_{\mbox{even}} = \hat{K}^{(e)} +\ \mbox{sech}[2\mu t + 2p] [\Pi_n]^{-1}\left ( \begin{array}{c} +\hat{X}^{(1,e)}_n \cr - \hat{X}^{(2,e)}_n \cr \end{array} \right ) \ , +\label{emelv} +\end{equation} +where +\[ +\hat{K}^{(e)}= -2N (1-z^4) [8a\rho^{3/2} z^2]^{-1} +\sqrt{\rho \cos^2\be - 1}\ , +\] +\begin{eqnarray*} +\Pi_n &=& \bigg [ \cos \be \pm \cos \varphi \ \mbox{sech}[2\mu t +2p] +\cos[2(n-1)\be]\bigg ] \times \\ +& &\bigg [ \cos \be \pm \cos \varphi \ \mbox{sech}[2\mu t +2p] +\cos[2(n+1)\be]\bigg ]\ , +\end{eqnarray*} +\begin{eqnarray*} +\hat{X}^{(1,e)}_n &=& \bigg [ \cos \be \ \mbox{sech}\ [ 2\mu t +2p] +\pm (\cos \varphi \\ +& & -i \sin \varphi \tanh [ 2\mu t +2p]) \cos [2n\be]\bigg ] e^{i2\th(t)}\ , +\end{eqnarray*} +\begin{eqnarray*} +\hat{X}^{(2,e)}_n &=& \bigg [ \cos \be \ \mbox{sech}\ [ 2\mu t +2p] +\pm (\cos \varphi \\ +& & +i \sin \varphi \tanh [ 2\mu t +2p]) \cos [2n\be]\bigg ] e^{-i2\th(t)}\ . +\end{eqnarray*} + +\section{Davey-Stewartson II Equations} + +The DSII (\ref{fDS2}) can be written in the Hamiltonian form, +\begin{equation} + \left\{ + \begin{array}{ccc} + i q_t &=& \delta H / \delta \overline{q} \ ,\cr + i \overline{q}_t &=& - \delta H / \delta q \ , \cr + \end{array} \right. +\label{fhDS2} +\end{equation} +where +\begin{displaymath} + H= \int^{2 \pi}_0 \int^{2 \pi}_0 + [\left| q_y \right|^2 - \left| q_x \right|^2 + + \frac{1}{2} (r_2-r_1) \left| q \right|^2] \, dx \, dy \, . +\end{displaymath} +We have the lemma \cite{Li00a}. +\begin{lemma} +The inner product of the vector +\begin{equation} + \U= \left( + \begin{array}{c} + \psi_2 \hat{\psi}_2 \cr + \psi_1 \hat{\psi}_1 + \end{array}\right)^- +S \left( + \begin{array}{c} + \psi_2 \hat{\psi}_2 \\ + \psi_1 \hat{\psi}_1 + \end{array}\right) \, , +\nonumber +\end{equation} +where $\psi = (\psi_1 , \psi_2)$ is an eigenfunction solving the +Lax pair (\ref{LP1}, \ref{LP2}), and $\hat{\psi} = (\hat{\psi}_1 , +\hat{\psi}_2)$ is an eigenfunction solving the corresponding +congruent Lax pair (\ref{CLP1}, \ref{CLP2}), +and $S= \displaystyle{\left( + \begin{array}{ccc} + 0 & 1 \\ 1 & 0 + \end{array} +\right)}$, with the vector field $J \na H$ given by the +right hand side of (\ref{fhDS2}) vanishes, +\begin{displaymath} + \langle \U\, , \, J \na H \rangle =0 \, , +\end{displaymath} +where $J= \displaystyle{\left( + \begin{array}{ccc} + 0 & 1 \\ -1 & 0 + \end{array} +\right)}$. +\label{melem2} +\end{lemma} +If we only consider even functions, i.e., $q$ and $u=r_2 -r_1$ +are even functions in both $x$ and $y$, then we can split $\U$ +into its even and odd parts, +\begin{displaymath} + \U=\U^{(e,x)}_{(e,y)} + \U^{(e,x)}_{(o,y)} + \U^{(o,x)}_{(e,y)} + + \U^{(o,x)}_{(o,y)} \, , +\end{displaymath} +where +\begin{eqnarray} + \U^{(e,x)}_{(e,y)} &=& \frac{1}{4} \bigg [ + \U(x,y) + \U(-x,y) + \U(x,-y) + \U(-x,-y) \bigg ] \, , \nonumber\\ + \U^{(e,x)}_{(o,y)} &=& \frac{1}{4} \bigg [ + \U(x,y) + \U(-x,y) - \U(x,-y) - \U(-x,-y) \bigg ] \, , \nonumber\\ + \U^{(o,x)}_{(e,y)} &=& \frac{1}{4} \bigg [ + \U(x,y) - \U(-x,y) + \U(x,-y) - \U(-x,-y) \bigg ] \, , \nonumber\\ + \U^{(o,x)}_{(o,y)} &=& \frac{1}{4} \bigg [ + \U(x,y) - \U(-x,y) - \U(x,-y) + \U(-x,-y) \bigg ] \, . \nonumber +\end{eqnarray} +Then we have the lemma \cite{Li00a}. +\begin{lemma} + When $q$ and $u=r_2-r_1$ are even functions in both $x$ and + $y$, we have +\begin{displaymath} + \langle \U^{(e,x)}_{(e,y)} \, , \, J \na H \rangle =0 \, . +\end{displaymath} +\label{melem3} +\end{lemma} + +\subsection{Melnikov Integrals} + +Consider the perturbed DSII equation, +\begin{equation} +\left\{ + \begin{array}{l} + i \partial_t q = [\partial^2_x - \partial^2_y] q + + [2 ( \left| q \right|^2 - \om^2) + u_y] q + + \e i f \, , \\[1ex] +[\partial^2_x + \partial^2_y] u + = -4 \partial_y \left| q \right|^2 \, , + \end{array} \right. +\label{PDS2} +\end{equation} +where $q$ and $u$ are respectively complex-valued and +real-valued functions of three variables $(t,x,y)$, and $G += (f, \overline{f})$ are the perturbation terms which can +depend on $q$ and $\overline{q}$ and their derivatives and $t$, $x$ +and $y$. The Melnikov integral is given by \cite{Li00a}, +\begin{eqnarray} + M &=& \int^{\infty}_{- \infty} \langle \U , + G \rangle \, dt \nonumber \\[1ex] +&=& 2 \int^{\infty}_{- \infty} \int^{2 \pi}_0 \int^{2 \pi}_0 + R_e \left\{(\psi_2 \hat{\psi}_2) f + (\psi_1 \hat{\psi}_1) + \overline{f} \right\} \, dx \, dy \, dt \, , \label{mlf2} +\end{eqnarray} +where the integrand is evaluated on an unperturbed homoclinic +orbit in certain center-unstable ($=$ center-stable) manifold, +and such orbit can be obtained through the B\"acklund-Darboux +transformations given in Theorem \ref{DSTH}. A concrete example +is given in section \ref{dsex}. When we only +consider even functions, i.e., $q$ and $u$ are even functions in +both $x$ and $y$, the corresponding Melnikov function is given +by \cite{Li00a}, +\begin{eqnarray} +M^{(e)} &=& \int^{\infty}_{- \infty} + \langle \U^{(e,x)}_{(e,y)} , \vec{G} \rangle \, dt \nonumber\\[1ex] +&=& \int^{\infty}_{- \infty} + \langle \U , \vec{G} \rangle \, dt \, , \label{mlf3} +\end{eqnarray} +which is the same as expression (\ref{mlf2}). + +\subsection{An Example \label{mids}} + +We continue from the example in section \ref{dsex}. +We generate the following eigenfunctions +corresponding to the potential $Q$ given in (\ref{newsl}) through +the iterated \BD, +\begin{eqnarray} +\Psi^{\pm} &=& \Gamma^{(I)} \Gamma \psi^{\pm} \, , \ \ \ \ \mbox{ at } +\ \lambda = \lambda_0 = \sqrt{\eta^2- \frac{1}{4}\k_1^2} \, , +\label{nwef1} \\[1ex] +\Phi_{\pm} &=& \Gamma^{(I)} \Gamma \phi_{\pm} \, , \ \ \ \ +\mbox{ at } \lambda =0 \, , \label{nwef2} +\end{eqnarray} +where +\[ + \Gamma = \left[ + \begin{array}{cc} +\Lambda +a & b\\ \\ +\overline{b} & \Lambda -a + \end{array} \right] \, , \ \ \ \Gamma^{(I)} = \left[ + \begin{array}{cc} +\Lambda + a^{(I)} & b^{(I)} \\ \\ +\overline{b^{(I)}} & \Lambda - \overline{a^{(I)}} + \end{array} \right] \, , +\] +where $\Lambda = \alpha \partial_y - \lambda$ for general $\lambda$. +\begin{lemma}[see \cite{Li00a}] +The eigenfunctions $\Psi^{\pm}$ and $\Phi_{\pm}$ defined in +(\ref{nwef1}) and (\ref{nwef2}) have the representations, +\begin{eqnarray} + \Psi^{\pm} &=& \frac{\pm 2 \lambda_0 W(\psi^+, + \psi^-)}{\Delta}\left[ + \begin{array}{c} + (- \lambda_0 + a^{(I)}) \overline{\psi}_2 + -b^{(I)} \overline{\psi}_1 \\ \\ +\overline{b^{(I)}} \overline{\psi}_2 + + (\lambda_0 + \overline{a^{(I)}}) \overline{\psi}_1 + \end{array}\right] \, ,\label{rnef1} \\ +\Phi_{\pm} &=& \frac{\mp i \alpha \k_2}{\Delta^{(I)}} +\left[ + \begin{array}{c} + \Xi_1 \\[1ex] \Xi_2 + \end{array} \right] \, , \label{rnef2} +\end{eqnarray} +where +\begin{eqnarray*} +& & W(\psi^+,\psi^-) = \left| + \begin{array}{cc} +\psi^+_1 & \psi^-_1 \\ \\ +\psi^+_2 & \psi^-_2 + \end{array} \right| = + -i \k_1 c^+_0 c^-_0 q_c \exp \left\{ i2r_2t \right\} \, , \\ +& & \Delta = - \Bigl[ \left| \psi_1 \right|^2 + + \left| \psi_2 \right|^2 \Bigr] \, , \\ +& & \psi = \psi^+ + \psi^- \, , \\ +& & \Delta^{(I)} = - \Bigl[ \left| \varphi_1 \right|^2 + + \left| \varphi_2 \right|^2 \Bigr]\, , \\ +& & \varphi = \varphi^+ + \varphi^- \, , \\ +& & \varphi^{\pm} = \Gamma \phi_{\pm} \hbox{ at } + \lambda =0 \, , \\ +& & \Xi_1 = \overline{\varphi}_1 (\varphi^+_1 \varphi^-_1) ++ \overline{\varphi^+_2} (\varphi^+_1 \varphi^-_2) + +\overline{\varphi^-_2} (\varphi^-_1 \varphi^+_2) \, , \\ +& & \Xi_2 = \overline{\varphi}_2 (\varphi^+_2 \varphi^-_2) ++ \overline{\varphi^+_1} (\varphi^-_1 \varphi^+_2) + +\overline{\varphi^-_1} (\varphi^+_1 \varphi^-_2) \, . +\end{eqnarray*} +If we take $r_2$ to be real (in the Melnikov vectors, $r_2$ +appears in the form $r_2-r_1=2(\left|q_c \right|^2 - \omega^2))$, then +\begin{equation} + \Psi^{\pm} \to 0 \, , \ \ \ \ \Phi_{\pm} \to 0 \, , \, + \hbox{ as } t \to \pm \infty \, . +\label{anef} +\end{equation} +\end{lemma} +Next we generate eigenfunctions solving the corresponding +congruent Lax pair (\ref{CLP1}, \ref{CLP2}) with the potential $Q$, through +the iterated \BD ~and the symmetry transformation (\ref{IT}) \cite{Li00a}. +\begin{lemma} +Under the replacements +\begin{displaymath} + \alpha \longrightarrow - \alpha \, \ \ (\vth_2 \longrightarrow +\pi - \vth_2 ), \ \ \ + \hvth \longrightarrow \hvth +\pi -2 \vth_2 \, , \ \ \ + \hat{\rho} \longrightarrow - \hat{\rho}\, , +\end{displaymath} +the coefficients in the iterated \BD ~are transformed as follows, +\[ +a^{(I)} \longrightarrow \overline{a^{(I)}} \, , \ \ +b^{(I)} \longrightarrow b^{(I)} \, , \, +\] +\[ +\bigg (c^{(I)} = \overline{b^{(I)}}\bigg ) \longrightarrow +\bigg (c^{(I)} = \overline{b^{(I)}}\bigg ) \, , \, +\bigg (d^{(I)} = - \overline{a^{(I)}}\bigg ) \longrightarrow +\bigg (\overline{d^{(I)}} =-a^{(I)}\bigg ) \, . +\] +\label{rpl} +\end{lemma} +\begin{lemma}[see \cite{Li00a}] +Under the replacements +\begin{eqnarray} +& & \alpha \mapsto - \alpha \, \ \ (\vth_2 \longrightarrow \pi - +\vth_2 ), \ \ r_1 \mapsto -r_2 \, , \nonumber \\ + \label{rptr} \\ +& & r_2 \mapsto -r_1 \, , \ \ +\hvth \longrightarrow \hvth +\pi -2 \vth_2 \, , \ \ \ + \hat{\rho} \longrightarrow - \hat{\rho}\, , \nonumber +\end{eqnarray} +the potentials are transformed as follows, +\begin{eqnarray*} +& & Q \longrightarrow Q \, , \\ +& & (R=\overline{Q}) \longrightarrow (R= \overline{Q}) \, , \\ +& & R_1 \longrightarrow -R_2 \, , \\ +& & R_2 \longrightarrow -R_1 \, . +\end{eqnarray*} +\label{rppl} +\end{lemma} +The eigenfunctions $\Psi^{\pm }$ and $\Phi_{\pm}$ given in +(\ref{rnef1}) and (\ref{rnef2}) depend on the variables in the +replacement (\ref{rptr}): +\begin{eqnarray*} + \Psi^{\pm} &=& \Psi^{\pm} (\alpha , r_1 , r_2 , \hvth , \hrho ) \, , \\ +\Phi_{\pm} &=& \Phi_{\pm} (\alpha , r_1 , r_2 , \hvth , \hrho ) \, . +\end{eqnarray*} +Under replacement (\ref{rptr}), $\Psi^{\pm}$ and $\Phi_{\pm}$ are +transformed into +\begin{eqnarray} + \widehat{\Psi}^{\pm} &=& \Psi^{\pm} + (- \alpha , -r_2 , -r_1 , \hvth + \pi -2 \vth_2 , - \hrho ) + \, , \label{cref1}\\ + \widehat{\Phi}_{\pm} &=& \Phi_{\pm} + (- \alpha , -r_2 , -r_1 , \hvth + \pi -2 \vth_2 , - \hrho ) + \, . \label{cref2} +\end{eqnarray} +\begin{corollary}[see \cite{Li00a}] +$\widehat{\Psi}^{\pm}$ and $ \widehat{\Phi}_{\pm}$ solve the +congruent Lax pair (\ref{CLP1}, \ref{CLP2}) at $(Q, \overline{Q}, +R_1, R_2; \lambda_0)$ and $(Q, \overline{Q}, R_1, R_2; 0)$, respectively. +\label{ccor} +\end{corollary} +Notice that as a function of $\eta$, $\xi^0_1$ has two (plus and +minus) branches. In order to construct Melnikov vectors, we need +to study the effect of the replacement $\xi^0_1 \longrightarrow +-\xi^0_1$. +\begin{lemma}[see \cite{Li00a}] +Under the replacements +\begin{equation} + \xi^0_1 \longrightarrow - \xi^0_1 \, \ \ (\vth_2 \longrightarrow +-\vth_2 ), \ \ + \hvth \longrightarrow \hvth + \pi -2 \vth_2 , \ \ +\hrho \longrightarrow - \hrho , +\label{krptr} +\end{equation} +the coefficients in the iterated \BD ~are invariant, +\[ + a^{(I)} \mapsto a^{(I)} \, , \ \ \ \ b^{(I)} \mapsto b^{(I)} \, , +\] +\[ +( c^{(I)}= \overline{b^{(I)}}) \mapsto (c^{(I)} = \overline{b^{(I)}}) +\, , \ \ \ \ +( d^{(I)}= - \overline{a^{(I)}}) \mapsto (d^{(I)} = -\overline{a^{(I)}}) +\, ; +\] +thus the potentials are also invariant, +\[ +Q \longrightarrow Q \, , \ \ \ \ (R=\overline{Q}) \longrightarrow +(R=\overline{Q}) \, , +\] +\[ +R_1 \longrightarrow R_1 \, , \ \ \ \ R_2 \longrightarrow R_2 \, . +\] +\label{lektr} +\end{lemma} +The eigenfunction $\Phi_{\pm}$ given in (\ref{rnef2}) depends on +the variables in the replacement (\ref{krptr}): +\begin{displaymath} + \Phi_{\pm} = \Phi_{\pm} (\xi^0_1 , \hvth , \hrho ) \, . +\end{displaymath} +Under the replacement (\ref{krptr}), $\Phi_{\pm}$ is transformed +into +\begin{equation} + \widetilde{\Phi}_{\pm} = \Phi_{\pm} + (-\xi^0_1 , \hvth + \pi -2 \vth_2, -\hrho )\ . +\label{kfr} +\end{equation} +\begin{corollary} +$\widetilde{\Phi}_{\pm}$ solves the Lax pair (\ref{LP1},\ref{LP2}) at +$(Q , \overline{Q} , R_1 , R_2 \, ; \, 0)$. +\label{kcor} +\end{corollary} + +In the construction of the Melnikov vectors, we need to replace +$\Phi_{\pm}$ by $ \widetilde{\Phi}_{\pm}$ to guarantee the +periodicity in $x$ of period $L_1 =\frac{2\pi}{\k_1}$. + +The Melnikov vectors for the Davey-Stewartson~II equations are +given by, +\begin{eqnarray} + \U^{\pm} &=& \left( \begin{array}{c} +\Psi^{\pm}_2 \widehat{\Psi}^{\pm}_2 \\[1ex] +\Psi^{\pm}_1 \widehat{\Psi}^{\pm}_1 + \end{array}\right)^- +S \left( +\begin{array}{c} + \Psi^{\pm}_2 \widehat{\Psi}^{\pm}_2 \\[1ex] +\Psi^{\pm}_1 \widehat{\Psi}^{\pm}_1 + \end{array} \right) \, , \label{mv1}\\[2ex] + \U_{\pm} &=& \left( +\begin{array}{c} + \widetilde{\Phi}_{\pm}^{(2)} \widehat{\Phi}_{\pm}^{(2)} \\[1ex] +\widetilde{\Phi}_{\pm}^{(1)} \widehat{\Phi}_{\pm}^{(1)} +\end{array}\right)^- +S \left( +\begin{array}{c} + \widetilde{\Phi}_{\pm}^{(2)} \widehat{\Phi}_{\pm}^{(2)} \\[1ex] +\widetilde{\Phi}_{\pm}^{(1)} \widehat{\Phi}_{\pm}^{(1)} + \end{array} \right) \, , \label{mv2} +\end{eqnarray} +where $S = \left ( \begin{array}{lr} 0 & 1 \\ 1 & 0 \end{array} \right )$. +The corresponding Melnikov functions (\ref{mlf2}) are given by, +\begin{eqnarray} +M^{\pm} &=& \int^{\infty}_{- \infty} \langle \U^{\pm} \, , \, + \vec{G} \rangle \, dt \nonumber \\ +&=& 2 \int^{\infty}_{- \infty} \int^{2\pi}_0 \int^{2\pi}_0 + R_e \bigg \{[ \Psi^{\pm}_2 \widehat{\Psi}^{\pm}_2 ] + f(Q, \overline{Q}) \nonumber \\ +& & + [ \Psi^{\pm}_1 + \widehat{\Psi}^{\pm}_1] + \overline{f}(Q, \overline{Q}) \bigg \} \, dx \, dy \, dt \, + , \label{mf1}\\ +M_{\pm} &=& \int^{\infty}_{- \infty} \langle \U_{\pm} \, , \, + \vec{G} \rangle \, dt \nonumber \\ +&=& 2 \int^{\infty}_{- \infty} \int^{2\pi}_0 \int^{2\pi}_0 +R_e \bigg \{ [ \widetilde{\Phi}^{(2)}_{\pm} + \widehat{\Phi}^{(2)}_{\pm}] f(Q, \overline{Q}) \nonumber \\ +& & + [ \widetilde{\Phi}^{(1)}_{\pm} + \widehat{\Phi}^{(1)}_{\pm}] \overline{f}(Q, \overline{Q}) + \bigg \}\, dx \, dy \, dt \, , \label{mf2} +\end{eqnarray} +where $Q$ is given in (\ref{newsl}), $\Psi^{\pm}$ is given in +(\ref{rnef1}), $\widetilde{\Phi}_{\pm}$ is given in (\ref{rnef2}) +and (\ref{kfr}), $\widehat{\Psi}^{\pm}$ is given in (\ref{rnef1}) +and (\ref{cref1}), and $\widehat{\Phi}_{\pm}$ is given in +(\ref{rnef2}) and (\ref{cref2}). As given in (\ref{mlf3}), the +above formulas also apply when we consider even function $Q$ in +both $x$ and $y$. + + + + + + + +\clearpage{} +\clearpage{}\chapter{Invariant Manifolds} + +Invariant manifolds have attracted intensive studies which led +to two main approaches: Hadamard's method \cite{Had01} \cite{Fen71} +and Perron's method \cite{Per30} \cite{CLL91}. For example, for a partial +differential equation of the form +\[ +\pa_t u = L u + N(u)\ , +\] +where $L$ is a linear operator and $N(u)$ is the nonlinear term, if the +following two ingredients +\begin{enumerate} +\item the gaps separating the unstable, center, and stable spectra +of $L$ are large enough, +\item the nonlinear term $N(u)$ is Lipschitzian in $u$ with small +Lipschitz constant, +\end{enumerate} +are available, then establishing the existence of unstable, center, and +stable manifolds is rather straightforward. Building invariant manifolds +when any of the above conditions fails, is a very challenging and interesting +problem \cite{Li01b}. + +There has been a vast literature on invariant manifolds. A good starting +point of reading can be from the references \cite{Kel67} \cite{Fen71} +\cite{CLL91}. Depending upon the emphasis on the specific problem, +one may establish invariant manifolds for a specific flow, or investigate +the persistence of existing invariant manifolds under perturbations to +the flow. + +In specific applications, most of the problems deal with manifolds with +boundaries. In this context, the relevant concepts are overflowing invariant, +inflowing invariant, and locally invariant submanifolds, defined in the +Chapter on General Setup and Concepts. + +\section{Nonlinear Schr\"odinger Equation Under Regular Perturbations +\label{rpsec}} + +Persistence of invariant manifolds depends upon the nature of the +perturbation. Under the so-called regular perturbations, i.e., the +perturbed evolution operator is $C^1$ close to the unperturbed one, +for any fixed time; invariant manifolds persist ``nicely''. Under +other singular perturbations, this may not be the case. + +Consider the regularly perturbed nonlinear Schr\"odinger (NLS) equation +\cite{LMSW96} \cite{LW97b}, +\begin{equation} +iq_t = q_{xx} +2 [ |q|^2 - \om^2] q +i \e [\hat{\pa}^2_xq - \al q +\be ] \ , +\label{rpnls} +\end{equation} +where $q = q(t,x)$ is a complex-valued function of the two real +variables $t$ and $x$, $t$ represents time, and $x$ represents +space. $q(t,x)$ is subject to periodic boundary condition of period +$2 \pi$, and even constraint, i.e., +\[ +q(t,x + 2 \pi) = q(t,x)\ , \ \ q(t,-x) = q(t,x)\ . +\] +$\om$ is a positive constant, $\al >0$ and $\be >0$ are constants, +$\hat{\pa}^2_x$ is a bounded Fourier multiplier, +\[ +\hat{\pa}^2_x q = -\sum_{k=1}^{N}k^2 \xi_k \tq_k \cos kx\ , +\] +$\xi_k = 1$ when $k \leq N$, $\xi_k = 8k^{-2}$ when $k>N$, for some +fixed large $N$, and $\e > 0$ is the perturbation parameter. + +One can prove the following theorems \cite{LMSW96} \cite{LW97b}. +\begin{theorem}[Persistence Theorem] +For any integers $k$ and $n$ ($1 \leq k,n <\infty$), there exist a +positive constant $\e_0$ and a neighborhood $\U$ of the circle +$S_\om = \{ q \ | \ \pa_x q = 0, \ |q| = \om, \ 1/2 < \om <1 \}$ in +the Sobolev space $H^k$, such +that inside $\U$, for any $\e \in (-\e_0, \e_0)$, there exist $C^n$ +locally invariant submanifolds $W^{cu}_\e$ and $W^{cs}_\e$ of +codimension 1, and $W^c_\e$ ($=W^{cu}_\e \cap W^{cs}_\e$) +of codimension 2 under the evolution operator $F^t_\e$ given by +(\ref{rpnls}). When $\e =0$, $W^{cu}_0$, $W^{cs}_0$, and $W^c_0$ are +tangent to the center-unstable, center-stable, and center subspaces +of the circle of fixed points $S_\om$, respectively. +$W^{cu}_\e$, $W^{cs}_\e$, and $W^c_\e$ +are $C^n$ smooth in $\e$ for $\e \in (-\e_0,\e_0)$. +\label{Persthm} +\end{theorem} +$W^{cu}_\e$, $W^{cs}_\e$, and $W^c_\e$ are called persistent +center-unstable, center-stable, and center submanifolds near $S_\om$ +under the evolution operator $F^t_\e$ given by (\ref{rpnls}). +\begin{theorem}[Fiber Theorem] +Inside the persistent center-unstable submanifold $W^{cu}_\e$ near $S_\om$, +there exists a family of $1$-dimensional $C^n$ smooth submanifolds (curves) +$\{ \F^{(u,\e)}(q): q \in W^c_\e \}$, called unstable fibers: +\begin{itemize} +\item $W^{cu}_\e$ can be represented as a union of these fibers, +\[ +W^{cu}_\e =\bigcup _{q \in W^c_\e} \F^{(u,\e)}(q). +\] +\item $\F^{(u,\e)}(q)$ depends $C^{n-1}$ smoothly on both $\e$ and $q$ +for $\e \in (-\e_0,\e_0)$ and $q \in W^c_\e$, in the sense that $\W$ defined +by +\[ +\W= \bigg \{ (q_1,q,\e)\ \bigg | \ q_1 \in \F^{(u,\e)}(q), +\ q \in W^c_\e,\ \e \in (-\e_0,\e_0) \bigg \} +\] +is a $C^{n-1}$ smooth submanifold of $H^k \times H^k \times (-\e_0,\e_0)$. +\item Each fiber $\F^{(u,\e)}(q)$ intersects $W^c_\e$ transversally at +$q$, two fibers $\F^{(u,\e)}(q_1)$ and $\F^{(u,\e)}(q_2)$ are either +disjoint or identical. +\item The family of unstable fibers $\{ \F^{(u,\e)}(q): q \in W^c_\e \}$ is +negatively invariant, in the sense that the family of fibers commutes +with the evolution operator $F^t_\e$ in the following way: +\[ +F^t_\e(\F^{(u,\e)}(q)) \subset \F^{(u,\e)}(F^t_\e(q)) +\] +for all $q \in W^c_\e$ and all $t \leq 0$ such that +$\bigcup_{\tau \in [t,0]}F^\tau_\e(q) \subset W^c_\e$. +\item There are positive constants $\k$ +and $C$ which are independent of $\e$ such that if $q \in W^c_\e$ and +$q_1 \in \F^{(u,\e)}(q)$, then +\[ +\bigg \| F^t_\e(q_1) - F^t_\e(q) \bigg \| \leq C e^{\k t} +\bigg \| q_1 -q \bigg \|, +\] +for all $t \leq 0$ such that $\bigcup_{\tau \in [t,0]}F^\tau_\e(q) +\subset W^c_\e$, where $\| \ \|$ denotes $H^k$ norm of periodic +functions of period $2\pi$. +\item For any $q, p \in W^c_\e$, $q \neq p$, any $q_1 \in +\F^{(u,\e)}(q)$ and any $p_1 \in \F^{(u,\e)}(p)$; if +\[ +F^t_\e(q), F^t_\e(p) \in W^c_\e,\ \ \forall t \in (-\infty, 0], +\] +and +\[ +\| F^t_\e(p_1) -F^t_\e(q)\| \ra 0,\ \ \mbox{as}\ t\ra -\infty; +\] +then +\[ +\bigg \{ {\| F^t_\e(q_1) -F^t_\e(q)\| \over \| F^t_\e(p_1) -F^t_\e(q)\|} +\bigg \} \bigg / e^{ {1\over 2} \k t} \ra 0, \ \ \mbox{as}\ t\ra -\infty. +\] +\end{itemize} +Similarly for $W^{cs}_\e$. +\label{fiberthm} +\end{theorem} +When $\e =0$, certain low-dimensional invariant submanifolds of +the invariant manifolds, have explicit representations through +Darboux transformations. Specifically, the periodic orbit (\ref{4.7}) +where $1/2 < c < 1$, has two-dimensional stable and unstable manifolds +given by (\ref{4.13}). Unstable and stable fibers with bases along +the periodic orbit also have expressions given by (\ref{4.13}). + +\section{Nonlinear Schr\"odinger Equation Under Singular Perturbations +\label{spsec}} + +Consider the singularly perturbed nonlinear Schr\"odinger equation +\cite{Li01b}, +\begin{equation} +iq_t = q_{xx} +2 [|q|^2 - \om^2] q +i \e [q_{xx} - \al q +\be ] \ , +\label{spnls} +\end{equation} +where $q = q(t,x)$ is a complex-valued function of the two real +variables $t$ and $x$, $t$ represents time, and $x$ represents +space. $q(t,x)$ is subject to periodic boundary condition of period +$2 \pi$, and even constraint, i.e., +\[ +q(t,x + 2 \pi) = q(t,x)\ , \ \ q(t,-x) = q(t,x)\ . +\] +$\om \in (1/2, 1)$ is a positive constant, $\al >0$ and $\be >0$ +are constants, and $\e > 0$ is the perturbation parameter. + +Here the perturbation +term $\e \pa_x^2$ generates the semigroup $e^{\e t \pa_x^2}$, the +regularity of the invariant mainfolds with respect to $\e$ at $\e =0$ +will be closely related to the regularity of the semigroup $e^{\e t \pa_x^2}$ +with respect to $\e$ at $\e =0$. Also the singular perturbation +term $\e \pa_x^2 q$ breaks the spectral gap separating the center +spectrum and the stable spectrum. Therefore, standard invariant manifold +results do not apply. Invariant manifolds do not persist ``nicely''. +Nevertheless, certain persistence results do hold. + +One can prove the following unstable fiber theorem and center-stable +manifold theorem \cite{Li01b}. +\begin{theorem}[Unstable Fiber Theorem] +Let ${\mathcal A}$ be the annulus: +${\mathcal A} = \{ q \ | \ \pa_x q = 0, \ 1/2 < |q| < 1 \}$, for any +$p\in {\mathcal A}$, there is an unstable +fiber ${\mathcal F}^+_p$ which is a curve. ${\mathcal F}^+_p$ has +the following properties: +\begin{enumerate} +\item ${\mathcal F}^+_p$ is a $C^1$ smooth in +$H^k$-norm, $k\geq 1$. +\item ${\mathcal F}^+_p$ is also $C^1$ smooth in $\epsilon$, $\alpha$, +$\beta$, $\omega$, and $p$ in $H^k$-norm, $k \geq 1$, +$\epsilon \in [0,\epsilon_0)$ for some $\epsilon_0>0$. +\item ${\mathcal F}^+_p$ has the exponential decay property: $\forall +p_1\in {\mathcal F}^+_p$, +\[ +\frac{\| F^tp_1-F^tp\|_k}{\| p_1-p\|_k}\leq +Ce^{\mu t},\quad \forall t\leq 0, +\] +where $F^t$ is the evolution operator, $\mu > 0$. +\item $\{ {\mathcal F}^+_p\}_{p\in {\mathcal A}}$ forms an invariant +family of unstable fibers, +\[ +F^t{\mathcal F}^+_p\subset {\mathcal F}^+_{F^tp}\ ,\quad +\forall t\in [-T,0], +\] +and $\forall T>0$ ($T$ can be $+\infty$), such that $F^tp\in +{\mathcal A}$, $\forall t\in [-T,0]$. +\end{enumerate} +\label{UFT} +\end{theorem} +\begin{theorem}[Center-Stable Manifold Theorem] +There exists a $C^1$ smooth codimension 1 +locally invariant center-stable manifold $W^{cs}_k$ in a neighborhood of +the annulus ${\mathcal A}$ (Theorem \ref{UFT}) in $H^k$ for any $k\geq 1$. +\begin{enumerate} +\item At points in the subset $W^{cs}_{k+4}$ of +$W^{cs}_k$, $W^{cs}_k$ is $C^1$ smooth in $\epsilon$ in $H^k$-norm +for $\epsilon \in [0,\epsilon_0)$ and some $\epsilon_0 >0$. +\item $W^{cs}_k$ is $C^1$ smooth in ($\alpha ,\beta ,\omega$). +\end{enumerate} +\label{CSM} +\end{theorem} +\begin{remark}\label{csnr} $C^1$ regularity in $\epsilon$ is +crucial in locating a homoclinic orbit. As can be seen later, one +has detailed information on certain unperturbed (i.e. $\epsilon=0$) +homoclinic orbit, which will be used in tracking candidates for a +perturbed homoclinic orbit. In particular, Melnikov measurement will +be needed. Melnikov measurement measures zeros of +$\mathcal{O}(\epsilon)$ signed distances, thus, the perturbed orbit +needs to be $\mathcal{O}(\epsilon)$ close to the unperturbed orbit in +order to perform Melnikov measurement.\end{remark} + +\section{Proof of the Unstable Fiber Theorem \ref{UFT}} + +Here we give the proof of the unstable fiber theorem \ref{UFT}, +proofs of other fiber theorems in this chapter are easier. + +\subsection{The Setup of Equations} + +First, write $q$ as +\begin{equation} +q(t,x)=[\rho(t)+f(t,x)]e^{i\theta (t)}, +\label{pcd} +\end{equation} +where $f$ has zero spatial mean. We use the notation $\langle \cdot +\rangle$ to denote spatial mean, +\begin{equation}\label{mean} \langle q\rangle +=\frac{1}{2\pi}\int^{2\pi}_0qdx.\end{equation} +Since the $L^2$-norm is an action variable when $\epsilon =0$, it is +more convenient to replace $\rho$ by: +\begin{equation}\label{L2n} I=\langle |q|^2\rangle =\rho^2+\langle +|f|^2\rangle.\end{equation} +The final pick is +\begin{equation}\label{Jpc}J=I-\omega^2.\end{equation} +In terms of the new variables $(J,\theta, f)$, Equation \eqref{spnls} +can be rewritten as +\begin{align}\label{nc1} \dot{J}&=\epsilon \left[-2\alpha +(J+\omega^2)+2\beta \sqrt{J+\omega^2}\cos \theta \right]+\epsilon +\mathcal{R}^J_2,\\ +\label{nc2} \dot{\theta}&=-2J-\epsilon \beta \frac{\sin +\theta}{\sqrt{J+\omega^2}}+\mathcal{R}^\theta_2,\\ +\label{nc3}f_t&=L_\epsilon f+V_\epsilon +f-i\mathcal{N}_2-i\mathcal{N}_3,\end{align} +where +\begin{align}\label{wnc1} L_\epsilon f&=-if_{xx}+\epsilon (-\alpha +f+f_{xx})-i2\omega ^2(f+\bar{f}),\\ +\label{wnc2}V_\epsilon f&=-i2J(f+\bar{f})+i\epsilon \beta f\frac{\sin +\theta}{\sqrt{J+\omega^2}},\\ +\label{wnc3}\mathcal{R}^J_2&=-2\langle |f_x|^2\rangle +2\beta \cos +\theta \left[ \sqrt{J+\omega^2-\langle +|f|^2\rangle}-\sqrt{J+\omega^2}\right],\\ +\begin{split}\label{wnc4}\mathcal{R}^\theta_2&=-\langle +(f+\bar{f})^2\rangle -\frac{1}{\rho}\langle |f|^2(f+\bar{f})\rangle\\ +&\quad -\epsilon +\beta\sin +\theta +\left[ +\frac{1}{\sqrt{J+\omega^2-\langle +|f|^2\rangle}}-\frac{1}{\sqrt{J+\omega^2}}\right],\end{split}\\ +\label{wnc5} \mathcal{N}_2&=2\rho [2(|f|^2-\langle |f|^2\rangle +)+(f^2-\langle f^2\rangle )],\\ +\begin{split}\label{wnc6} \mathcal{N}_3&=-\langle +f^2+\bar{f}^2+6|f|^2\rangle f+2(|f|^2f-\langle |f|^2f\rangle )\\ +&\quad -\frac{1}{\rho} \langle |f|^2(f+\bar{f})\rangle f-2\langle +|f|^2\rangle \bar{f}\\ +&\quad -\epsilon \beta \sin \theta \left[ \frac{1}{\sqrt{J+\omega^2-\langle +|f|^2\rangle}}-\frac{1}{\sqrt{J+\omega^2}}\right]f.\end{split}\end{align} + +\begin{remark} The singular perturbation term ``$\epsilon \pa_x^2q$" +can be seen at two locations, $L_\epsilon$ and $\mathcal{R}^J_2$ +(\ref{wnc1},\ref{wnc3}). The singular perturbation term $\langle +|f_x|^2\rangle $ in $\mathcal{R}^J_2$ does not create any difficulty. +Since $H^1$ is a Banach algebra~\cite{Ada75}, this term is still of +quadratic order, $\langle |f_x|^2\rangle \sim \mathcal{O}(\| +f\|^2_1)$.\end{remark} +\begin{lemma} The nonlinear terms have the orders: +\begin{equation*}\begin{split}&|\mathcal{R}^J_2|\sim \mathcal{O}(\| +f\|^2_s),\quad |\mathcal{R}^\theta_2|\sim \mathcal{O}(\| +f\|^2_s),\\ +& \|\mathcal{N}_2\|_s\sim +\mathcal{O} (\| f\|^2_s),\quad \| \mathcal{N}_3\|_s\sim +\mathcal{O}(\| f\|^3_s),\quad (s\geq +1).\end{split}\end{equation*}\end{lemma} + +Proof: The proof is an easy direct verification. Q.E.D. + +\subsection{The Spectrum of $L_\epsilon$} + +The spectrum of $L_\epsilon$ consists of only point spectrum. The +eigenvalues of $L_\epsilon$ are: +\begin{equation}\label{leev} \mu^\pm_k=-\epsilon (\alpha+k^2)\pm +k\sqrt{4\omega^2-k^2},\quad (k=1,2,\ldots );\end{equation} +where $\omega \in \left( \frac{1}{2},1\right)$, only $\mu^\pm_1$ are +real, and $\mu^\pm_k$ are complex for $k>1$. + +\setlength{\unitlength}{0.8in} +\begin{figure} +\begin{picture}(6.5,2.5) +\put(0,1.25){\line(2,0){2.5}} +\put(3.5,1.25){\line(2,0){2.5}} +\put(1.25,0){\line(0,1){2.5}} +\put(4.75,0){\line(0,1){2.5}} +\put(1.75,.5){$\epsilon =0$} +\put(5.25,.5){$\epsilon >0$} +\put(1.375,2.375){$\mu$} +\put(4.875,2.375){$\mu$} +\put(.5,1.21){$\bullet$} +\put(2,1.21){$\bullet$} +\put(4,1.21){$\bullet$} +\put(5.5,1.21){$\bullet$} +\put(1.21,.25){$\bullet$} +\put(1.21,.5){$\bullet$} +\put(1.21,.75){$\bullet$} +\put(1.21,1){$\bullet$} +\put(1.21,1.5){$\bullet$} +\put(1.21,1.75){$\bullet$} +\put(1.21,2){$\bullet$} +\put(1.21,2.25){$\bullet$} +\put(4,.25){$\bullet$} +\put(4.125,.5){$\bullet$} +\put(4.25,.75){$\bullet$} +\put(4.375,1){$\bullet$} +\put(4,2.25){$\bullet$} +\put(4.125,2){$\bullet$} +\put(4.25,1.75){$\bullet$} +\put(4.375,1.5){$\bullet$} +\end{picture} +\caption{The point spectra of the linear operator $L_\e$.} +\label{sgcb} +\end{figure} + +The main difficulty introduced by the singular +perturbation $\epsilon \partial^2_x f$ is the breaking of the +spectral +gap condition. Figure~\ref{sgcb} shows the distributions of the +eigenvalues when $\epsilon =0$ and $\epsilon \neq 0$. It clearly shows +the breaking of the stable spectral gap condition. As a result, +center and center-unstable manifolds do not necessarily +persist. On the other hand, the unstable spectral gap condition is +not broken. This gives the hope for the persistence of center-stable +manifold. Another case of persistence can be described as follows: +Notice that the plane $\Pi$, +\begin{equation}\label{Pi} \Pi=\{ q\mid \ \partial_xq=0\},\end{equation} +is invariant under the flow \eqref{spnls}. When $\epsilon =0$, there is an +unstable fibration with base points in a neighborhood of the circle +$S_\omega$ of fixed points, +\begin{equation} +S_\om=\{ q\in \Pi \mid \ |q|=\om \}, +\label{rcl} +\end{equation} +in $\Pi$, as an invariant sub-fibration of the +unstable Fenichel fibration with base points in the center manifold. +When $\epsilon >0$, the center manifold may not persist, but $\Pi$ +persists, moreover, the unstable spectral gap condition is not +broken, therefore, the unstable sub-fibration with base points in +$\Pi$ may persist. Since the semiflow generated by \eqref{spnls} +is not a $C^1$ perturbation of +that generated by the unperturbed NLS due to the singular +perturbation $\epsilon \partial^2_x$, standard results on +persistence can not be applied. + +The eigenfunctions corresponding to the real eigenvalues are: +\begin{equation} +\label{leef} +e^\pm_1=e^{\pm i\vth_1}\cos x,\quad e^{\pm i\vth_1}=\frac{1\mp +i\sqrt{4\omega^2-1}}{2\omega}. +\end{equation} +Notice that they are independent of $\epsilon$. +The eigenspaces corresponding to the complex conjugate pairs of +eigenvalues are given by: +\begin{equation*}E_1=\ \mbox{span}_{\CC}\{ \cos x \}.\end{equation*} +and have real dimension $2$. + +\subsection{The Re-Setup of Equations} + +For the goal of this subsection, we need to single out the +eigen-directions \eqref{leef}. Let +\begin{equation*}f=\sum_{\pm }\xi_1^\pm e^\pm_1+h,\end{equation*} +where $\xi^\pm_1$ are real variables, and +\begin{equation*}\langle h\rangle=\langle h\cos x \rangle =0.\end{equation*} +In terms of the coordinates $(\xi_1^\pm , J,\theta ,h)$, +\eqref{nc1}-\eqref{nc3} can be rewritten as: +\begin{align}\label{eig1} +\dot{\xi}^+_1&=\mu^+_1\xi^+_1+V^+_1\xi^+_1+\mathcal{N}^+_1, \\ +\label{eig2} \dot{J}&=\epsilon \left[ -2\alpha (J+\omega^2)+2\beta +\sqrt{J+\omega^2}\cos \theta \right] +\epsilon \mathcal{R}^J_2,\\ +\label{eig3} \dot{\theta} & = -2J-\epsilon \beta \frac{\sin +\theta}{\sqrt{J+\omega^2}}+\mathcal{R}^\theta_2,\\ +\label{eig4} h_t&=L_\epsilon h+V_\epsilon h+\tilde{\mathcal{N}},\\ +\label{eig5} +\dot{\xi}^-_1&=\mu^-_1\xi^-_1+V^-_1\xi^-_1+\mathcal{N}^-_1,\end{align} +where $\mu^\pm_1$ are given in \eqref{leev}, $\mathcal{N}^\pm_1$ and +$\tilde{\mathcal{N}}$ are projections of $-i\mathcal{N}_2-i\mathcal{N}_3$ +to the corresponding directions, and +\begin{align*}V^+_1\xi^+_1&=2c_1J(\xi^+_1+\xi^-_1)+\epsilon \beta +\frac{\sin \theta}{\sqrt{J+\omega^2}}(c^+_1\xi^+_1-c^-_1\xi^-_1),\\ +V^-_1\xi^-_1&=-2c_1J(\xi^+_1+\xi^-_1)+\epsilon \beta \frac{\sin +\theta}{\sqrt{J+\omega^2}}(c_1^-\xi^+_1-c^+_1\xi^-_1),\\ +c_1&=\frac{1}{\sqrt{4\omega^2-1}},\quad +c^+_1=\frac{2\omega^2-1}{\sqrt{4\omega^2-1}},\quad +c^-_1=\frac{2\omega^2}{\sqrt{4\omega^2-1}}.\end{align*} + +\subsection{A Modification} + +\begin{definition} For any $\delta >0$, we define the annular +neighborhood of the circle $S_\omega$ \eqref{rcl} as +\begin{equation*}\mathcal{A}(\delta)=\{ (J,\theta )\mid \ |J|<\delta +\}.\end{equation*} +\end{definition} + +To apply the Lyapunov-Perron's method, it is standard and necessary to +modify the $J$ equation so that $\mathcal{A}(4\delta)$ is +overflowing invariant. Let $\eta \in C^\infty (R,R)$ be a ``bump" function: +\begin{equation*}\eta=\begin{cases} 0, & \text{in } (-2,2)\cup +(-\infty,-6)\cup (6,\infty),\\ +1, & \text{in } (3,5),\\ +-1, & \text{in } (-5,-3),\end{cases}\end{equation*} +and $|\eta'|\leq 2$, $|\eta''|\leq C$. We +modify the $J$ equation \eqref{eig2} as follows: +\begin{equation}\label{meig2}\dot{J} =\epsilon b\eta ( +J/\delta)+\epsilon \left[ -2\alpha (J+\omega^2)+2\beta +\sqrt{J+\omega^2}\cos +\theta \right] +\epsilon \mathcal{R}^J_2,\end{equation} +where $b>2 (2\alpha \omega^2+2\beta \omega)$. Then +$\mathcal{A}(4\delta)$ is overflowing invariant. There are two main +points in +adopting the bump function: +\begin{enumerate}\item One needs $\mathcal{A}(4\delta)$ to be +overflowing invariant so that a Lyapunov-Perron type integral +equation can be set up along orbits in $\mathcal{A}(4\delta)$ for +$t\in (-\infty ,0)$. +\item One needs the vector field inside $\mathcal{A}(2\delta)$ to be +unchanged so that results for the modified system can be claimed +for the original system in $\mathcal{A}(\delta)$.\end{enumerate} + +\begin{remark} Due to the singular perturbation, the real part of +$\mu^\pm_k$ approaches $-\infty$ as $k\to \infty$. Thus the $h$ +equation \eqref{eig4} can not be modified to give overflowing flow. +This rules out the construction of unstable fibers with base points +having general $h$ coordinates.\end{remark} + +\subsection{Existence of Unstable Fibers} + +For any $(J_0,\theta_0)\in \mathcal{A}(4\delta)$, let +\begin{equation}\label{borb} J=J_*(t),\quad \theta=\theta_*(t), \quad +t\in (-\infty ,0],\end{equation} +be the backward orbit of the modified system \eqref{meig2} and +\eqref{eig3} with the initial point $(J_0,\theta_0)$. If +\begin{equation*}(\xi^+_1(t),J_*(t)+\tilde{J}(t),\theta_*(t)+\tilde{\theta}(t),h(t),\xi^-_1(t))\end{equation*} +is a solution of the modified full system, then one has +\begin{eqnarray}\dot{\xi}^+_1 &=& \mu^+_1\xi^+_1+F^+_1, +\label{meq1}\\ +u_t &=& Au+F,\label{meq2} +\end{eqnarray} +where +\begin{align*} u&=\begin{pmatrix} \tilde{J} \\ \tilde{\theta}\\ h\\ +\xi^-_1 \end{pmatrix}, \quad A=\begin{pmatrix} 0 & 0& 0 & 0 +\\ +-2 & 0 & 0 & 0 \\ 0 & 0 & L_\epsilon & 0 \\ +0 & 0 & 0 & \mu^-_1 \end{pmatrix},\quad F=\begin{pmatrix} F_J\\ +F_\theta\\ F_h\\ +F^-_1\end{pmatrix},\\ +&F^+_1 = V^+_1\xi^+_1+\mathcal{N}^+_1,\\ +&F_J=\epsilon b\left[ \eta (J/\delta)-\eta +(J_*(t)/\delta)\right]+\epsilon \left[ +-2\alpha \tilde{J} +2\beta \sqrt{J+\omega^2}\cos \theta \right.\\ +&\quad -\left. 2\beta \sqrt{J_*(t)+\omega^2}\cos \theta_*(t)\right] +\epsilon +\mathcal{R}^J_2,\\ +&F_\theta = -\epsilon \beta \frac{\sin +\theta}{\sqrt{J+\omega^2}}+\epsilon \beta \frac{\sin +\theta_*(t)}{\sqrt{J_*(t)+\omega^2}}+\mathcal{R}^\theta_2,\\ +&F_h=V_\epsilon h+\tilde{\mathcal{N}},\\ +&F_1^-=V^-_1\xi^-_1+\mathcal{N}^-_1,\\ +&J=J_*(t)+\tilde{J},\quad \theta=\theta_*(t)+\tilde{\theta}. +\end{align*} +\nid +System \eqref{meq1}-\eqref{meq2} can be written in the equivalent +integral equation form: +\begin{align}\label{eit1} \xi^+_1(t)&= +\xi^+_1(t_0)e^{\mu^+_1(t-t_0)}+\int^t_{t_0}e^{\mu^+_1(t-\tau)}F^+_1(\tau +)d\tau, \\ +\label{eit2} u(t)&= +e^{A(t-t_0)}u(t_0)+\int^t_{t_0}e^{A(t-\tau)}F(\tau )d\tau.\end{align} +By virtue of the gap between $\mu^+_1$ and the real parts of the +eigenvalues of $A$, one can introduce the following space: For $\sigma +\in \left( \frac{\mu^+_1}{100},\frac{\mu^+_1}{3}\right)$, and $n\geq 1$, let +\begin{equation*}\begin{split} G_{\sigma ,n}&=\bigg \{ +g(t)=(\xi^+_1(t),u(t))\bigg | \ t \in (-\infty ,0], \ g(t)\text{ is +continuous} \\ &\quad \text{in } t\text{ in } H^n\text{ norm }, +\ \| g\|_{\sigma ,n}=\sup_{t\leq 0}e^{-\sigma t} [ +|\xi^+_1(t)|+\| u(t)\|_n ] <\infty\bigg \}\ . +\end{split}\end{equation*} +$G_{\sigma ,n}$ is a Banach space under the norm $\| \cdot \|_{\sigma +,n}$. Let $\mathcal{B}_{\sigma ,n}(r)$ denote the ball in +$G_{\sigma ,n}$ centered at the origin with radius $r$. Since $A$ +only has point spectrum, the spectral mapping theorem is valid. It is +obvious that for $t\geq 0$, +\begin{equation*}\| e^{At}u\|_n\leq C(1+t)\| u\|_n,\end{equation*} +for some constant $C$. Thus, if $g(t)\in \mathcal{B}_{\sigma +,n}(r)$, $r<\infty$ is a solution of \eqref{eit1}-\eqref{eit2}, by +letting $t_0\to -\infty$ in \eqref{eit2} and setting $t_0=0$ in +\eqref{eit1}, one has +\begin{align}\label{per1} +\xi^+_1(t)&=\xi^+_1(0)e^{\mu^+_1t}+\int^t_0e^{\mu^+_1(t-\tau)}F^+_1(\tau +)d\tau,\\ +\label{per2} u(t)&=\int^t_{-\infty}e^{A(t-\tau)}F(\tau )d\tau.\end{align} +For $g(t)\in \mathcal{B}_{\sigma ,n}(r)$, let $\Gamma(g)$ be the +map defined by the right hand side of \eqref{per1}-\eqref{per2}. +Then a solution of \eqref{per1}-\eqref{per2} is a fixed point of +$\Gamma$. For any $n\geq 1$ and $\epsilon <\delta^2$, and $\delta$ and +$r$ are small enough, $F^+_1$ and $F$ are Lipschitz in $g$ with small +Lipschitz constants. Standard arguments of the Lyapunov-Perron's +method readily imply the existence of a fixed point $g_*$ of $\Gamma$ +in $\mathcal{B}_{\sigma ,n}(r)$. + +\subsection{Regularity of the Unstable Fibers in $\e$} + +The difficulties lie in the investigation on the regularity of $g_*$ with +respect to $(\epsilon ,\alpha ,\beta ,\omega ,J_0,\theta_0,\xi^+_1(0))$. +The most difficult one is the regularity with respect to +$\epsilon$ due to the singular perturbation. +Formally differentiating $g_*$ in \eqref{per1}-\eqref{per2} with +respect to $\epsilon$, one gets +\begin{align}\label{dper1} \xi^+_{1,\epsilon}(t)&= +\int^t_0e^{\mu^+_1(t-\tau)}\left[ \partial_uF^+_1\cdot u_\epsilon ++\partial_{\xi_1^+}F^+_1\cdot +\xi^+_{1,\epsilon}\right](\tau )d\tau ++\mathcal{R}^+_1(t),\\ +u_\epsilon (t)&=\int^t_{-\infty}e^{A(t-\tau)}\left[ \partial_uF\cdot +u_\epsilon +\partial_{\xi_1^+}F\cdot +\xi^+_{1,\epsilon}\right](\tau )d\tau ++\mathcal{R}(t),\label{dper2}\end{align} +where +\begin{align}\begin{split}\label{wdp1} +\mathcal{R}^+_1(t)&=\xi^+_1(0)\mu^+_{1,\epsilon}te^{\mu^+_1t}+\int^t_0\mu^+_{1,\epsilon}(t-\tau)e^{\mu^+_1(t-\tau)}F^+_1(\tau +)d\tau\\ +&\quad + \int^t_0e^{\mu^+_1(t-\tau)}[\partial_\epsilon +F^+_1+\partial_{u_*}F^+_1\cdot u_{*,\epsilon}](\tau )d\tau,\end{split}\\ +\begin{split}\label{wdp2} \mathcal{R}(t)&= +\int^t_{-\infty}(t-\tau)A_\epsilon e^{A(t-\tau)}F(\tau )d\tau\\ +&\quad +\int^+_{-\infty}e^{A(t-\tau)}[\partial_\epsilon +F+\partial_{u_*}F\cdot u_{*,\epsilon}](\tau)d\tau,\end{split}\\ +\label{wdp3} \mu^+_{1,\epsilon}&=-(\alpha +1),\\ +A_\epsilon &=\begin{pmatrix} 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 \\ +0 & 0 & -\alpha+\partial^2_x & 0 \\ +0 & 0 & 0 & -(\alpha+1) \end{pmatrix},\\ +u_*&=(J_*,\theta_*,0,0)^T,\label{wdp4}\end{align} +where $T=$\ transpose, and $(J_*,\theta_*)$ are given in \eqref{borb}. +The troublesome terms are the ones containing $A_\epsilon $ or +$u_{*,\epsilon}$ in \eqref{wdp1}-\eqref{wdp2}. +\begin{equation} +\| A_\epsilon F\|_n\leq C\| F\|_{n+2}\leq \tilde{c} +( \|u\|_{n+2}+|\xi^+_1|), +\label{upre} +\end{equation} +where $\tilde{c}$ is small when $(\ \cdot \ )$ on the right hand side +is small. +\begin{align*}\begin{split} \partial_{J_*}F_J\cdot J_{*,\epsilon}&= +\frac{\epsilon}{\delta}b[ \eta'( +J/\delta)-\eta'( J_*/\delta)] +J_{*,\epsilon}\\ +&\quad + \epsilon [ \beta \frac{\cos +\theta}{\sqrt{J+\omega^2}}-\beta\frac{\cos +\theta_*}{\sqrt{J_*+\omega^2}}] +J_{*,\epsilon}\\ +&\quad +\text{easier terms}.\end{split}\\ +\begin{split} |\partial_{J_*}F_J\cdot J_{*,\epsilon}|&\leq +\frac{\epsilon}{\delta^2}b\sup_{0\leq \hat{\gamma}\leq 1}\left| \eta''( +[\hat{\gamma} J_*+(1-\hat{\gamma})J]/\delta)\right | \ |\tilde{J}| +\ |J_{*,\epsilon}|\\ +&\quad + \epsilon \beta +C(|\tilde{J}|+|\tilde{\theta}|)|J_{*,\epsilon}|+\text{ easier terms}\\ +& \leq C_1(|\tilde{J}|+|\tilde{\theta}|)|J_{*,\epsilon}|+\text{ +easier terms,}\end{split}\\ +\begin{split} \sup_{t\leq 0} e^{-\sigma t} |\partial_{J_*}F_J\cdot +J_{*,\epsilon}|&\leq C_1\sup_{t\leq 0}[ e^{-(\sigma ++\tilde{\nu} )t}(|\tilde{J}|+|\tilde{\theta}|)]\sup_{t\leq 0}[e^{\tilde{\nu} +t}|J_{*,\epsilon}|]\\ +&\quad + \text{easier terms},\end{split}\end{align*} +where $\sup_{t\leq 0}e^{\tilde{\nu} t}|J_{*,\epsilon}|$ can be bounded when +$\epsilon$ is sufficiently small for any fixed $\tilde{\nu} >0$, through a +routine estimate on Equations~\eqref{meig2} and \eqref{eig3} for +$(J_*(t),\theta_*(t))$. Other terms involving $u_{*,\epsilon}$ can be +estimated similarly. Thus, the $\| \ \|_{\sigma ,n}$ norm of terms +involving $u_{*,\epsilon}$ has to be bounded by $\|\ \|_{\sigma +\tilde{\nu} +,n}$ norms. This leads to the standard rate condition for the +regularity of invariant manifolds. That is, the regularity is +controlled by +the spectral gap. The $\| \ \|_{\sigma ,n}$ norm of the term involving +$A_\epsilon$ has to be bounded by $\| \ \|_{\sigma ,n+2}$ norms. This +is a new phenomenon caused by the singular perturbation. This problem +is resolved by virtue of a special property of the fixed point $g_*$ +of $\Gamma$. +Notice that if $\sigma_2\geq \sigma_1$, $n_2\geq n_1$, then +$G_{\sigma_2,n_2}\subset G_{\sigma_1,n_1}$. Thus by the uniqueness of +the +fixed point, if $g_*$ is the fixed point of $\Gamma$ in +$G_{\sigma_2,n_2}$, $g_*$ is also the fixed point of $\Gamma$ in +$G_{\sigma_1,n_1}$. Since $g_*$ exists in $G_{\sigma ,n}$ for an +fixed $n\geq 1$ and $\sigma \in ( +\frac{\mu^+_1}{100},\frac{\mu^+_1}{3}-10\tilde{\nu})$ where +$\tilde{\nu}$ is small enough, +\begin{align*} \| \mathcal{R}^+_1\| _{\sigma,n}&\leq C_1+C_2\| +g_*\|_{\sigma+\tilde{\nu},n},\\ +\| \mathcal{R}\|_{\sigma ,n}&\leq C_3\| g_*\|_{\sigma ,n+2}+C_4\| +g_*\|_{\sigma+\tilde{\nu},n}+C_5,\end{align*} +where $C_j\ (1\leq j\leq 5)$ depend upon $\| g_*(0)\|_n$ and $\| +g_*(0)\|_{n+2}$. Let +\begin{equation*}M=2(\| \mathcal{R}^+_1\|_{\sigma +,n}+\|\mathcal{R}\|_{\sigma, n}),\end{equation*} +and $\Gamma'$ denote the linear map defined by the right hand sides +of \eqref{dper1} and \eqref{dper2}. Since the terms +$\partial_uF^+_1$, $\partial_{\xi^+_1}F^+_1$, $\partial_uF$, and +$\partial_{\xi^+_1}F$ all have small $\|\ \|_n$ norms, $\Gamma '$ +is a contraction map on $\mathcal{B}(M)\subset L(\mathcal{R}, G_{\sigma +,n})$, where $\mathcal{B}(M)$ is the ball of radius $M$. Thus +$\Gamma '$ has a unique fixed point $g_{*,\epsilon}$. Next one needs +to show that $g_{*,\epsilon}$ is indeed the partial derivative of +$g_*$ with respect to $\epsilon$. That is, one needs to show +\begin{equation}\label{ddef} \lim_{\Delta \epsilon \to 0}\frac{\| +g_*(\epsilon +\Delta \epsilon)-g_*(\epsilon)-g_{*,\epsilon} \Delta +\epsilon \|_{\sigma ,n}}{\Delta \epsilon}=0.\end{equation} +\nid +This has to be accomplished directly from Equations +\eqref{per1}-\eqref{per2}, \eqref{dper1}-\eqref{dper2} satisfied by +$g_*$ and +$g_{*,\epsilon}$. The most troublesome estimate is still the one +involving $A_\epsilon$. First, notice the fact that $e^{\epsilon +\partial^2_x}$ is holomorphic in $\epsilon$ when $\epsilon>0$, and +not differentiable at $\epsilon=0$. Then, notice that $g_*\in +G_{\sigma ,n}$ for any $n\geq 1$, thus, $e^{\epsilon +\partial_x^2}g_*$ is differentiable, up to certain order $m$, in +$\epsilon$ at +$\epsilon=0$ from the right, i.e. +\begin{equation*}(d^+/d\epsilon)^me^{\epsilon +\partial^2_x}g_*|_{\epsilon =0}\end{equation*} +exists in $H^n$. Let +\begin{equation*}\begin{split} z(t,\Delta \epsilon)&= e^{(\epsilon ++\Delta \epsilon)t\partial^2_x}g_*-e^{\epsilon +t\partial^2_x}g_*-(\Delta \epsilon) t\partial ^2_x e^{\epsilon +t\partial _x^2}g_*\\ +&= e^{\epsilon t\partial^2_x}w(\Delta \epsilon),\end{split}\end{equation*} +where $t\geq 0$, $\Delta \epsilon >0$, and +\begin{equation*}w(\Delta \epsilon) =e^{(\Delta \epsilon ) t\partial +_x^2}g_*-g_*-(\Delta \epsilon ) t\partial^2_x g_*.\end{equation*} +Since $w(0)=0$, by the Mean Value Theorem, one has +\begin{equation*}\| w(\Delta \epsilon )\|_n=\| w(\Delta \epsilon +)-w(0)\|_n\leq \sup_{0\leq \lambda \leq 1}\| \frac{dw}{d\Delta +\epsilon}(\lambda +\Delta \epsilon)\|_n|\Delta \epsilon|,\end{equation*} +where at $\lambda=0$, $\frac{d}{d\Delta \epsilon}=\frac{d^+}{d\Delta +\epsilon}$, and +\begin{equation*}\frac{dw}{d\Delta \epsilon}=t[e^{(\Delta \epsilon ) +t\partial^2_x}\partial^2_x g_*-\partial^2_x g_*]. +\end{equation*} +Since $\frac{dw}{d\Delta \epsilon}(0)=0$, by the Mean Value Theorem +again, one has +\[ +\| \frac {dw}{d\Dl \e }(\la \Dl \e )\|_n = +\| \frac {dw}{d\Dl \e }(\la \Dl \e ) - +\frac {dw}{d\Dl \e }(0)\|_n \leq \sup_{0 \leq \la_1 \leq 1} +\| \frac {d^2w}{d\Dl \e^2 }(\la_1 \la \Dl \e )\|_n |\la | |\Dl \e |\ , +\] +where +\[ +\frac {d^2w}{d\Dl \e^2 } = t^2 [e^{(\Dl \e )t \pa_x^2} \pa_x^4 g_* ]\ . +\] +Therefore, one has the estimate +\begin{equation} +\| z(t,\Dl \e )\|_n \leq |\Dl \e |^2 t^2 \| g_* \|_{n+4} \ . +\label{defes} +\end{equation} +This estimate is sufficient for handling the estimate involving +$A_\e$. The estimate involving $u_{*,\e}$ can be handled in a similar +manner. For instance, let +\[ +\tilde{z}(t,\Dl \e ) = F(u_*(t,\e +\Dl \e ))- F(u_*(t,\e )) +-\Dl \e \pa_{u_*}F \cdot u_{*,\e}\ , +\] +then +\[ +\| \tilde{z}(t,\Dl \e )\|_{\sg , n} \leq |\Dl \e |^2 +\sup_{0 \leq \la \leq 1} \| [u_{*,\e} \cdot \pa^2_{u_*}F \cdot u_{*,\e} ++\pa_{u_*}F \cdot u_{*,\e \e}](\la \Dl \e )\|_{\sg , n} \ . +\] +From the expression of $F$ (\ref{meq2}), one has +\begin{equation*}\begin{split} & \| u_{*,\epsilon}\cdot +\partial^2_{u_*}F\cdot u_{*,\epsilon}+\partial_{u_*}F\cdot +u_{*,\epsilon\epsilon}\|_{\sigma,n}\\ +&\quad \leq C_1\| g_*\|_{\sigma +2\tilde{\nu} ,n} [( \sup_{t\leq +0}e^{\tilde{\nu} t}|u_{*,\epsilon}|)^2+\sup_{t\leq 0}e^{2\tilde{\nu} +t}|u_{*,\epsilon \epsilon}|],\end{split}\end{equation*} +and the term $[ \ ]$ on the right hand side can be easily shown to be +bounded. In conclusion, let +\begin{equation*}h=g_*(\epsilon +\Delta +\epsilon)-g_*(\epsilon)-g_{*,\epsilon}\Delta +\epsilon,\end{equation*} +one has the estimate +\begin{equation*}\| h\|_{\sigma ,n}\leq \tilde{\k} \| +h\|_{\sigma,n}+|\Delta \epsilon|^2\tilde{C}(\| g_*\|_{\sigma ,n+4};\| +g_*\|_{\sigma +2\tilde{\nu} ,n}),\end{equation*} +where $\tilde{\k}$ is small, thus +\begin{equation*}\| h\|_{\sigma ,n}\leq 2 |\Delta \epsilon +|^2\tilde{C}(\| g_*\|_{\sigma,n+4};\| g_*\|_{\sigma +2\tilde{\nu} +,n}).\end{equation*} This implies that +\begin{equation*}\lim_{\Delta \epsilon \to 0}\frac{\| h\|_{\sigma +,n}}{|\Delta \epsilon|}=0,\end{equation*} +which is \eqref{ddef}. + +Let $g_*(t)=(\xi^+_1(t),u(t))$. First, let me comment on $\left. +\frac{\partial u}{\partial +\xi^+_1(0)}\right|_{\xi^+_1(0)=0,\epsilon =0}=0$. From \eqref{per2}, one has +\begin{equation*}\| \frac{\partial u}{\partial +\xi^+_1(0)}\| _{\sigma, n}\leq \k_1\| \frac{\partial +g_*}{\partial \xi^+_1(0)}\|_{\sigma ,n},\end{equation*} +where by letting $\xi^+_1(0)\to 0$ and $\epsilon \to 0^+$, $\k_1\to 0$. Thus +\begin{equation*}\left. \frac{\partial u}{\partial +\xi^+_1(0)}\right|_{\xi^+_1(0)=0,\epsilon =0}=0.\end{equation*} +I shall also comment on ``exponential decay" property. Since $\| +g_*\|_{\frac{\mu^+_1}{3},n}\leq r$, +\begin{equation*}\| g_*(t)\|_n\leq re^{\frac{\mu^+_1}{3}t},\quad +\forall t\leq 0.\end{equation*} +\begin{definition} Let $g_*(t)=(\xi^+_1(t), u(t))$, where +\begin{equation*}u(0)=\int^0_{-\infty}e^{A(t-\tau)}F(\tau)d\tau\end{equation*} +depends upon $\xi^+_1(0)$. Thus +\begin{equation*}u^0_*:\xi^+_1(0)\mapsto u(0),\end{equation*} +defines a curve, which we call an unstable fiber denoted by +$\mathcal{F}^+_p$, where $p=(J_0,\theta_0)$ is the base point, +$\xi^+_1(0)\in [-r,r]\times [-r,r]$. +\end{definition} +Let $S^t$ denote the evolution operator of \eqref{meq1}-\eqref{meq2}, then +\begin{equation*}S^t\mathcal{F}_p^t\subset \mathcal{F}^t_{S^tp},\quad +\forall t\leq 0.\end{equation*} +That is, $\{ \mathcal{F}^+_p\}_{p\in \mathcal{A}(4\delta)}$ is an +invariant family of unstable fibers. The proof of the Unstable Fiber +Theorem is finished. Q.E.D. + +\begin{remark} If one replaces the base orbit $(J_*(t),\theta_*(t))$ +by a general orbit for which only $\|\ \|_n$ norm is bounded, then +the estimate \eqref{upre} will not be possible. The $\|\ \|_{\sigma +,n+2}$ norm of the fixed point $g_*$ will not be bounded either. In +such case, $g_*$ may not be smooth in $\epsilon$ due to the singular +perturbation.\end{remark} + +\begin{remark} Smoothness of $g_*$ in $\epsilon$ at $\epsilon =0$ is +a key point in locating homoclinic orbits as discussed in later sections. +From integrable theory, information is known at $\epsilon =0$. This +key point will +link ``$\epsilon =0$" information to ``$\epsilon \neq 0$" studies. +Only continuity in $\epsilon $ at $\epsilon =0$ is not enough for the +study. The beauty of the entire theory is reflected by the fact +that although $e^{\epsilon \partial^2_x}$ is not holomorphic at +$\epsilon =0$, $e^{\epsilon \partial^2_x}g_*$ can be smooth at +$\epsilon =0$ from the right, up to certain order depending upon the +regularity of $g_*$. This is the beauty of the singular +perturbation. +\end{remark} + +\section{Proof of the Center-Stable Manifold Theorem \ref{CSM}} + +Here we give the proof of the center-stable manifold theorem \ref{CSM}, +proofs of other invariant manifold theorems in this chapter are easier. + +\subsection{Existence of the Center-Stable Manifold} + +We start with Equations \eqref{eig1}-\eqref{eig5}, let +\begin{equation}\label{2cr} v=\begin{pmatrix} J\\ \theta \\ h \\ +\xi^-_1\end{pmatrix},\quad \tilde{v}=\begin{pmatrix} +J\\ h\\ \xi^-_1\end{pmatrix},\end{equation} +and let $E_n(r)$ be the tubular neighborhood of $S_\om$ \eqref{rcl}: +\begin{equation}\label{defEn} E_n(r)=\{ (J,\theta +,h,\xi^-_1)\in H^n\mid \ \| \tilde{v}\|_n\leq r\}.\end{equation} +$E_n(r)$ is of codimension $1$ in the entire phase space +coordinatized by $(\xi^+_1,J,\theta +,h,\xi^-_1)$. + +Let $\chi\in C^\infty(R,R)$ be a ``cut-off" function: +\begin{equation*}\chi=\begin{cases} 0, & \text{in } (-\infty ,-4)\cup +(4,\infty),\\ +1, & \text{in } (-2,2).\end{cases}\end{equation*} +We apply the cut-off \begin{equation*}\chi_\delta =\chi ( +\|\tilde{v}\|_n/\delta) \chi ( +\xi^+_1/\delta)\end{equation*} to +Equations~\eqref{eig1}-\eqref{eig5}, so that the equations in a +tubular neighborhood +of the circle +$S_\omega$ \eqref{rcl} are unchanged, and linear outside a +bigger tubular neighborhood. The modified equations take the form: +\begin{align}\label{ceq1} +\dot{\xi}_1^+&=\mu^+_1\xi^+_1+\tilde{F}^+_1, \\ +\label{ceq2}v_t&=Av+\tilde{F},\end{align} +where $A$ is given in \eqref{meq2}, +\begin{align*} \tilde{F}^+_1&=\chi_\delta[V^+_1\xi^+_1+\mathcal{N}^+_1],\\ +\tilde{F}&=(\tilde{F}_J,\tilde{F}_\theta +,\tilde{F}_h,\tilde{F}^-_1)^T,\quad +T=\text{transpose},\\ +\tilde{F}_J&=\chi_\delta \ \e \left[-2\alpha (J+\omega^2)+2\beta +\sqrt{J+\omega^2}\cos \theta+\mathcal{R}^J_2\right],\\ +\tilde{F}_\theta&=\chi _\delta \left[ -\epsilon \beta \frac{\sin +\theta}{\sqrt{J+\omega^2}}+\mathcal{R}^\theta_2\right],\\ +\tilde{F}_h&=\chi_\delta [V_\epsilon h+\tilde{\mathcal{N}}], \\ +\tilde{F}_1^-&=\chi_\delta [V^-_1\xi^-_1+ \mathcal{N}^-_1],\end{align*} +Equations~\eqref{ceq1}-\eqref{ceq2} can be written in the equivalent +integral equation form: +\begin{align}\label{csit1} +\xi^+_1(t)&=\xi^+_1(t_0)e^{\mu^+_1(t-t_0)}+\int^t_{t_0}e^{\mu^+_1(t-\tau +)}\tilde{F}^+_1(\tau +)d\tau,\\ +\label{csit2} +v(t)&=e^{A(t-t_0)}v(t_0)+\int^t_{t_0}e^{A(t-\tau)}\tilde{F}(\tau +)d\tau.\end{align} +We introduce the following space: For $\sigma \in \left( +\frac{\mu^+_1}{100},\frac{\mu^+_1}{3}\right)$, +and $n\geq 1$, let +\begin{equation*}\begin{split} \tilde{G}_{\sigma ,n}&= \bigg \{ +g(t)=(\xi^+_1(t),v(t)) \bigg | \ t\in [0,\infty),g(t)\text{ is continuous +in } t\\ +&\quad \text{in } H^n \ \mbox{norm} ,\| g\|_{\sigma ,n}=\sup_{t\geq +0}e^{-\sigma t}[ |\xi^+_1(t)|+\| v(t)\|_n ] +<\infty \bigg \}\ .\end{split}\end{equation*} +$\tilde{G}_{\sigma,n}$ is a Banach space under the norm $\| \cdot +\|_{\sigma ,n}$. Let $\tilde{\mathcal{A}}_{\sigma ,n}(r)$ denote the +closed tubular neighborhood of $S_\omega$ \eqref{rcl}: +\begin{equation*} \tilde{\mathcal{A}}_{\sigma ,n}(r)= \bigg \{ g(t) +=(\xi^+_1(t),v(t))\in \tilde{G}_{\sigma,n}\bigg | \ \sup_{t\geq +0}e^{-\sigma t}[ |\xi^+_1(t)|+\| +\tilde{v}(t)\|_n] \leq r\bigg \}\ ,\end{equation*} +where $\tilde{v}$ is defined in \eqref{2cr}. If $g(t)\in +\tilde{\mathcal{A}}_{\sigma ,n}(r)$, $r<\infty$, is a solution of +\eqref{csit1}-\eqref{csit2}, by letting $t_0\to +\infty$ in +\eqref{csit1} and setting $t_0=0$ in \eqref{csit2}, one has +\begin{align}\label{cspe1} +\xi^+_1(t)&=\int^t_{+\infty}e^{\mu^+_1(t-\tau)}\tilde{F}^+_1(\tau +)d\tau, \\ +\label{cspe2}v(t)&=e^{At}v(0)+\int^t_0e^{A(t-\tau)}\tilde{F}(\tau +)d\tau .\end{align} +For any $g(t)\in \tilde{\mathcal{A}}_{\sigma ,n}(r)$, let +$\tilde{\Gamma}(g)$ be the map defined by the right hand side of +\eqref{cspe1}-\eqref{cspe2}. In contrast to the map $\Gamma$ defined +in \eqref{per1}-\eqref{per2}, $\tilde{\Gamma}$ contains constant +terms of order $\mathcal{O}(\epsilon)$, e.g. $\tilde{F}_J$ and +$\tilde{F}_\theta$ both contain such terms. Also, +$\tilde{\mathcal{A}}_{\sigma, n}(r)$ is a tubular neighborhood of the +circle $S_\omega$ \eqref{rcl} instead of the ball +$\mathcal{B}_{\sigma ,n}(r)$ for $\Gamma$. Fortunately, these facts +will not create any difficulty in showing $\tilde{\Gamma}$ is a +contraction on $\tilde{\mathcal{A}}_{\sigma ,n}(r)$. For any $n\geq +1$ and $\epsilon <\delta^2$, and $\delta $ and $r$ are small enough, +$\tilde{F}_1^+$ and $\tilde{F}$ are Lipschitz in $g$ with small +Lipschitz constants. $\tilde{\Gamma}$ has a unique fixed point +$\tilde{g}_*$ in $\tilde{\mathcal{A}}_{\sigma ,n}(r)$, following from +standard arguments. + +\subsection{Regularity of the Center-Stable Manifold in $\e$} + +For the regularity of $\tilde{g}_*$ with +respect to $(\epsilon, \alpha ,\beta ,\omega ,v(0))$, the most +difficult one is of course with respect to $\epsilon$ due to the +singular +perturbation. Formally differentiating $\tilde{g}_*$ in +\eqref{cspe1}-\eqref{cspe2} with respect to $\epsilon$, one gets +\begin{align}\label{dcsp1} +\xi^+_{1,\epsilon}(t)&=\int^t_{+\infty}e^{\mu^+_1(t-\tau)}\left[ +\partial_{\xi^+_1}\tilde{F}^+_1\cdot +\xi^+_{1,\epsilon}+\partial_v\tilde{F}^+_1\cdot v_\epsilon +\right](\tau +)d\tau +\tilde{R}^+_1(t),\\ +\label{dcsp2}v(t)&=\int^t_0e^{A(t-\tau)}\left[ +\partial_{\xi^+_1}\tilde{F}\cdot +\xi^+_{1,\epsilon}+\partial_v\tilde{F}\cdot v_\epsilon +\right](\tau )d\tau +\tilde{R}(t),\end{align} +where +\begin{align}\label{wdcs1} +\tilde{R}_1^+(t)&=\int^t_{+\infty}\mu^+_{1,\epsilon}(t-\tau) +e^{\mu^+_1(t-\tau)}\tilde{F}_1^+(\tau )d\tau ++\int^t_{+\infty}e^{\mu^+_1(t-\tau)}\partial _\epsilon +\tilde{F}^+_1(\tau )d\tau, \\ +\label{wdcs2} \tilde{R}(t)&= tA_\epsilon +e^{At}v(0)+\int^t_0(t-\tau)A_\epsilon e^{A(t-\tau)}\tilde{F}(\tau +)d\tau + \int^t_0e^{A(t-\tau )}\partial_\epsilon \tilde{F}(\tau +)d\tau,\end{align} +and $\mu^+_{1,\epsilon}$ and $A_\epsilon$ are given in +\eqref{wdp3}-\eqref{wdp4}. The troublesome terms are the ones +containing +$A_\epsilon$ in \eqref{wdcs2}. These terms can be handled in the same +way as in the Proof of the Unstable Fiber Theorem. The crucial +fact utilized is that if $v(0)\in H^{n_1}$, then $\tilde{g}_*$ is the +unique fixed point of $\tilde{\Gamma}$ in both +$\tilde{G}_{\sigma,n_1}$ and $\tilde{G}_{\sigma ,n_2}$ for any $n_2\leq n_1$. + +\begin{remark} In the Proof of the Unstable Fiber Theorem, the +arbitrary initial datum in \eqref{per1}-\eqref{per2} is $\xi^+_1(0)$ +which is a scalar. Here the arbitrary initial datum in +\eqref{cspe1}-\eqref{cspe2} is $v(0)$ which is a function of $x$. +If $v(0)\in H^{n_2}$ but not $H^{n_1}$ for some $n_1>n_2$, +then $\tilde{g}_*\notin \tilde{G}_{\sigma ,n_1}$, in contrast to the +case of +\eqref{per1}-\eqref{per2} where $g_*\in G_{\sigma ,n}$ for any fixed +$n\geq 1$. The center-stable manifold $W^{cs}_n$ stated in the +Center-Stable Manifold Theorem will be defined through $v(0)$. This +already illustrates why $W^{cs}_n$ has the regularity in $\epsilon$ +as stated in the theorem.\end{remark} + +We have +\begin{align*} &\| \tilde{R}^+_1\|_{\sigma ,n}\leq \tilde{C_1},\\ +& \| \tilde{R}\|_{\sigma ,n}\leq \tilde{C}_2\| \tilde{g}_*\|_{\sigma +,n+2}+\tilde{C}_3,\end{align*} +for $\tilde{g}_*\in \tilde{\mathcal{A}}_{\sigma , n+2}(r)$, where +$\tilde{C}_j\ (j=1,2,3)$ are constants depending in particular upon the +cut-off in $\tilde{F}^+_1$ and $\tilde{F}$. Let $\tilde{\Gamma}'$ +denote the linear map defined by the right hand sides of +\eqref{dcsp1}-\eqref{dcsp2}. If $v(0)\in H^{n+2}$ and $\tilde{g}_*\in +\tilde{\mathcal{A}}_{\sigma,n+2}(r)$, standard argument shows that +$\tilde{\Gamma}'$ is a contraction map on a closed ball in +$L(R,\tilde{G}_{\sigma,n})$. Thus $\tilde{\Gamma}'$ has a unique +fixed point +$\tilde{g}_{*,\epsilon}$. Furthermore, if $v(0)\in H^{n+4}$ and +$\tilde{g}_*\in \tilde{\mathcal{A}}_{\sigma ,n+4}(r)$, one has that +$\tilde{g}_{*,\epsilon}$ is indeed the derivative of $\tilde{g}_*$ in +$\epsilon$, following the same argument as in the Proof of the +Unstable Fiber Theorem. Here one may be able to replace the +requirement $v(0)\in H^{n+4}$ and $\tilde{g}_*\in +\tilde{\mathcal{A}}_{\sigma ,n+4}(r)$ by just $v(0)\in H^{n+2}$ and +$\tilde{g}_*\in \tilde{\mathcal{A}}_{\sigma ,n+2}(r)$. But we are +not interested in sharper results, and the current result is +sufficient for our purpose. + +\begin{definition} For any $v(0)\in E_n(r)$ where $r$ is sufficiently +small and $E_n(r)$ is defined in \eqref{defEn}, let +$\tilde{g}_*(t)=(\xi^+_1(t),v(t))$ be the fixed point of +$\tilde{\Gamma}$ in $\tilde{G}_{\sigma ,n}$, where one has +\begin{equation*}\xi^+_1(0)=\int^0_{+\infty}e^{\mu^+_1(t-\tau)} +\tilde{F}^+_1(\tau )d\tau ,\end{equation*} +which depend upon $v(0)$. Thus +\begin{equation*} \xi^+_*:v(0)\mapsto \xi^+_1(0),\end{equation*} +defines a codimension $1$ surface, which we call center-stable +manifold denoted by $W^{cs}_n$.\end{definition} + +The regularity of the fixed point $\tilde{g}_*$ immediately implies +the regularity of $W^{cs}_n$. We have sketched the proof of the most +difficult regularity, i.e. with respect to $\epsilon$. Uniform +boundedness of $\partial_\epsilon \xi^+_*$ in $v(0)\in E_{n+4}(r)$ and +$\epsilon \in [0,\epsilon _0)$, is obvious. Other parts of the +detailed proof is completely standard. We have that $W^{cs}_n$ is a +$C^1$ +locally invariant submanifold which is $C^1$ in $(\alpha ,\beta, +\omega)$. $W^{cs}_n$ is $C^1$ in $\epsilon$ at point in the subset +$W^{cs}_{n+4}$. Q.E.D. + +\begin{remark}\label{reop} Let $S^t$ denote the evolution +operator of the perturbed nonlinear Schr\"odinger equation +\eqref{spnls}. The proofs of the Unstable Fiber Theorem and the +Center-Stable Manifold Theorem also imply the following: $S^t$ is a +$C^1$ +map on $H^n$ for any fixed $t>0$, $n\geq 1$. $S^t$ is also $C^1$ in +$(\alpha ,\beta ,\omega)$. $S^t$ is $C^1$ in $\epsilon$ as a map +from $H^{n+4}$ to $H^n$ for any fixed $n\geq 1$, $\epsilon \in +[0,\epsilon_0)$, $\epsilon_0>0$.\end{remark} + +\section{Perturbed Davey-Stewartson II Equations \label{invds}} + +Invariant manifold results in Sections \ref{spsec} and \ref{rpsec} also hold +for perturbed Davey-Stewartson II equations \cite{Li02b}, +\begin{eqnarray*} +iq_t &=& \Upsilon q+ \bigg [2(|q|^2-\omega^2)+ u_y \bigg ]q ++i\epsilon f \ , \\ +& & \ \ \Delta u = -4\partial_y |q|^2 \ , +\end{eqnarray*} +where $q$ is a complex-valued function of the three variables ($t,x,y$), +$u$ is a real-valued function of the three variables ($t,x,y$), +$\Upsilon =\partial_{xx}-\partial_{yy}$, $\Delta=\partial_{xx} ++\partial_{yy}$, $\omega >0$ is a constant, and $f$ is the perturbation. +We also consider periodic boundary conditions. + +Under singular perturbation +\[ +f = \Dl q - \al q + \be \ , +\] +where $\alpha >0$, $\beta >0$ are constants, Theorems \ref{UFT} and +\ref{CSM} hold for the perturbed Davey-Stewartson II equations \cite{Li02b}. + +When the singular perturbation $\Dl$ is mollified into a bounded +Fourier multiplier +\[ +\hat{\Dl} q = -\sum_{k \in Z^2} \be_k |k|^2 \tilde{q}_k \cos k_1 x +\cos k_2 y \ , +\] +in the case of periods ($2\pi , 2\pi$), +\[ +\be_k = 1, \ \ |k| \leq N , \quad \be_k = |k|^{-2},\ \ |k| > N, +\] +for some large $N$, $|k|^2 = k_1^2 + k_2^2$, Theorems \ref{Persthm} +and \ref{fiberthm} hold for the perturbed Davey-Stewartson II equations +\cite{Li02b}. + +\section{General Overview \label{GOV}} + +For discrete systems, i.e., the flow is given by a map, it is more +convenient to use Hadamard's method +to prove invariant manifold and fiber theorems \cite{HPS77}. +Even for continuous systems, Hadamard's method was often utilized +\cite{Fen71}. On the other hand, Perron's method provides shorter +proofs. It involves manipulation of integral equations. This method +should be a favorite of analysts. Hadamard's method deals with graph +transform. The proof is often much longer, with clear geometric intuitions. +It should be a favorite of geometers. For finite-dimensional +continuous systems, N. Fenichel proved persistence of invariant +manifolds under $C^1$ perturbations of flow in a very general setting +\cite{Fen71}. He then went on to prove the fiber theorems in \cite{Fen74} +\cite{Fen77} also in this general setting. Finally, he applied this +general machinery to a general system of ordinary differential equations +\cite{Fen79}. As a result, Theorems \ref{Persthm} and \ref{fiberthm} +hold for the following perturbed discrete cubic nonlinear Schr\"odinger +equations \cite{LM97}, +\begin{eqnarray} +i\dot{q_n}&=&{1 \over h^2}\bigg[q_{n+1}-2q_n+q_{n-1}\bigg]+|q_n|^2(q_{n+1}+ + q_{n-1})-2\om^2 q_n \nonumber \\ + & &+i\e \bigg[-\al q_n +{1 \over h^2}(q_{n+1}-2q_n+q_{n-1}) + + \be \bigg], \label{PDNLS} +\end{eqnarray} +\nid +where $i=\sqrt{-1}$, $q_n$'s are complex variables, +\[ +q_{n+N}=q_n, \ \ (\mbox{periodic}\ \mbox{condition}); \quad \mbox{and}\ +q_{-n}=q_n, \ \ (\mbox{even}\ \mbox{condition}); +\] +$h={1\over N}$, and +\begin{eqnarray*} +& & N\tan{\pi \over N}< \om 3,\\ +& & 3\tan{\pi \over 3}< \om < \infty, \ \ \mbox{for}\ N=3. \\ +& & \e\in[0,\e_1),\ \alpha\ (>0), \ \be\ (>0) \ +\mbox{are}\ \mbox{constants.} +\end{eqnarray*} +This is a $2(M+1)$ dimensional system, where +\[ +M=N/2,\ \ (N\ \mbox{even}); \quad \mbox{and}\ +M=(N-1)/2, \ \ (N\ \mbox{odd}). +\] +This system is a finite-difference discretization of the perturbed NLS +(\ref{spnls}). + +For a general system of ordinary differential equations, +Kelley \cite{Kel67} used the Perron's method to give a very short proof +of the classical unstable, stable, and center manifold theorem. This +paper is a good starting point of reading upon Perron's method. In the book +\cite{HPS77}, Hadamard's method is mainly employed. This book is an +excellent starting point for a comprehensive reading on invariant +manifolds. + +There have been more and more invariant manifold results for infinite +dimensional systems \cite{LW97b}. For the employment of Perron's +method, we refer the readers to \cite{CLL91}. For the employment of +Hadamard's method, we refer the readers to \cite{BLZ98} \cite{BLZ99} +\cite{BLZ00} which are terribly long papers. + + + + + + + + + + +\clearpage{} +\clearpage{}\chapter{Homoclinic Orbits} + +In terms of proving the existence of a homoclinic orbit, +the most common tool is the so-called Melnikov integral method \cite{Mel63} +\cite{Arn64}. This method was subsequently developed by Holmes and Marsden +\cite{GH83}, and most recently by Wiggins \cite{Wig88}. For partial +differential equations, this method was mainly developed by Li et al. +\cite{LMSW96} \cite{Li01b} \cite{LM97} \cite{Li02f}. + +There are two derivations for the Melnikov integrals. One is the +so-called geometric argument \cite{GH83} \cite{Wig88} \cite{LM97} +\cite{LMSW96} \cite{Li01b}. +The other is the so-called Liapunov-Schmitt argument \cite{CHMP80} +\cite{CH82}. The Liapunov-Schmitt argument is a fixed-point type argument +which directly leads to the existence of a homoclinic orbit. The condition +for the existence of a fixed point is the Melnikov integral. +The geometric argument is a signed distance argument which applies +to more general situations than the Liapunov-Schmitt argument. It turns out +that the geometric argument is a much more powerful machinary than the +Liapunov-Schmitt argument. In particular, the geometric argument can handle +geometric singular perturbation problems. I shall also mention an +interesting derivation in \cite{Arn64}. + +In establishing the existence of homoclinic orbits in high dimensions, one +often needs other tools besides the Melnikov analysis. For example, when +studying orbits +homoclinic to fixed points created through resonances in +($n \geq 4$)-dimensional near-integrable systems, one often needs tools +like Fenichel fibers, as presented in previous chapter, to set up geometric +measurements for locating such +homoclinci orbits. Such homoclinic orbits often have a geometric singular +perturbation nature. In such cases, the Liapunov-Schmitt argument can not +be applied. For such works on finite dimensional systems, see for +example \cite{Kov92a} \cite{Kov92b} \cite{LM97}. +For such works on infinite dimensional systems, see for example +\cite{LMSW96} \cite{Li01b}. + +\section{Silnikov Homoclinic Orbits in NLS Under Regular +Perturbations \label{horrnls}} + +We continue from section \ref{rpsec} and consider the regularly perturbed +nonlinear Schr\"odinger (NLS) equation (\ref{rpnls}). The following theorem +was proved in \cite{LMSW96}. +\begin{theorem} +There exists a $\e_0 > 0$, such that for any $\e \in (0, \e_0)$, there +exists a codimension 1 surface in the external parameter space +$(\alpha,\beta, \om) \in \RR^+\times \RR^+\times \RR^+$ where $\om \in +(\frac{1}{2}, 1)$, and $\al \om < \be$. For any +$(\alpha ,\beta, \omega)$ on the codimension 1 +surface, the regularly perturbed nonlinear Schr\"odinger equation +(\ref{rpnls}) possesses a symmetric pair of Silnikov homoclinic orbits +asymptotic +to a saddle $Q_\epsilon$. The codimension 1 surface has the approximate +representation given by $\al = 1/\k(\om)$, where $\k(\om)$ is plotted +in Figure \ref{kappa}. +\label{rhorbit} +\end{theorem} +The proof of this theorem is easier than that given in later sections. +\begin{figure} +\includegraphics{e-fig7-1.eps} +\caption{The graph of $\k(\om)$.} +\label{kappa} +\end{figure} +To prove the theorem, one starts from the invariant plane +\[ +\Pi=\{ q\mid \ \partial_x q=0 \}. +\] +On $\Pi$, there is a saddle $Q_\e =\sqrt{I} e^{i\th}$ to which +the symmetric pair of Silnikov homoclinic orbits will be asymptotic to, where +\begin{equation} +I=\omega^2-\epsilon \frac{1}{2\omega}\sqrt{\beta^2-\alpha^2\omega^2}+\cdots , +\quad \cos \theta =\frac{\alpha \sqrt{I}}{\beta}, \quad \theta \in +(0,\frac{\pi}{2}). +\label{Qec} +\end{equation} +Its eigenvalues are +\begin{equation} +\la_n^\pm = -\e [\al +\xi_n n^2]\pm 2 \sqrt{(\frac{n^2}{2} + +\om^2-I)(3I -\om^2 -\frac{n^2}{2} )}\ , +\label{Qev} +\end{equation} +where $n=0,1,2, \cdots $, $\om \in (\frac{1}{2}, 1)$, $\xi_n = 1$ when +$n \leq N$, $\xi_n = 8n^{-2}$ when $n>N$, for some fixed large $N$, +and $I$ is given in +(\ref{Qec}). The crucial points to notice are: (1). only $\la_0^+$ and +$\la_1^+$ have positive real parts, $\mbox{Re}\{ \la_0^+\} < \mbox{Re}\{ +\la_1^+\} $; (2). all the other eigenvalues have negative real parts among +which the absolute value of $\mbox{Re}\{ \la_2^+\}=\mbox{Re}\{ \la_2^-\}$ +is the smallest; (3). $|\mbox{Re}\{ \la_2^+\}| < \mbox{Re}\{ \la_0^+\}$. +Actually, items (2) and (3) are the main characteristics of Silnikov +homoclinic orbits. + +The unstable manifold $W^u(Q_\e)$ of $Q_\e$ has a fiber representation +given by Theorem \ref{fiberthm}. The Melnikov measurement measures the +signed distance between $W^u(Q_\e)$ and the center-stable manifold +$W^{cs}_\e$ proved in Thoerem \ref{Persthm}. By virtue of the Fiber Theorem +\ref{fiberthm}, one can show that, to the leading order in +$\e$, the signed distance is given by the Melnikov integral +\begin{eqnarray*} +M &=& \int^{+\infty}_{-\infty}\int^{2\pi}_0 + [\partial_qF_1(q_0(t))(\hat{\partial}^2_xq_0(t)-\alpha +q_0(t)+\beta ) \\ +& & \quad \quad + \partial_{\bar{q}}F_1(q_0(t)) +(\hat{\partial}^2_x\overline{q_0(t)}-\alpha +\overline{q_0(t)}+\beta )]dxdt, +\end{eqnarray*} +where $q_0(t)$ is given in section \ref{1DCNSE}, equation (\ref{4.13}); +and $\partial_qF_1$ and $\partial_{\bar{q}}F_1$ are +given in section \ref{MVNLS}, equation (\ref{6.5}). The zero of the +signed distance implies the existence of an orbit in $W^u(Q_\e)\cap +W^{cs}_\e$. The stable manifold $W^s(Q_\e)$ of $Q_\e$ is a codimension +1 submanifold in $W^{cs}_\e$. To locate a homoclinic orbit, one needs +to set up a second measurement measuring the signed distance between +the orbit in $W^u(Q_\e)\cap W^{cs}_\e$ and $W^s(Q_\e)$ inside $W^{cs}_\e$. +To set up this signed distance, first one can rather easily track +the (perturbed) orbit by an unperturbed orbit to an $\O (\e)$ neighborhood +of $\Pi$, then one needs to prove the size of $W^s(Q_\e)$ to be $\O (\e^\nu)$ +($\nu <1$) with normal form transform. To the leading order in +$\e$, the zero of the second signed distance is given by +\[ +\be \cos \ga = \frac {\al \om (\Dl \ga )} {2 \sin \frac {\Dl \ga }{2}} \ , +\] +where $\Dl \ga = -4 \vth_0$ and $\vth_0$ is given in (\ref{4.13}). +To the leading order in $\e$, the common zero of the two second signed +distances satisfies $\al = 1/\k(\om)$, where $\k(\om)$ is plotted +in Figure \ref{kappa}. Then the claim of the theorem is proved by virtue +of the implicit function theorem. For rigorous details, see \cite{LMSW96}. +In the singular perturbation case as discussed in next section, +the rigorous details are given in later sections. + +\section{Silnikov Homoclinic Orbits in NLS Under Singular +Perturbations \label{horsnls}} + +We continue from section \ref{spsec} and consider the singularly perturbed +nonlinear Schr\"odinger (NLS) equation (\ref{spnls}). The following theorem +was proved in \cite{Li01b}. +\begin{theorem} +There exists a $\e_0 > 0$, such that for any $\e \in (0, \e_0)$, there +exists a codimension 1 surface in the external parameter space +$(\alpha,\beta, \om) \in \RR^+\times \RR^+\times +\RR^+$ where $\om \in (\frac{1}{2}, 1)/S$, $S$ is a finite subset, and +$\al \om < \be$. For any $(\alpha ,\beta, \omega)$ on the codimension 1 +surface, the singularly perturbed nonlinear Schr\"odinger equation +(\ref{spnls}) possesses a symmetric pair of Silnikov homoclinic orbits +asymptotic +to a saddle $Q_\epsilon$. The codimension 1 surface has the approximate +representation given by $\al = 1/\k(\om)$, where $\k(\om)$ is plotted +in Figure \ref{kappa}. +\label{shorbit} +\end{theorem} +In this singular perturbation case, the persistence and fiber theorems +are given in section \ref{spsec}, Theorems \ref{CSM} and \ref{UFT}. The +normal form transform for proving the size estimate of the stable manifold +$W^s(Q_\e)$ is still achievable. The proof of the theorem is also completed +through two measurements: the Melnikov measurement and the second measurement. + +\section{The Melnikov Measurement} + +\subsection{Dynamics on an Invariant Plane} + +The 2D subspace $\Pi$, +\begin{equation}\label{sPi} \Pi=\{ q\mid \ \partial_xq=0\},\end{equation} +is an invariant plane under the flow (\ref{spnls}). The +governing equation in $\Pi$ is +\begin{equation} i\dot{q}=2[|q|^2-\om^2]q+i\epsilon [-\alpha +q+\beta],\label{Pie1}\end{equation} +where $\cdot =\frac{d}{dt}\ $. +Dynamics of this equation is shown in Figure \ref{figPi}. +Interesting dynamics is created through resonance in the neighborhood of the +circle $S_\om$: +\begin{equation}\label{srcl} S_\om=\{ q\in \Pi \mid \ |q|=\om \}.\end{equation} +When $\epsilon =0$, $S_\om$ consists of fixed points. To explore the +dynamics in this neighborhood better, one can make a series of +changes of coordinates. Let $q=\sqrt{I}e^{i\theta}$, +then \eqref{Pie1} can be rewritten as +\begin{align} \dot{I}&= \epsilon ( -2\alpha I+2\beta \sqrt{I}\cos +\theta )\ ,\label{Ithe1}\\ +\dot{\theta} &=-2(I-\om^2)-\epsilon \beta \frac{\sin +\theta}{\sqrt{I}}\ .\label{Ithe2}\end{align} +There are three fixed points: +\begin{enumerate}\item The focus $O_\epsilon$ in the neighborhood of +the origin, +\begin{equation}\begin{cases} I=\epsilon ^2\frac{\beta^2}{4\omega^4}+\cdots ,\\ +\cos \theta=\frac{\alpha \sqrt{I}}{\beta}, & \theta \in \left( +0,\frac{\pi}{2}\right).\end{cases}\label{Oec}\end{equation} +Its eigenvalues are +\begin{equation}\label{Oee} \mu_{1,2}=\pm +i\sqrt{4(\omega^2-I)^2-4\epsilon \sqrt{I}\beta \sin \theta}-\epsilon +\alpha,\end{equation} +where $I$ and $\theta$ are given in \eqref{Oec}. +\item The focus $P_\epsilon$ in the neighborhood of $S_\omega$ \eqref{srcl}, +\begin{equation}\label{Pec}\begin{cases} I=\omega^2+\epsilon +\frac{1}{2\omega}\sqrt{\beta^2-\alpha^2\omega^2}+\cdots ,\\ +\cos \theta=\frac{\alpha \sqrt{I}}{\beta},& \theta \in \left( +-\frac{\pi}{2}, 0\right).\end{cases}\end{equation} +Its eigenvalues are +\begin{equation}\label{Pee}\mu_{1,2}=\pm +i\sqrt{\epsilon}\sqrt{-4\sqrt{I}\beta \sin \theta+\epsilon \left( +\frac{\beta \sin +\theta}{\sqrt{I}}\right)^2}-\epsilon \alpha,\end{equation} +where $I$ and $\theta$ are given in \eqref{Pec}. +\item The saddle $Q_\epsilon$ in the neighborhood of $S_\omega$ \eqref{srcl}, +\begin{equation}\label{Qecs}\begin{cases}I=\omega^2-\epsilon +\frac{1}{2\omega}\sqrt{\beta^2-\alpha^2\omega^2}+\cdots ,\\ +\cos \theta =\frac{\alpha \sqrt{I}}{\beta},&\theta \in \left( +0,\frac{\pi}{2}\right).\end{cases}\end{equation} +Its eigenvalues are +\begin{equation}\label{Qee} \mu_{1,2}=\pm +\sqrt{\epsilon}\sqrt{4\sqrt{I}\beta\sin \theta -\epsilon \left( +\frac{\beta \sin +\theta}{\sqrt{I}}\right)^2}-\epsilon \alpha,\end{equation} +where $I$ and $\theta $ are given in \eqref{Qecs}. +\end{enumerate} +\begin{figure} +\includegraphics{fig2.eps} +\caption{Dynamics on the invariant plane $\Pi$.} +\label{figPi} +\end{figure} +\nid +Now focus our attention to order $\sqrt{\epsilon}$ neighborhood of +$S_\omega$ \eqref{srcl} and let +\begin{equation*}J=I-\omega^2,\quad J=\sqrt{\epsilon}j,\quad +\tau=\sqrt{\epsilon}t,\end{equation*} +we have +\begin{align}\label{je1} j'&= 2\left[ -\alpha +(\omega^2+\sqrt{\epsilon}j) +\beta +\sqrt{\omega^2+\sqrt{\epsilon}j}\cos \theta \right],\\ +\label{je2}\theta '&= -2j-\sqrt{\epsilon}\beta \frac{\sin +\theta}{\sqrt{\omega^2+\sqrt{\epsilon}j}},\end{align} +where $'=\frac{d}{d\tau}\ $. To leading order, we get +\begin{align}\label{je3} j'&=2[-\alpha \omega^2+\beta \omega \cos \theta]\ ,\\ +\label{je4}\theta' & =-2j\ .\end{align} +There are two fixed points which are the counterparts of $P_\epsilon$ +and $Q_\epsilon$ \eqref{Pec} and \eqref{Qecs}: +\begin{enumerate}\item The center $P_*$, +\begin{equation}\label{Pc} j=0,\quad \cos \theta=\frac{\alpha +\omega}{\beta},\quad \theta \in +\left(-\frac{\pi}{2},0\right).\end{equation} +Its eigenvalues are +\begin{equation}\label{Pe} \mu_{1,2}=\pm +i2\sqrt{\omega}(\beta^2-\alpha^2\omega^2)^{\frac{1}{4}}.\end{equation} +\item The saddle $Q_*$, +\begin{equation}\label{Qc} j=0,\quad \cos \theta=\frac{\alpha +\omega}{\beta},\quad \theta \in \left( +0,\frac{\pi}{2}\right).\end{equation} Its eigenvalues are +\begin{equation}\label{Qe} \mu_{1,2}=\pm +2\sqrt{\omega}(\beta^2-\alpha ^2\omega^2)^{\frac{1}{4}}.\end{equation} +\end{enumerate} +\nid +In fact, \eqref{je3} and \eqref{je4} form a Hamiltonian system with the +Hamiltonian +\begin{equation}\label{fham} \mathcal{H}=j^2+2\omega (-\alpha \omega +\theta+\beta \sin \theta).\end{equation} +Connecting to $Q_*$ is a fish-like singular level set of +$\mathcal{H}$, which intersects the axis $j=0$ at $Q_*$ and +$\hat{Q}=(0,\hat{\theta})$, +\begin{equation}\label{head} \alpha \omega +(\hat{\theta}-\theta_*)=\beta (\sin \hat{\theta}-\sin \theta_*),\quad +\hat{\theta}\in (-\frac{3\pi}{2}, 0),\end{equation} +where $\theta_*$ is given in \eqref{Qc}. See Figure \ref{fish} for an +illustration of the dynamics of \eqref{je1}-\eqref{je4}. +\begin{figure} +\includegraphics{fig3.eps} +\caption{The fish-like dynamics in the neighborhood of the resonant circle +$S_\om$.} +\label{fish} +\end{figure} +\nid +For later use, we define +a piece of each of the stable and unstable manifolds of $Q_*$, +\begin{equation*}j=\phi_*^u(\theta),\quad j=\phi^s_*(\theta),\quad +\theta\in [\hat{\theta}+\hat{\delta}, \theta_*+2\pi],\end{equation*} +for some small $\hat{\delta}>0$, and +\begin{equation}\begin{split}\label{cur} +\phi^u_*(\theta)&=-\frac{\theta-\theta_*}{|\theta +-\theta_*|}\sqrt{2\omega [\alpha \omega +(\theta-\theta_*)-\beta (\sin \theta-\sin \theta_*)]},\\ + \phi^s_*(\theta )&=-\phi^u_*(\theta).\end{split}\end{equation} +$\phi^u_*(\theta)$ and $\phi^s_*(\theta)$ perturb smoothly in $\theta +$ and $\sqrt{\epsilon}$ into $\phi^u_{\sqrt{\epsilon}}$ and +$\phi^s_{\sqrt{\epsilon}}$ for +\eqref{je1} and \eqref{je2}. + +The homoclinic orbit to be located will take off from $Q_\epsilon$ +along its unstable curve, flies away from and returns to $\Pi$, lands +near the stable curve of $Q_\epsilon$ and approaches $Q_\epsilon$ spirally. + +\subsection{A Signed Distance} + +Let $p$ be any point on $\phi^u_{\sqrt{\e}}$ \eqref{cur} which is the +unstable curve of $Q_\epsilon $ in $\Pi$ (\ref{sPi}). Let $q_\epsilon (0)$ +and $q_0(0)$ be any two points on the unstable fibers +$\mathcal{F}_p^+\mid_\epsilon$ and +$\mathcal{F}_p^+\mid_{\epsilon=0}$, with the same $\xi^+_1$ +coordinate. See Figure \ref{mms} for an illustration. +\begin{figure} +\includegraphics{fig4.eps} +\caption{The Melnikov measurement.} +\label{mms} +\end{figure} +By the Unstable Fiber Theorem \ref{UFT}, $\mathcal{F}^+_p$ is +$C^1$ in $\epsilon $ for $\epsilon \in [0,\epsilon_0)$, $\epsilon_0 +>0$, thus +\begin{equation*}\| q_\epsilon (0)-q_0(0)\|_{n+8}\leq C\epsilon.\end{equation*} +The key point here is that $\mathcal{F}^+_p\subset H^s$ for any fixed +$s\geq 1$. By Remark~\ref{reop}, the evolution operator of the +perturbed NLS equation \eqref{spnls} $S^t$ is $C^1$ in $\epsilon$ as a +map from $H^{n+4}$ to $H^n$ for any fixed $n\geq 1$, $\epsilon \in +[0,\epsilon_0)$, $\epsilon _0>0$. Also $S^t$ is a $C^1$ map on $H^n$ +for any fixed $t>0$, $n\geq 1$. Thus +\begin{equation*}\| q_\epsilon (T)-q_0(T)\| _{n+4}=\| S^T(q_\epsilon +(0))-S^T(q_0(0))\|_{n+4}\leq C_1\epsilon,\end{equation*} +where $T>0$ is large enough so that +\begin{equation*}q_0(T)\in W^{cs}_{n+4}\mid_{\epsilon=0}.\end{equation*} +Our goal is to determine when $q_\epsilon (T)\in W^{cs}_n$ through +Melnikov measurement. Let $q_\epsilon (T)$ and $q_0(T)$ have the coordinate +expressions +\begin{equation}\label{sme} q_\epsilon +(T)=(\xi^{+,\epsilon}_1,v_\epsilon),\quad +q_0(T)=(\xi^{+,0}_1,v_0).\end{equation} +Let $\tilde{q}_\epsilon (T)$ be the unique point on +$W^{cs}_{n+4}$, +which has the same $v$-coordinate as $q_\epsilon (T)$, +\begin{equation*}\tilde{q}_\epsilon +(T)=(\tilde{\xi}^{+,\epsilon}_1,v_\epsilon)\in +W^{cs}_{n+4}.\end{equation*} +By the Center-Stable Manifold Theorem, at points in the subset +$W^{cs}_{n+4}$, $W^{cs}_n$ is $C^1$ smooth in $\epsilon$ for $\epsilon +\in [0,\epsilon_0)$, $\epsilon_0 >0$, thus +\begin{equation}\label{sme1} \| q_\epsilon (T)-\tilde{q}_\epsilon +(T)\|_n\leq C_2\epsilon .\end{equation} +Also our goal now is to determine when the signed distance +\begin{equation*}\xi^{+,\epsilon}_1-\tilde{\xi}^{+,\epsilon}_1,\end{equation*} +is zero through Melnikov measurement. Equivalently, one can define +the signed distance +\begin{equation*}\begin{split} d_1&=\langle \nabla +F_1(q_0(T)),q_\epsilon (T)-\tilde{q}_\epsilon (T)\rangle\\ +&= \partial_qF_1(q_0(T)) (q_\epsilon (T)-\tilde{q}_\epsilon (T)) ++ \partial_{\bar{q}}F_1(q_0(T)) (q_\epsilon +(T)-\tilde{q}_\epsilon (T))^-, \end{split}\end{equation*} +where $\nabla F_1$ is given in (\ref{mvnls1}), +and $q_0(t)$ is the homoclinic orbit given in (\ref{4.13}). +In fact, $q_\epsilon (t)$, $\tilde{q}_\epsilon (t)$, +$q_0(t)\in H^n$, for any fixed $n\geq 1$. The rest of the derivation +for Melnikov integrals is completely standard. For details, see +\cite{LMSW96} \cite{LM97}. +\begin{equation}\label{Meln1} d_1=\epsilon M_1+o(\epsilon ), +\end{equation} +where +\begin{equation*}\begin{split} +M_1&=\int^{+\infty}_{-\infty}\int^{2\pi}_0 +[\partial_qF_1(q_0(t))(\partial^2_xq_0(t)-\alpha +q_0(t)+\beta)\\ +&\quad + +\partial_{\bar{q}}F_1(q_0(t))(\partial^2_x\overline{q_0(t)}-\alpha +\overline{q_0(t)}+\beta )]dxdt,\end{split}\end{equation*} +where again $q_0(t)$ is given in (\ref{4.13}), and +$\partial_qF_1$, and $\partial_{\bar{q}}F_1$ are +given in (\ref{mvnls1}). + +\section{The Second Measurement} + +\subsection{The Size of the Stable Manifold of $Q_\e$} + +Assume that the Melnikov measurement is successful, that is, the +orbit $q_\epsilon(t)$ is in the intersection of the unstable manifold +of $Q_\e$ and the center-stable manifold $W^u(Q_\epsilon )\cap W^{cs}_n$. +The goal of the second measurement is to determine when +$q_\epsilon (t)$ is also in the codimension 2 stable manifold of $Q_\e$, +$W^s_n(Q_\epsilon)$, in $H^n$. The existence of $W^s_n(Q_\epsilon)$ +follows from standard stable manifold theorem. +$W^s_n(Q_\epsilon)$ can be visualized as a codimension 1 wall in +$W^{cs}_n$ with base curve $\phi^u_{\sqrt{\e}}$ \eqref{cur} in $\Pi$. +\begin{theorem} \cite{Li01b} The size of $W^s_n(Q_\epsilon)$ off +$\Pi$ is of order $\mathcal{O}(\sqrt{\epsilon})$ for $\omega \in +\left(\frac{1}{2},1 \right)/S$, where $S$ is a finite +subset.\label{heiwall} +\end{theorem} +Starting from the system +\eqref{nc1}-\eqref{nc3}, one can only get the size of +$W^s_n(Q_\epsilon)$ off $\Pi$ +to be $\mathcal{O}(\epsilon)$ from the standard stable manifold theorem. +This is not enough for the second measurement. An estimate of order +$\mathcal{O}(\epsilon^\k)$, +$\k<1$ can be achieved if the quadratic term $\mathcal{N}_2$ +\eqref{wnc5} in \eqref{nc3} can be removed through a normal form +transformation. Such a normal form transformation has been achieved +\cite{Li01b}, see also the later section on Normal Form Transforms. + +\subsection{An Estimate} + +From the explicit expression of $q_0(t)$ (\ref{4.13}), we know that +$q_0(t)$ approaches $\Pi$ at the +rate $\mathcal{O}(e^{- \sqrt{4\omega^2-1}t})$. Thus +\begin{equation}\label{sarg1} \text{distance} \left\{ +q_0(T+\frac{1}{\mu}|\ln \epsilon |), \Pi\right\}T)$ be a time such that +\begin{equation} \| q_\epsilon (t)-q_0(t)\| _n\leq \tilde{C}_2 \epsilon +|\ln \epsilon |^2,\end{equation} +for all $t\in [T,T_1]$, where $\tilde{C}_2 =\tilde{C}_2(T)$ is +independent of $\epsilon$. + From \eqref{sme1}, such a $T_1$ exists. The proof will be completed +through a continuation argument. For $t\in [T,T_1]$, +\begin{equation}\begin{split}\label{smcd}& +(|\xi^{+,0}_1(t)|+|\xi^{-,0}_1(t)|)+\|h^0(t)\|_n\leq +C_3re^{-\frac{1}{2}\mu +(t-T)},\quad \ |J^0(t)|\leq C_4\sqrt{\epsilon},\\ +& |J^\epsilon (t)|\leq |J^0(t)|+|J^\epsilon(t)-J^0(t)| \leq +|J^0(t)|+\tilde{C}_2\epsilon |\ln +\epsilon |^2\leq C_5\sqrt{\epsilon},\\ +&(|\xi^{+,\epsilon}_1(t)|+|\xi^{-,\epsilon}_1(t)|)+\|h^\epsilon +(t)\|_n\leq C_3re^{-\frac{1}{2}\mu (t-T)} +\tilde{C}_2\epsilon +|\ln \epsilon |^2,\end{split}\end{equation} +where $r$ is small. Since actually $q_\epsilon (t)$, $q_0(t)\in H^n$ +for any fixed $n\geq 1$, by Theorem~\ref{CSM}, +\begin{equation} |\xi^{+,\epsilon}_1(t)-\xi^{+,0}_1(t)|\leq C_6\| +v_\epsilon (t)-v_0(t)\|_n+C_7\epsilon ,\end{equation} +whenever $v_\epsilon (t)$, $v_0(t)\in E_{n+4}(r)$, where $v_\epsilon +(T)=v_\epsilon $ and $v_0(T)=v_0$ are defined in \eqref{sme}. Thus +we only need to estimate $\| v_\epsilon (t)-v_0(t)\|_n$. From +\eqref{cspe2}, we have for $t\in [T, T_1]$ that +\begin{equation}\label{smeq1} +v(t)=e^{A(t-T)}v(T)+\int^t_Te^{A(t-\tau)}\tilde{F}(\tau +)d\tau.\end{equation} +Let $\Delta v(t)=v_\epsilon (t)-v_0(t)$. Then +\begin{equation}\begin{split} \Delta +v(t)&=[e^{A(t-T)}-e^{A\mid_{\epsilon +=0}(t-T)}]v_0(T)+e^{A(t-T)}\Delta v(T)\\ +&\quad + \int^t_Te^{A(t-\tau )}[\tilde{F}(\tau )-\tilde{F}(\tau +)|_{\epsilon =0}]d\tau \\ +&\quad + \int^t_T[e^{A(t-\tau )}-e^{A\mid_{\epsilon =0}(t-\tau +)}]\ \tilde{F}(\tau )|_{\epsilon =0}d\tau.\end{split}\end{equation} +By the condition \eqref{smcd}, we have for $t\in [T, T_1]$ that +\begin{equation}\| +\tilde{F}(t)-\tilde{F}(t)|_{\epsilon=0}\|_n\leq +[C_8\sqrt{\epsilon}+C_9 re^{-\frac{1}{2}\mu (t-T)} ]\epsilon |\ln +\epsilon |^2.\end{equation} +Then +\begin{equation} \| \Delta v(t)\|_n\leq C_{10}\epsilon +(t-T)+C_{11}r\epsilon |\ln \epsilon +|^2+C_{12}\sqrt{\epsilon}(t-T)^2\epsilon +|\ln \epsilon |^2.\end{equation} +Thus by the continuation argument, for $t\in [T, T+\frac{1}{\mu}|\ln +\epsilon |]$, there is a constant $\hat{C}_1=\hat{C}_1(T)$, +\begin{equation} \| \Delta v(t)\|_n\leq \hat{C}_1\epsilon |\ln +\epsilon |^2.\end{equation} +Q.E.D. + +By Lemma~\ref{lesm} and estimate \eqref{sarg1}, +\begin{equation}\label{sarg2} \text{distance} \left\{ q_\epsilon +( T+\frac{1}{\mu}|\ln \epsilon |), \Pi\right\}<\tilde{C}\epsilon +|\ln \epsilon |^2. +\end{equation} +By Theorem \ref{heiwall}, the height of the wall $W^s_n(Q_\epsilon)$ +off $\Pi$ is larger than the distance between $q_\epsilon +( T+\frac{1}{\mu}|\ln \epsilon |)$ and $\Pi$. Thus, if $q_\epsilon +( T+\frac{1}{\mu}|\ln \epsilon |)$ can move from one side of the wall +$W^s_n(Q_\epsilon)$ to the other side, then by continuity $q_\epsilon +( T+\frac{1}{\mu}|\ln \epsilon |)$ has to be on the wall +$W^s_n(Q_\epsilon)$ at some values of the parameters. + +\subsection{Another Signed Distance} + +Recall the fish-like singular level set given by $\mathcal{H}$ +\eqref{fham}, the width of the fish is of order +$\mathcal{O}(\sqrt{\epsilon })$, and the length of the fish is of +order $\mathcal{O}(1)$. Notice also that $q_0(t)$ has a phase shift +\begin{equation} \theta^0_1=\theta^0( T+\frac{1}{\mu}|\ln +\epsilon |)-\theta^0(0).\end{equation} +For fixed $\beta$, changing $\alpha$ can induce $\mathcal{O}(1)$ +change in the length of the fish, $\mathcal{O}(\sqrt{\epsilon})$ +change +in $\theta^0_1$, and $\mathcal{O}(1)$ change in $\theta^0(0)$. See +Figure~\ref{supsm} for an illustration. +The leading order signed distance from $q_\epsilon ( +T+\frac{1}{\mu}|\ln \epsilon |)$ to $W^s_n(Q_\epsilon )$ can be +defined +as +\begin{equation}\begin{split} \tilde{d} +&=\mathcal{H}(j_0,\theta^0(0))-\mathcal{H}(j_0,\theta^0(0)+\theta^0_1)\\ +&=2\omega \left [\alpha \omega \theta^0_1+\beta [\sin \theta^0(0)-\sin +(\theta^0(0)+\theta^0_1)]\right ],\end{split}\label{dtd}\end{equation} +where $\mathcal{H}$ is given in \eqref{fham}. The common zero of +$M_1$ \eqref{Meln1} and $\tilde{d}$ and the implicit function +theorem +imply the existence of a homoclinic orbit asymptotic to $Q_\epsilon$. +The common roots to $M_1$ and $\tilde{d}$ are given by $\al = 1/\k(\om)$, +where $\k(\om)$ is plotted in Figure \ref{kappa}. +\begin{figure} +\includegraphics{fig5.eps} +\caption{The second measurement.} +\label{supsm} +\end{figure} + +\section{Silnikov Homoclinic Orbits in Vector NLS Under Perturbations} + +In recent years, novel results have been obtained on the solutions of +the vector nonlinear Schr\"odinger equations \cite{AOT99} \cite{AOT00} +\cite{YT01}. Abundant ordinary integrable results have been carried through +\cite{WF00} \cite{FSW00}, including linear stability calculations +\cite{FMMW00}. Specifically, the vector nonlinear +Schr\"odinger equations can be written as +\begin{eqnarray*} +& & ip_t + p_{xx} + \frac{1}{2} (|p|^2 + \chi |q|^2) p = 0 , \\ +& & iq_t + q_{xx} + \frac{1}{2} (\chi |p|^2 + |q|^2) q = 0 , +\end{eqnarray*} +where $p$ and $q$ are complex valued functions of the two real variables +$t$ and $x$, and $\chi$ is a positive constant. These equations describe +the evolution of two orthogonal pulse envelopes in birefringent optical +fibers \cite{Men87} \cite{Men89}, with industrial applications in fiber +communication systems \cite{HK95} and all-optical switching devices +\cite{Isl92}. For linearly birefringent fibers \cite{Men87}, $\chi =2/3$. +For elliptically birefringent fibers, $\chi$ can take other positive +values \cite{Men89}. When $\chi = 1$, these equations are first +shown to be integrable by S. Manakov \cite{Man74}, and thus called Manakov +equations. When $\chi$ is not 1 or 0, these equations are +non-integrable. Propelled by the industrial applications, extensive +mathematical studies on the vector nonlinear Schr\"odinger equations +have been conducted. Like the scalar nonlinear Schr\"odinger equation, the +vector nonlinear Schr\"odinger equations also possess figure eight +structures in their phase space. Consider the singularly perturbed vector +nonlinear Schr\"odinger equations, +\begin{eqnarray} +& & ip_t + p_{xx} + \frac{1}{2} [(|p|^2 + |q|^2)-\om^2] p = +i \e [ p_{xx} -\al p - \be ]\ , \label{pnls1}\\ +& & iq_t + q_{xx} + \frac{1}{2} [(|p|^2 + |q|^2)-\om^2] q = +i \e [ q_{xx} -\al q - \be ]\ , \label{pnls2} +\end{eqnarray} +where $p(t,x)$ and $q(t,x)$ are subject to periodic boundary condition +of period $2\pi$, and are even in $x$, i.e. +\[ +p(t,x + 2 \pi) = p(t,x)\ , \ \ p(t,-x) = p(t,x)\ , +\] +\[ +q(t,x + 2 \pi) = q(t,x)\ , \ \ q(t,-x) = q(t,x)\ , +\] +$\om \in (1,2)$, $\al > 0$ and $\be$ are real constants, and +$\e > 0$ is the perturbation parameter. We have +\begin{theorem}[\cite{Li02d}] +There exists a $\e_0 > 0$, such that +for any $\e \in (0, \e_0)$, there exists +a codimension 1 surface in the space of $(\alpha,\beta, \om) \in +\RR^+\times \RR^+\times \RR^+$ where +$\om \in (1, 2)/S$, $S$ is a finite subset, and +$\al \om < \sqrt{2} \be$. For any $(\alpha ,\beta, \omega)$ on the +codimension 1 surface, the singularly perturbed vector nonlinear +Schr\"odinger equations (\ref{pnls1})-(\ref{pnls2}) possesses a +homoclinic orbit asymptotic to a saddle +$Q_\epsilon$. This orbit is also the homoclinic orbit +for the singularly perturbed scalar nonlinear Schr\"odinger equation +studied in last section, and is the only one asymptotic +to the saddle $Q_\epsilon$ for the singularly perturbed +vector nonlinear Schr\"odinger equations (\ref{pnls1})-(\ref{pnls2}). +The codimension 1 surface has the +approximate representation given by $\al = 1/\k(\om)$, where $\k(\om)$ +is plotted in Figure \ref{kappa}. +\end{theorem} + +\section{Silnikov Homoclinic Orbits in Discrete +NLS Under Perturbations \label{hordnls}} + +We continue from section \ref{GOV} and consider the perturbed discrete +nonlinear Schr\"odinger equation (\ref{PDNLS}). The following theorem +was proved in \cite{LM97}. + +Denote by $\Sg_N\ (N\geq 7)$ the external parameter space, +\begin{eqnarray*} +\Sg_N&=&\bigg\{ (\om,\al,\be)\ \bigg | \ \om \in (N\tan{\pi \over N}, + N\tan{2\pi \over N}),\\ + & &\al\in (0,\al_0), \be\in (0,\be_0); \\ + & &\mbox{where}\ \al_0\ \mbox{and}\ \be_0\ \mbox{are}\ \mbox{any} + \ \mbox{fixed}\ \mbox{positive}\ \mbox{numbers}. \bigg\} +\end{eqnarray*} +\begin{theorem} +For any $N$ ($7\leq N<\infty$), there exists a positive number $\e_0$, +such that for any $\e \in (0,\e_0)$, there exists a +codimension $1$ surface $E_\e$ in $\Sg_N$; for any +external parameters ($\om,\al,\be$) on $E_\e$, there exists a +homoclinic orbit asymptotic to a saddle $Q_\e$. +The codimension $1$ surface $E_\e$ has the approximate expression +$\al=1/\k$, where $\k=\k(\om;N)$ +is shown in Fig.\ref{nkappa}. +\label{dhorbit} +\end{theorem} +In the cases ($3\leq N \leq 6$), $\k$ is always negative. For +$N \geq 7$, $\k$ +can be positive as shown in Fig.\ref{nkappa}. When $N$ is even and +$\geq 7$, there is in fact a symmetric pair of homoclinic orbits asymptotic to +a fixed point $Q_\e$ at the same values of the external parameters; +since for even $N$, we have the symmetry: +If $q_n=f(n,t)$ solves (\ref{PDNLS}), +then $q_n=f(n+N/2,t)$ also solves (\ref{PDNLS}). When $N$ is odd +and $\geq 7$, the study can not guarantee that two homoclinic orbits +exist at the same value of the external parameters. +\begin{figure} +\includegraphics{e-fig7-3.eps} +\includegraphics{e-fig7-4.eps} +\caption{The graph of $\k(\om;N)$.} +\label{nkappa} +\end{figure} + +\section{Comments on DSII Under Perturbations} + +We continue from section \ref{invds}. Under both regular and singular +perturbations, the rigorous Melnikov measurement can be established +\cite{Li02b}. It turns out that only local well-posedness is necessary +for rigorously setting up the Melnikov measurement, thanks to the fact +that the unperturbed homoclinic orbits given in section \ref{dsex} are +classical solutions. Thus the Melnikov integrals given in section \ref{mids} +indeed rigorously measure signed distances. For details, see \cite{Li02b}. + +The obstacle toward proving the existence of homoclinic +orbits comes from a technical difficulty in solving a linear system +to get the normal form for proving the size estimate of the stable +manifold of the saddle. For details, see \cite{Li02b}. + +\section{Normal Form Transforms} + +Consider the singularly perturbed Davey-Stewartson II equation in section +\ref{invds}, +\begin{equation} +\left \{ \begin{array}{l} iq_t=\Upsilon q+ [2(|q|^2-\omega^2)+ +u_y ]q +i\epsilon [\Delta q-\alpha q+\beta ]\ , \cr +\Delta u = -4\partial_y |q|^2 \ . \cr \end{array} \right. +\label{PDS} +\end{equation} +Let +\[ +q(t,x,y)=[\rho(t) +f(t,x,y)]e^{i\theta(t)},\quad \lag f \rag = 0, +\] +where $\lag \ , \ \rag$ denotes spatial mean. Let +\[ +I = \lag |q|^2 \rag = \rho^2 + \lag |f|^2 \rag , \quad J=I-\om^2. +\] +In terms of the new variables ($J, \th, f$), Equation (\ref{PDS}) can +be rewritten as +\begin{eqnarray} +\dot{J} &=& \epsilon \bigg [ -2\alpha(J+\omega^2)+2\beta\sqrt{J+\omega^2} +\cos \theta \bigg ] +\epsilon \R_2^J, \label{cc1} \\ +\dot{\th} &=& -2J - \epsilon \beta \frac {\sin \theta}{\sqrt{J+\omega^2}} ++\R_2^\th, \label{cc2} \\ +f_t &=& L_\epsilon f+V_\epsilon f-i \N_2 -i \N_3, \label{cc3} +\end{eqnarray} +where +\begin{eqnarray*} +L_\epsilon f &=& -i\Upsilon f+\epsilon (\Delta-\alpha )f-2i\omega^2 +\Delta^{-1}\Upsilon (f+\bar f), \\ +\N_2 &=& 2\om \bigg [ \Delta^{-1}\Upsilon |f|^2 ++f\Delta^{-1} \Upsilon (f+\bar f) +- \lag f\Delta^{-1}\Upsilon (f+\bar f)\rag \bigg ], +\end{eqnarray*} +and $\R_2^J$, $\R_2^\th$, $V_\epsilon f$, and $\N_3$ are higher order +terms. It is the quadratic term $\N_2$ that blocks the size estimate +of the stable manifold of the saddle \cite{Li02b}.Thus, our goal is to find a normal form transform $g = f + K(f,f)$ +where $K$ is a bilinear form, that transforms the equation +\[ +f_t=L_\epsilon f-i\N_2, +\] +into an equation with a cubic nonlinearity +\[ +g_t=L_\epsilon g+{\mathcal O}(\| g\|^3_s),\quad +(s\geq 2), +\] +where $L_\epsilon$ is given in (\ref{cc3}). In terms of Fourier transforms, +\[ +f=\sum_{k\neq 0}\hat{f}(k)e^{ik\cdot \xi },\quad +\bar{f}=\sum_{k\neq 0}\overline{\hat{f}(-k)}e^{ik\cdot \xi }\ , +\] +where $k=(k_1,k_2) \in \ZZ^2$, $\xi = (\k_1x, \k_2y)$. The terms in +$\N_2$ can be written as +\[ +\Dl^{-1}\Upsilon |f|^2 = \frac {1}{2} \sum_{k+\ell \neq 0} a(k+\ell) +\bigg [ \hat{f}(k)\overline{\hat{f}(-\ell )}+\hat{f}(\ell ) +\overline{\hat{f}(-k)} \bigg ] e^{i(k+ \ell ) \cdot \xi }\ , +\] +\[ +f\Delta^{-1} \Upsilon f - \lag f\Delta^{-1}\Upsilon f \rag += \frac {1}{2} \sum_{k+\ell \neq 0} [a(k)+a( \ell )] \hat{f}(k)\hat{f}(\ell) +e^{i(k+ \ell ) \cdot \xi }\ , +\] +\[ +f\Delta^{-1} \Upsilon \bar{f} - \lag f\Delta^{-1}\Upsilon \bar{f} \rag += \frac {1}{2} \sum_{k+\ell \neq 0} \bigg [ a(\ell)\hat{f}(k) +\overline{\hat{f}(-\ell )}+a(k)\hat{f}(\ell) +\overline{\hat{f}(-k)}\bigg ] e^{i(k+ \ell ) \cdot \xi }\ , +\] +where +\[ +a(k) = \frac {k_1^2 \k_1^2 - k_2^2 \k_2^2}{k_1^2 \k_1^2 + k_2^2 \k_2^2}\ . +\] +We will search for a normal form transform of the +general form, +\[ +g=f+K(f,f), +\] +where +\begin{eqnarray*} +K(f,f) &=& \sum_{k+\ell\neq 0}\left[ +\hat{K}_1(k,\ell)\hat{f}(k) +\hat{f}(\ell)+\hat{K}_2(k,\ell)\hat{f}(k)\overline{\hat{f}(-\ell)}\right.\\ +& &\quad \left.+\hat{K}_2(\ell,k)\overline{\hat{f}(-k)}\hat{f}(\ell)+ +\hat{K}_3(k,\ell)\overline{\hat{f}(-k)}\overline{\hat{f}(-\ell)}\right] +e^{i(k+\ell)x}, +\end{eqnarray*} +where $\hat{K}_j(k,\ell)$, $(j=1,2,3)$ are the unknown coefficients to +be determined, and $\hat{K}_j(k,\ell)=\hat{K}_j(\ell,k)$, $(j=1,3)$. +To eliminate the quadratic terms, we first need to set +\[ +iL_\epsilon K(f,f)-iK(L_\epsilon +f,f)-iK(f,L_\epsilon f)=\N_2, +\] +which takes the explicit form: +\begin{eqnarray} +& &(\sigma_1+i\sigma)\hat{K}_1(k,\ell)+B(\ell)\hat{K}_2(k,\ell)+B(k) +\hat{K}_2(\ell,k)\nonumber \\ +& & \quad +B(k+\ell)\overline{\hat{K}_3(k,\ell)}=\frac{1}{2\om} +[B(k)+B(\ell)],\label{nore1}\\ +& &-B(\ell)\hat{K}_1(k,\ell)+(\sigma_2+i\sigma)\hat{K}_2(k,\ell)+ B(k+\ell) +\overline{\hat{K}_2(\ell,k)}\nonumber \\ +& & \quad +B(k)\hat{K}_3(k,\ell)=\frac{1}{2\om} +[B(k+\ell)+B(\ell)],\label{nore2}\\ +& &-B(k)\hat{K}_1(k,\ell)+B(k+\ell) +\overline{\hat{K}_2(k,\ell)}+(\sigma_3+i\sigma)\hat{K}_2(\ell,k)\nonumber \\ +& & \quad +B(\ell)\hat{K}_3(k,\ell)= \frac{1}{2\om} +[B(k+\ell)+B(k)],\label{nore3}\\ +& &B(k+\ell) \overline{\hat{K}_1(k,\ell)}-B(k)\hat{K}_2(k,\ell) +-B(\ell)\hat{K}_2(\ell,k)\nonumber \\ +& & \quad +(\sigma _4+i\sigma )\hat{K}_3(k,\ell)=0,\label{nore4} +\end{eqnarray} +where $B(k)=2\om^2 a(k)$, and +\begin{eqnarray*} +& & \sg = \e \bigg [ \al -2(k_1\ell_1\k_1^2+k_2\ell_2\k_2^2)\bigg ]\ , \\ +& & \sg_1= 2(k_2\ell_2\k_2^2-k_1\ell_1\k_1^2) +B(k+\ell)-B(k)-B(\ell)\ , \\ +& & \sg_2= 2[(k_2+\ell_2)\ell_2\k_2^2-(k_1+\ell_1)\ell_1\k_1^2] ++B(k+\ell)-B(k)+B(\ell)\ , \\ +& & \sg_3= 2[(k_2+\ell_2)k_2\k_2^2-(k_1+\ell_1)k_1\k_1^2] ++B(k+\ell)+B(k)-B(\ell)\ , \\ +& & \sg_4= 2[(k_2^2+k_2\ell_2+\ell_2^2)\k_2^2-(k_1^2+k_1\ell_1+\ell_1^2) +\k_1^2] +B(k+\ell)+B(k)+B(\ell)\ . +\end{eqnarray*} +Since these coefficients are even in $(k,\ell)$, we will search for +even solutions, i.e. +\[ +\hat{K}_j(-k,-\ell)=\hat{K}_j(k,\ell),\quad j=1,2,3. +\] + +The technical difficulty in the normal form transform comes from not +being able to answer the following two questions in solving the linear +system (\ref{nore1})-(\ref{nore4}): +\begin{enumerate} +\item Is it true that for all $k, \ell \in \ZZ^2/\{ 0\}$, there +is a solution ? +\item What is the asymptotic behavior of the solution as $k$ and/or +$\ell \ra \infty$ ? In particular, is the asymptotic behavior like +$k^{-m}$ and/or $\ell^{-m}$ ($m\geq 0$) ? +\end{enumerate} + +\nid +Setting $\pa_y = 0$ in (\ref{PDS}), the singularly perturbed +Davey-Stewartson II equation reduces to the singularly perturbed +nonlinear Schr\"odinger (NLS) equation (\ref{spnls}) in section \ref{spsec}. +Then the above two questions can be answered \cite{Li01b}. For the +regularly perturbed nonlinear Schr\"odinger (NLS) equation (\ref{rpnls}) +in section \ref{rpsec}, the answer to the above two questions is even +simplier \cite{LMSW96} \cite{Li01b}. + +\section{Transversal Homoclinic Orbits in a Periodically Perturbed +SG \label{PPSGE}} + +Transversal homoclinic orbits in continuous systems often appear +in two types of systems: (1). periodic systems where the Poincar\'e +period map has a transversal homoclinic orbit, (2). autonomous +systems where the homoclinic orbit is asymptotic to a hyperbolic +limit cycle. + +Consider the periodically perturbed sine-Gordon (SG) equation, +\begin{equation} +u_{tt}=c^2 u_{xx}+\sin u+\epsilon [-a u_t+u^3 \chi(\| u\|)\cos t], +\label{PSG} +\end{equation} +where +\[ +\chi(\| u\|)=\left\{ \begin{array}{ll} 1, & \| +u\| \leq M,\\ 0, & \| u\| \geq 2M,\end{array}\right. +\] +for $M<\| u\| <2M$, $\chi (\| u\|)$ is a smooth bump function, +under odd periodic boundary condition, +\[ +u(x+2\pi ,t)=u(x,t),\quad +u(x,t)=-u(x,t), +\] +$\frac{1}{4}0$, $\epsilon$ is a small +perturbation parameter. +\begin{theorem}[\cite{LMSW96}, \cite{SZ00}] There exists an +interval $I\subset \RR^{+}$ such that for any $a\in I$, there exists a +transversal homoclinic orbit +$u=\xi (x,t)$ asymptotic to $0$ in $H^{1}$. +\end{theorem} + +\section{Transversal Homoclinic Orbits in a Derivative NLS \label{ADNS}} + +Consider the derivative nonlinear Schr\"odinger equation, +\begin{equation} +i q_t = q_{xx} + 2 |q|^2 q +i \e \bigg [ (\frac{9}{16}-|q|^2 )q +\mu +|\hat{\pa}_x q|^2 \bar{q} \bigg ]\ , \label{derNLS} +\end{equation} +where $q$ is a complex-valued function of two real variables $t$ and $x$, +$\e > 0$ is the perturbation parameter, $\mu$ is a real constant, and +$\hat{\pa}_x $ is a bounded Fourier multiplier, +\[ +\hat{\pa}_x q = -\sum_{k=1}^K k \tq_k \sin kx\ , \quad +\mbox{for} \ q = \sum_{k=0}^\infty \tq_k \cos kx\ , +\] +and some fixed $K$. Periodic boundary condition +and even constraint are imposed, +\[ +q(t,x+2\pi ) = q(t,x)\ , \ \ q(t,-x)=q(t,x) \ . +\] +\begin{theorem}[\cite{Li02a}] +There exists a $\e_0 > 0$, such that +for any $\e \in (0, \e_0)$, and $|\mu | > 5.8$, +there exist two transversal homoclinic orbits asymptotic to +the limit cycle $q_c = \frac{3}{4} \exp \{ -i [ \frac{9}{8} t + \ga ]\}$. +\label{thmdns} +\end{theorem} + + + + + + + + + + + + + + + + + + +\clearpage{} +\clearpage{}\chapter{Existence of Chaos} + +The importance of homoclinic orbits with respect to chaotic dynamics was +first realized by Poincar{\'e} \cite{Poi99}. In 1961, Smale constructed +the well-known horseshoe in the neighborhood of a transversal homoclinic +orbit \cite{Sma61} \cite{Sma65} \cite{Sma67}. In particular, +Smale's theorem implies Birkhoff's theorem on the existence of a +sequence of structurely stable periodic orbits in the neighborhood +of a transversal homoclinic orbit \cite{Bir12}. In 1984 and 1988 \cite{Pal84} +\cite{Pal88}, Palmer gave a beautiful proof of Smale's theorem using a +shadowing lemma. Later, this proof was generalized to infinite dimensions +by Steinlein and Walther \cite{SW89} \cite{SW90} and Henry \cite{Hen94}. +In 1967, Silnikov proved Smale's theorem for autonomous systems in finite +dimensions using a fixed point argument \cite{Sil67b}. In 1996, Palmer proved +Smale's theorem for autonomous systems in finite dimensions using shadowing +lemma \cite{CKP95} \cite{Pal96}. In 2002, Li proved Smale's theorem for +autonomous systems in infinite dimensions using shadowing lemma \cite{Li02a}. + +For nontransversal homoclinic orbits, the most well-known type which leads +to the existence of Smale horseshoes is the so-called Silnikov homoclinic +orbit \cite{Sil65} \cite{Sil67a} \cite{Sil70} \cite{Den89} \cite{Den93}. +Existence of Silnikov homoclinic orbits and new constructions of Smale +horseshoes for concrete nonlinear wave systems have been established +in finite dimensions \cite{LM97} \cite{LW97} and in infinite dimensions +\cite{LMSW96} \cite{Li99a} \cite{Li01b}. + +\section{Horseshoes and Chaos} + +\subsection{Equivariant Smooth Linearization} + +Linearization has been a popular topics in dynamical systems. It asks the +question whether or not one can transform a nonlinear system into its +linearized system at a fixed point, in a small neighborhood of the fixed point. +For a sample of references, see \cite{Har63} \cite{Sel85}. Here we +specifically consider the singularly perturbed nonlinear Schr\"odinger (NLS) +equation (\ref{spnls}) in sections \ref{spsec} and \ref{horsnls}. +The symmetric pair of Silnikov homoclinic orbits is asymptotic to the +saddle $Q_\e =\sqrt{I} e^{i\th}$, where +\begin{equation} +I=\omega^2-\epsilon \frac{1}{2\omega}\sqrt{\beta^2-\alpha^2\omega^2}+\cdots , +\quad \cos \theta =\frac{\alpha \sqrt{I}}{\beta}, \quad \theta \in +(0,\frac{\pi}{2}). +\label{sQec} +\end{equation} +Its eigenvalues are +\begin{equation} +\la_n^\pm = -\e [\al +n^2]\pm 2 \sqrt{(\frac{n^2}{2} +\om^2-I)(3I -\om^2 - +\frac{n^2}{2} )}\ , +\label{sQev} +\end{equation} +where $n=0,1,2, \cdots $, $\om \in (\frac{1}{2}, 1)$, and $I$ is given in +(\ref{sQec}). +In conducting linearization, the crucial factor is the so-callled nonresonance +conditions. +\begin{lemma}[\cite{Li99a} \cite{Li02c}] +For any fixed $\e \in (0,\e_0)$, let $E_\e$ be the codimension 1 surface in +the external parameter space, on which the symmetric pair of Silnikov +homoclinic orbits are supported (cf: Theorem \ref{shorbit}). For almost +every $(\al,\be,\om)\in E_\e$, the eigenvalues $\la_n^\pm$ (\ref{Qev}) +satisfy the nonresonance condition of Siegel type: There exists a natural +number $s$ such that for any integer $n \geq 2$, +\[ +\bigg | \La_n - \sum_{j=1}^r \La_{l_j} \bigg | \geq 1/r^s\ , +\] +for all $r =2,3, \cdots, n$ and all $l_1, l_2, \cdots, l_r \in \ZZ$, +where $\La_n = \la_n^+$ for $n \geq 0$, and $\La_n = \la_{-n-1}^-$ for $n <0$. +\end{lemma} +Thus, in a neighborhood of $Q_\e$, the singularly perturbed NLS (\ref{spnls}) +is analytically equivalent to its linearization at $Q_\e$ \cite{Nik86}. +In terms of eigenvector basis, (\ref{spnls}) can be rewritten as +\begin{eqnarray} +\dot{x} &=& -a x - b y +\N_x(\vec{Q}), \nonumber \\ +\dot{y} &=& b x - a y +\N_y(\vec{Q}), \nonumber \\ +\dot{z}_1 &=& \ga_1 z_1 +\N_{z_1}(\vec{Q}), \label{nfeq} \\ +\dot{z}_2 &=& \ga_2 z_2 +\N_{z_2}(\vec{Q}), \nonumber \\ +\dot{Q} &=& LQ +\N_{Q}(\vec{Q}); \nonumber +\end{eqnarray} +where $a= -\mbox{Re}\{\la^+_2\}$, $b=\mbox{Im}\{\la^+_2\}$, +$\ga_1 = \la^+_0$, $\ga_2 = \la^+_1$; $\N$'s vanish identically +in a neighborhood $\Om$ of $\vec{Q} =0$, $\vec{Q} = (x,y,z_1,z_2,Q)$, $Q$ +is associated with the rest of eigenvalues, $L$ is given as +\[ +LQ = -i Q_{\z\z} -2 i [(2|Q_\e|^2-\om^2)Q+Q^2_\e\bar{Q}]+ +\e [-\al Q+Q_{\z\z}]\ , +\] +and $Q_\e$ is given in (\ref{sQec}). + +\subsection{Conley-Moser Conditions} + +We continue from last section. Denote by $h_k$ ($k=1,2$) the symmetric +pair of Silnikov homoclinic orbits. +The symmetry $\sg$ of half spatial period shifting has the new +representation in terms of the new coordinates +\begin{equation} +\sg \circ (x,y,z_1,z_2,Q) = (x,y,z_1,-z_2,\sg \circ Q). +\label{symm} +\end{equation} +\begin{definition} +The Poincar\'e section $\Sg_0$ is defined by the constraints: +\begin{eqnarray*} +& & y=0,\ \eta \exp \{ -2 \pi a/b \} < x < \eta; \\ +& & 0 < z_1 <\eta,\ -\eta < z_2 <\eta, \ \|Q\| < \eta; +\end{eqnarray*} +where $\eta$ is a small parameter. +\end{definition} +The horseshoes are going to be constructed on this Poincar\'e section. +\begin{definition} +The auxiliary Poincar\'e section $\Sg_1$ is defined by the constraints: +\begin{eqnarray*} +& & z_1 =\eta ,\quad -\eta < z_2 <\eta , \\ +& & \sqrt{x^2+y^2} < \eta , \quad \|Q\| < \eta . +\end{eqnarray*} +\end{definition} +Maps from a Poincar\'e section to a Poincar\'e section are induced by the +flow. Let $\vQ^0$ and $\vQ^1$ be the coordinates on $\Sg_0$ and $\Sg_1$ +respectively, then the map $P_0^1$ from $\Sg_0$ to $\Sg_1$ has the +simple expression: +\begin{eqnarray*} +x^1 &=& \bigg (\frac{z_1^0}{\eta}\bigg )^{\frac{a}{\ga_1}}x^0 \cos \bigg +[ \frac{b}{\ga_1}\ln \frac{\eta}{z_1^0} \bigg ]\ , \\ +y^1 &=& \bigg (\frac{z_1^0}{\eta}\bigg )^{\frac{a}{\ga_1}}x^0 \sin \bigg +[ \frac{b}{\ga_1}\ln \frac{\eta}{z_1^0} \bigg ]\ , \\ +z_2^1 &=& \bigg (\frac{\eta}{z_1^0}\bigg )^{\frac{\ga_2}{\ga_1}}z_2^0\ , \\ +Q^1 &=& e^{t_0 L} Q^0\ . +\end{eqnarray*} +Let $\vQ_*^0$ and $\vQ_*^1$ be the intersection points of the homoclinic +orbit $h_1$ with $\overline{\Sg_0}$ and $\Sg_1$ respectively, and let +$\vtQ^0 = \vQ^0 - \vQ^0_*$, and $\vtQ^1 = \vQ^1 - \vQ^1_*$. One can define +slabs in $\Sg_0$. +\begin{definition} +For sufficiently large natural number $l$, we define slab $S_l$ in $\Sg_0$ +as follows: +\begin{eqnarray*} +S_l &\equiv& \bigg \{ \vQ \in \Sg_0 \ \bigg | \ \eta \exp \{ -\ga_1 +(t_{0,2(l+1)}-{\pi \over 2b})\} \leq \\ +& & \tz^0_1(\vQ) \leq \eta \exp \{ -\ga_1 (t_{0,2l}-{\pi \over 2b})\}, \\ +& & |\tx^0(\vQ)| \leq \eta \exp \{ -{1\over 2}a \ t_{0,2l} \}, \\ +& & |\tz^1_2(P^1_0(\vQ))| \leq \eta \exp \{ -{1\over 2}a \ t_{0,2l} \}, \\ +& & \|\tQ^1(P^1_0(\vQ))\| \leq \eta \exp \{ -{1\over 2}a \ t_{0,2l} \} +\bigg \}, +\end{eqnarray*} +where +\[ +t_{0,l} = {1 \over b} [l\pi -\varphi_1] +o(1)\ , +\] +as $l \ra +\infty$, are the time that label the fixed points of the +Poincar\'e map $P$ from $\Sg_0$ to $\Sg_0$ \cite{Li99a} \cite{Li02c}, +and the notations $\tx^0(\vQ)$, $\tz^1_2(P^1_0(\vQ))$, etc. denote the +$\tx^0$ coordinate of the point $\vQ$, the $\tz^1_2$ coordinate of the +point $P^1_0(\vQ)$, etc.. +\label{dfslab} +\end{definition} +$S_l$ is defined so that it includes two fixed points $p^+_l$ and $p^-_l$ +of $P$. +Let $S_{l,\sg}= \sg \circ S_l$ where the symmetry $\sg$ is defined in +(\ref{symm}). We need to define a larger slab $\hS_l$ such that +$S_l \cup S_{l,\sg} \subset \hS_l$. +\begin{definition} +The larger slab $\hS_l$ is defined as +\begin{eqnarray*} +\hS_l &=& \bigg \{ \vQ \in \Sg_0 \ \bigg | \ \eta \exp \{ -\ga_1 +(t_{0,2(l+1)} -{\pi \over 2b})\} \leq \\ +& & z^0_1(\vQ) \leq \eta \exp \{ -\ga_1 (t_{0,2l} -{\pi \over 2b})\},\\ +& & |x^0(\vQ)-x^0_*| \leq \eta \exp \{ -{1\over 2} a\ t_{0,2l}\}, \\ +& & |z^1_2(P^1_0(\vQ))| \leq |z^1_{2,*}| + \eta \exp \{ -{1\over 2} +a\ t_{0,2l}\},\\ +& & \|Q^1(P^1_0(\vQ))\| \leq \eta \exp \{ -{1\over 2}a\ t_{0,2l}\} \bigg \}, +\end{eqnarray*} +where $z^1_{2,*}$ is the $z^1_2$-coordinate of $\vQ^1_*$. +\end{definition} +\begin{definition} +In the coordinate system $\{ \tx^0,\tz^0_1,\tz^0_2,\tQ^0 \}$, the stable +boundary of $\hS_l$, denoted by $\pa_s \hS_l$, is defined to be the boundary +of $\hS_l$ along ($\tx^0,\tQ^0$)-directions, and the unstable +boundary of $\hS_l$, denoted by $\pa_u \hS_l$, is defined to be the boundary +of $\hS_l$ along ($\tz^0_1,\tz^0_2$)-directions. A stable slice $V$ in +$\hS_l$ is a subset of $\hS_l$, defined as the region swept out through +homeomorphically moving and deforming +$\pa_s \hS_l$ in such a way that the part +\[ +\pa_s \hS_l \cap \pa_u \hS_l +\] +of $\pa_s \hS_l$ only moves and deforms inside $\pa_u \hS_l$. The +new boundary obtained through such moving and deforming of +$\pa_s \hS_l$ is called the stable boundary of $V$, which is denoted by +$\pa_s V$. The rest of the boundary of $V$ is called its unstable +boundary, which is denoted by $\pa_u V$. An unstable slice of $\hS_l$, +denoted by $H$, is defined similarly. +\end{definition} +As shown in \cite{Li99a} and \cite{Li02c}, under certain generic +assumptions, when $l$ is sufficiently +large, $P(S_l)$ and $P(S_{l,\sg})$ intersect $\hS_l$ into four disjoint +stable slices +$\{ V_1,V_2 \}$ and $\{ V_{-1},V_{-2}\}$ in $\hS_l$. $V_j$'s +($j=1,2,-1,-2$) do not intersect $\pa_s\hS_l$; moreover, +\begin{equation} +\pa_s V_i \subset P(\pa_sS_l), (i=1,2);\ +\pa_s V_i \subset P(\pa_sS_{l,\sg}), (i=-1,-2). +\label{bdc} +\end{equation} +Let +\begin{equation} +H_j = P^{-1}(V_j),\ \ (j=1,2,-1,-2), \label{defhs} +\end{equation} +where and for the rest of this article, $P^{-1}$ denotes preimage of $P$. +Then $H_j$ ($j=1,2,-1,-2$) are unstable slices. More importantly, the +Conley-Moser conditions are satisfied. Specifically, +Conley-Moser conditions are: +\framebox[1.8in][l]{Conley-Moser condition (i):} +\[ +\left \{ \begin{array}{l} V_j = P(H_j), \\ \pa_sV_j = P(\pa_sH_j), \ +\ (j=1,2,-1,-2) \\ \pa_uV_j = P(\pa_uH_j). +\end{array}\right. +\] +\framebox[1.8in][l]{Conley-Moser condition (ii):} There exists a constant +$0< \nu <1$, such that for any stable slice $V \subset V_j\ \ (j=1,2,-1,-2)$, +the diameter decay relation +\[ +d(\tilde{V}) \leq \nu d(V) +\] +holds, where $d(\cdot)$ denotes the diameter \cite{Li99a}, +and $\tilde{V}=P(V\cap H_k), \ \ (k=1,2,-1,-2)$; +for any unstable slice $H \subset H_j\ \ (j=1,2,-1,-2)$, the diameter +decay relation +\[ +d(\tilde{H}) \leq \nu d(H) +\] +holds, where $\tilde{H}=P^{-1}(H\cap V_k), \ \ (k=1,2,-1,-2)$. + +The Conley-Moser conditions are sufficient conditions for establishing +the topological conjugacy between the Poincare map $P$ restricted to +a Cantor set in $\Sg_0$, and the shift automorphism on symbols. It also +display the horseshoe nature of the intersection of $S_l$ and $S_{l,\sg}$ with +their images $P(S_l)$ and $P(S_{l,\sg})$. + +\subsection{Shift Automorphism} + +Let $\W$ be a set which consists of elements of the doubly infinite +sequence form: +\[ +a =(\cdot \cdot \cdot a_{-2} a_{-1} a_0, a_1 a_2 \cdot \cdot \cdot ), +\] +where $a_k \in \{ 1, 2, -1, -2\}$; $k\in \ZZ$. We introduce a topology in $\W$ +by taking as neighborhood basis of +\[ +a^* =( \cdot \cdot \cdot a^*_{-2} a^*_{-1} a^*_0, a^*_1 a^*_2 +\cdot \cdot \cdot ), +\] +the set +\[ +W_j = \bigg \{ a\in \W \ \bigg | \ a_k=a^*_k\ (|k|0$ sufficiently small there +exists $\dl >0$ such that every $\dl$ pseudo-orbit in $S$ has a +unique $\e$-shadowing orbit. +\label{shal} +\end{theorem} +The proof of this theorem by Palmer \cite{Pal88} is overall a +fixed point argument with the help of Green functions for linear maps. + +Let $y_0$ be a transversal homoclinic point asymptotic to a saddle $x_0$ +of a $C^1$ diffeomorphism $f : \RR^n \mapsto \RR^n$. Then the set +\[ +S= \{ x_0 \} \cup \{ f^j(y_0): j \in Z\} +\] +is hyperbolic. Denote by $A_0$ and $A_1$ the two orbit segments of +length $2m+1$ +\[ +A_0= \{ x_0,x_0, \cdots, x_0 \}\ , \quad +A_1= \{ f^{-m}(y_0), f^{-m+1}(y_0), \cdots, f^{m-1}(y_0), f^{m}(y_0) \} \ . +\] +Let +\[ +a=(\cdots, a_{-1},a_0, a_1, \cdots ) \ , +\] +where $a_j \in \{ 0,1\}$, be any doubly infinite binary sequence. Let +$A$ be the doubly infinite sequence of points in $S$, associated with +$a$ +\[ +A=\{ \cdots, A_{a_{-1}},A_{a_0}, A_{a_1}, \cdots \} \ . +\] +When $m$ is sufficiently large, $A$ is a $\dl$ pseudo-orbit in $S$. +By the shadowing lemma (Theorem \ref{shal}), there is a unique +$\e$-shadowing orbit that shadows $A$. In this manner, Palmer \cite{Pal88} +gave a beautiful proof of Smale's horseshoe theorem. +\begin{definition} +Denote by $\Sg$ the set of doubly infinite binary sequences +\[ +a=(\cdots, a_{-1},a_0, a_1, \cdots ) \ , +\] +where $a_j \in \{ 0,1\}$. We give the set $\{ 0,1\}$ the discrete topology +and $\Sg$ the product topology. The Bernoulli shift $\chi$ is defined by +\[ +[\chi(a)]_j = a_{j+1}\ . +\] +\end{definition} +\begin{theorem} +Let $y_0$ be a transversal homoclinic point asymptotic to a saddle $x_0$ +of a $C^1$ diffeomorphism $f : \RR^n \mapsto \RR^n$. Then there is a +homeomorphism $\phi$ of $\Sg$ onto a compact subset of $\RR^n$ which is +invariant under $f$ and such that when $m$ is sufficiently large +\[ +f^{2m+1} \circ \phi = \phi \circ \chi\ , +\] +that is, the action of $f^{2m+1}$ on $\phi(\Sg)$ is topologically conjugate +to the action of $\chi$ on $\Sg$. +\end{theorem} +Here one can define $\phi(a)$ to be the point on the shadowing orbit that +shadows the midpoint of the orbit segment $A_{a_0}$, which is either $x_0$ +or $y_0$. The topological conjugacy can be easily verified. For details, +see \cite{Pal88}. +Other references can be found in \cite{Pal84} \cite{Zen95}. + +\section{Shadowing Lemma and Chaos in Infinite-D Periodic Systems} + +An infinite-dimensional periodic system defined in a Banach space $X$ +can be written in the general form +\[ +\dot{x} = F(x,t)\ , +\] +where $x \in X$, and $F(x,t)$ is periodic in $t$. Let $f$ be the +Poincar\'e period map. When $f$ is a $C^1$ map which needs not to be +invertible, shadowing lemma and symbolic dynamics around a transversal +homoclinic orbit can both be established \cite{SW89} \cite{SW90} \cite{Hen94}. +Other references can be found in \cite{HL86} \cite{CLP89} \cite{Zen97} +\cite{Bla86}. There exists also a work on horseshoe construction without +shadowing lemma for +sinusoidally forced vibrations of buckled beam \cite{HM81}. + +\section{Periodically Perturbed Sine-Gordon (SG) Equation} + +We continue from section \ref{PPSGE}, and use the notations in +section \ref{Palmer}. For the periodically perturbed sine-Gordon +equation (\ref{PSG}), the Poincar\'e period map is a $C^1$ diffeomorphism +in $H^1$. As a corollary of the result in last section, we have the theorem +on the existence of chaos. +\begin{theorem} There is an integer $m$ and a homeomorphism $\phi$ +of $\Sigma$ onto a compact Cantor subset $\Lambda $ of +$H^{1}$. $\Lambda$ is invariant under the Poincar\'e period-$2\pi$ map $P$ of +the periodically perturbed sine-Gordon equation (\ref{PSG}). The +action of $P^{2m+1}$ on $\Lambda$ is topologically conjugate to +the action of $\chi$ on +$\Sigma: P^{2m+1} \circ \phi =\phi \circ +\chi$. That is, the following diagram commutes: +\[ +\begin{array}{ccc} +\Sg &\maprightu{\phi} & \Lambda\\ +\mapdownl{\chi} & & \mapdownr{P^{2m+1}}\\ +\Sg & \maprightd{\phi} & \Lambda +\end{array} +\] +\end{theorem} + +\section{Shadowing Lemma and Chaos in Finite-D Autonomous Systems} + +A finite-dimensional autonomous system can be written in the +general form +\[ +\dot{x} = F(x)\ , +\] +where $x \in \RR^n$. In this case, a transversal homoclinic orbit +can be an orbit asymptotic to a normally hyperbolic limit cycle. +That is, it is an orbit in the intersection of the stable and +unstable manifolds of a normally hyperbolic limit cycle. Instead of +the Poincar\'e period map as for periodic system, one may want +to introduce the so-called Poincar\'e return map which is a map induced by +the flow on a codimension 1 section which is transversal to the limit +cycle. Unfortunately, such a map is not even well-defined in the +neighborhood of the homoclinic orbit. This poses a challenging +difficulty in extending the arguments as in the case of a Poincar\'e +period map. In 1996, Palmer \cite{Pal96} completed a proof of a +shadowing lemma and existence of chaos using Newton's method. It will +be difficult to extend this method to infinite dimensions, since it +used heavily differentiations in time. +Other references can be found in \cite{CKP95} \cite{CKP97} \cite{Sil67b} +\cite{FS77}. + +\section{Shadowing Lemma and Chaos in Infinite-D Autonomous Systems} + +An infinite-dimensional autonomous system defined in a Banach space $X$ +can be written in the general form +\[ +\dot{x} = F(x)\ , +\] +where $x \in X$. In 2002, the author \cite{Li02a} completed a proof of a +shadowing lemma and existence of chaos using Fenichel's persistence of +normally hyperbolic invariant manifold idea. The setup is as follows, +\begin{itemize} +\item {\bf Assumption (A1):} There exist a hyperbolic limit cycle +$S$ and a transversal homoclinic orbit $\xi$ +asymptotic to $S$. As curves, $S$ and $\xi$ are $C^{3}$. +\item {\bf Assumption (A2):} The Fenichel fiber theorem is valid at $S$. +That is, there exist a family of unstable Fenichel fibers +$\{ {\mathcal F}^{u}(q): \ q \in S\}$ and a family of stable Fenichel +fibers $\{ {\mathcal F}^{s}(q): \ q\in S\}$. For each fixed $q\in S$, +${\mathcal F}^{u}(q)$ and ${\mathcal F}^{s}(q)$ are $C^{3}$ submanifolds. +${\mathcal F}^{u}(q)$ and ${\mathcal F}^{s}(q)$ are $C^{2}$ in $q, +\forall q\in S$. The unions $\bigcup_{q\in S}{\mathcal F}^{u}(q)$ and +$\bigcup_{q\in S}{\mathcal F}^{s}(q)$ are the unstable and stable +manifolds of $S$. Both families are invariant, i.e. +\[ +F^{t}({\mathcal F}^{u}(q))\subset +{\mathcal F}^{u}(F^{t}(q)), \forall\ t \leq 0, q\in S, +\] +\[ +F^{t}({\mathcal F}^{s}(q))\subset {\mathcal F}^{s}(F^{t}(q)), +\forall\ t \geq 0, q \in S, +\] +where $F^{t}$ is the evolution operator. There are positive constants +$\k$ and $\widehat{C}$ such that $\forall q\in S$, $\forall +q^{-}\in {\mathcal F}^{u}(q)$ and $\forall q^{+}\in +{\mathcal F}^{s}(q)$, +\[ +\| F^{t}(q^{-})-F^{t}(q)\| \leq +\widehat{C}e^{\k t}\| q^{-}-q\|, \forall \ t \leq 0\ , +\] +\[ +\| F^{t}(q^{+})-F^{t}(q)\| \leq \widehat{C}e^{-\k t}\| q^{+}-q\|, +\forall \ t \geq 0\ . +\] +\item {\bf Assumption (A3):} $F^{t}(q)$ is $C^{0}$ in $t$, for +$t\in (-\infty ,\infty)$, $q\in X$. For any fixed $t\in +(-\infty ,\infty )$, $F^{t}(q)$ is a $C^{2}$ diffeomorphism on +$X$. +\end{itemize} +\begin{remark} +Notice that we do not assume that as functions of time, $S$ and $\xi$ +are $C^3$ , and we only assume that as curves, $S$ and $\xi$ +are $C^3$. +\end{remark} + +Under the above setup, a shadowing lemma and existence of chaos can be +proved \cite{Li02a}. Another crucial element in the argument is the +establishment of a $\la$-lemma (also called inclination lemma) which +will be discussed in a later section. + +\section{A Derivative Nonlinear Schr\"odinger Equation} + +We continue from section \ref{ADNS}, and consider the derivative nonlinear +Schr\"odinger equation (\ref{derNLS}). The transversal homoclinic orbit +given in Theorem \ref{thmdns} is a classical solution. Thus, Assumption (A1) +is valid. Assumption (A2) follows from the standard arguments +in \cite{LW97} \cite{LMSW96} \cite{Li01b}. Since the perturbation +in (\ref{derNLS}) is bounded, Assumption (A3) follows from standard +arguments. Thus there exists chaos in the derivative nonlinear +Schr\"odinger equation (\ref{derNLS}) \cite{Li02a}. + +\section{$\la$-Lemma} + +$\la$-Lemma (inclination lemma) has been utilized in proving many +significant theorems in dynamical system \cite{PM82} \cite{Pal83} +\cite{Wal87}. +Here is another example, in \cite{Li02a}, it is shown that $\la$-Lemma +is crucial for proving a shadowing lemma in an infinite dimensional +autonomous system. Below we give a brief introduction to $\la$-Lemma +\cite{PM82}. + +Let $f$ be a $C^r$ ($r \geq 1$) diffeomorphism in $\RR^m$ with $0$ as +a saddle. Let $E^s$ and $E^u$ be the stable and unstable subspaces. +Through changing coordinates, one can obtain that for the local +stable and unstable manifolds, +\[ +W^s_{\mbox{loc}} \subset E^s\ , \quad W^u_{\mbox{loc}} \subset E^u\ . +\] +Let $B^s \subset W^s_{\mbox{loc}}$ and $B^u \subset W^u_{\mbox{loc}}$ +be balls, and $V=B^s \times B^u$. Let $q \in W^s_{\mbox{loc}}/\{0\}$, +and $D^u$ be a disc of the same dimension as $B^u$, which is transversal to +$W^s_{\mbox{loc}}$ at $q$. +\begin{lemma} [The $\la$-Lemma] +Let $D_n^u$ be the connected component of $f^n(D^u) \cap V$ to which +$f^n(q)$ belongs. Given $\e > 0$, there exists $n_0$ such that if $n > n_0$, +then $D_n^u$ is $\e$ $C^1$-close to $B^u$. +\end{lemma} +The proof of the lemma is not complicated, yet nontrivial \cite{PM82}. +In different situations, the claims of the $\la$-lemma needed may be +different \cite{PM82} \cite{Li02a}. Nevertheless, the above simple version +illustrated the spirit of $\la$-lemmas. + +\section{Homoclinic Tubes and Chaos Cascades} + + +When studying high-dimensional systems, +instead of homoclinic orbits one is more interested in the +so-called homoclinic tubes \cite{Li03j} \cite{Li03k} \cite{Li99b} +\cite{Li02f} \cite{Li02g}. +The concept of a homoclinic tube was introduced by Silnikov +\cite{Sil68b} in a study on the structure of the neighborhood +of a homoclinic tube asymptotic to an invariant torus $\sg$ +under a diffeomorphism $F$ in a finite dimensional phase space. +The asymptotic torus is of saddle type. The homoclinic tube +consists of a doubly infinite sequence of tori $\{ \sg_j,\ j=0, +\pm 1, \pm 2, \cdot \cdot \cdot \}$ in the transversal +intersection of the stable and unstable manifolds of $\sg$, +such that $\sg_{j+1} = F \circ \sg_j$ for any $j$. It is a +generalization of the concept of a transversal homoclinic +orbit when the points are replaced by tori. + +We are interested in homoclinic tubes for several reasons \cite{Li03j} +\cite{Li03k} \cite{Li99b} \cite{Li02f} \cite{Li02g}: 1. +Especially in high dimensions, dynamics inside each invariant +tubes in the neighborhoods of homoclinic tubes are often +chaotic too. We call such chaotic dynamics ``{\em{chaos in +the small}}'', and the symbolic dynamics of the invariant +tubes ``{\em{chaos in the large}}''. Such cascade structures +are more important than the structures in a neighborhood of a +homoclinic orbit, when high or infinite dimensional dynamical +systems are studied. 2. Symbolic dynamics structures in the +neighborhoods of homoclinic tubes are more observable than +in the neighborhoods of homoclinic orbits in numerical and +physical experiments. 3. When studying high or infinite +dimensional Hamiltonian system (for example, the cubic nonlinear +Schr\"odinger equation under Hamiltonian perturbations), each +invariant tube contains both KAM tori and stochastic layers +(chaos in the small). Thus, not only dynamics inside each +stochastic layer is chaotic, all these stochastic layers also +move chaotically under Poincar\'e maps. + +Using the shadowing lemma technique developed in \cite{Li02a}, +we obtained in \cite{Li03j} a theorem on the symbolic dynamics +of submanifolds in a neighborhood of a homoclinic tube under a +$C^3$-diffeomorphism defined in a Banach space. Such a proof removed +an uncheckable assumption in \cite{Sil68b}. The result of \cite{Sil68b} +gives a symbolic labeling of all the invariant tubes around a +homoclinic tube. Such a symbolic labeling does not imply the +symbolic dynamics of a single map proved in \cite{Li03j}. + +Then in \cite{Li03k}, as an example, the following sine-Gordon equation +under chaotic perturbation is studied, +\begin{equation} +u_{tt} = c^2 u_{xx} + \sin u +\e [ -a u + f(t, \th_2, \th_3, \th_4) +(\sin u - u)]\ , \label{ls1} +\end{equation} +which is subject to periodic boundary condition and odd constraint +\begin{equation} +u(t, x+2\pi ) = u(t, x)\ , \quad u(t, -x) = - u(t, x)\ , +\label{obc} +\end{equation} +where $u$ is a real-valued function of two real variables $t \geq 0$ +and $x$, c is a parameter, $\frac{1}{2} < c < 1$, $a>0$ is a parameter, +$\e \geq 0$ is a small parameter, $f$ is periodic in $t, \th_2, \th_3, +\th_4$, and $(\th_2, \th_3, \th_4) \in \mathbb{T}^{3}$, by introducing +the extra variable $\th_1 = \om_1 t +\th_1^0$, $f$ takes the form +\[ +f(t) = \sum_{n=1}^4 a_n \cos [ \th_n(t)]\ , +\] +and $a_n$'s are parameters. Let $\th_n = \om_n t + \th_n^0 +\e^\mu \vth_n$, +$n=2,3,4$, $\mu > 1$, and $\vth_n$'s are given by the ABC flow +\cite{DFGHMS86} which is verified numerically to be chaotic, +\begin{eqnarray} +\dot{\vth}_2 &=& A \sin \vth_4 + C \cos \vth_3 \ , \nonumber \\ +\dot{\vth}_3 &=& B \sin \vth_2 + A \cos \vth_4 \ , \nonumber \\ +\dot{\vth}_4 &=& C \sin \vth_3 + B \cos \vth_2 \ , \nonumber +\end{eqnarray} +where $A$, $B$, and $C$ are real parameters. Existence of a homoclinic tube +is proved. As a corollary of the theorem proved in \cite{Li03j}, Symbolic +dynamics of tori around the homoclinic tube is established, which is the +``chaos in the large''. The chaotic dynamics of ($\th_2, \th_3, \th_4$) +is the ``chaos in the small''. + +Here we see the embedding of smaller scale chaos in larger scale chaos. +By introducing more variables, such embedding can be continued with even +smaller scale chaos. This leads to a chain of embeddings. We call this +chain of embeddings of smaller scale chaos in larger scale chaos, a ``chaos +cascade''. We hope that such ``chaos cascade'' will be proved important. + + +\clearpage{} +\clearpage{}\chapter{Stabilities of Soliton Equations in $\RR^n$} + +Unlike soliton equations under periodic boundary conditions, phase space +structures, especially hyperbolic structures, of soliton equations under +decay boundary conditions are not well understood. The application of +B\"{a}cklund-Darboux transformations for generating hyperbolic foliations +is not known in decay boundary condition case. Nevertheless, we believe +that B\"{a}cklund-Darboux transformations may still have great potentials +for understanding phase space structures. This can be a great area for +interested readers to work in. The common use of B\"{a}cklund-Darboux +transformations is to generate multi-soliton solutions from single soliton +solutions. The relations between different soliton solutions in phase +spaces (for example certain Sobolev spaces) are not clear yet. There are +some studies on the linear and nonlinear stabilities of traveling-wave +(soliton) solutions \cite{PW92} \cite{Liu94}. Results concerning soliton +equations are only stability results. Instability results are obtained +for non-soliton equations. On the other hand, non-soliton equations are +not integrable, have no Lax pair structures, and their phase space +structures are much more difficult to be understood. The challenging +problem here is to identify which soliton equations possess traveling-wave +solutions which are unstable. The reason why we emphasize instabilities is +that they are the sources of chaos. Below we are going to review two types +of studies on the phase space structures of soliton equations under decay +boundary conditions. + +\section{Traveling Wave Reduction} + +If we only consider traveling-wave solutions to the soliton equations or +generalized ( or perturbed ) soliton equations, the resulting equations +are ordinary differential equations with parameters. These ODEs often +offer physically meaningful and mathematically pleasant problems to be +studied. One of the interesting features is that the traveling-wave +solutions correspond to homoclinic orbits to such ODEs. We also naturally +expect that such ODEs can have chaos. Consider the KdV equation +\begin{equation} +u_t +6uu_x +u_{xxx} = 0 \ , +\label{kdv} +\end{equation} +and let $u(x,t)= U(\xi)$, $\xi = x -c t$ ($c$ is a real parameter), we have +\[ +U''' + 6U U' -c U' = 0 \ . +\] +After one integration, we get +\begin{equation} +U''+3U^2-cU+c_1 = 0 \ , +\label{tveq} +\end{equation} +where $c_1$ is a real integration constant. This equation can be rewritten +as a system, +\begin{equation} +\left \{ \begin{array}{l} U'=V\ , \cr V'=-3U^2+cU-c_1\ . +\cr \end{array}\right . +\label{tvsys} +\end{equation} +For example, when $c_1=0$ and $c > 0$ we have the soliton, +\begin{equation} +U_s = {c \over 2} \ \mbox{sech}^2[\pm {\sqrt{c} \over 2} \xi ] \ , +\label{solit} +\end{equation} +which is a homoclinic orbit asymptotic to $0$. + +We may have an equation of the following form describing a real physical phenomenon, +\[ +u_t + [f(u)]_x +u_{xxx}+\e g(u) = 0\ , +\] +where $\e$ is a small parameter, and traveling waves are physically important. Then we have +\[ +U'''+[f(U)]'-cU'+\e g(U)=0\ , +\] +which can be rewritten as a system, +\[ +\left \{ \begin{array}{l} U'= V\ , \cr V'=W\ , \cr +W'=-f'V+cV-\e g(U)\ . +\cr \end{array}\right . +\] +This system may even have chaos. Such low dimensional systems are very popular in +current dynamical system studies and are mathematically very pleasant \cite{JKL91} \cite{JK91}. Nevertheless, +they represent a very narrow class of solutions to the original PDE, and in no way +they can describe the entire phase space structures of the original PDE. + +\section{Stabilities of the Traveling-Wave Solutions} + +The first step toward understanding the phase space structures of soliton equations +under decay boundary conditions is to study the linear or nonlinear stabilities of +traveling-wave solutions \cite{PW92} \cite{Liu94} \cite{GSS87} \cite{KS98}. Consider the KdV equation +(\ref{kdv}), and study the linear stability of the traveling wave (\ref{solit}). +First we change the variables from ($x,t$) into ($\xi,t$) where $\xi=x-ct$; then +in terms of the new variables ($\xi,t$), the traveling wave (\ref{solit}) is a fixed +point of the KdV equation (\ref{kdv}). Let $u(x,t)=U_s(\xi)+v(\xi,t)$ and linearize +the KdV equation (\ref{kdv}) around $U_s(\xi)$, we have +\[ +\pa_t v -c \pa_\xi v +6 [U_s \pa_\xi v +v \pa_\xi U_s]+\pa_\xi^3 v = 0\ . +\] +We seek solutions to this equation in the form $v= e^{\la t}Y(\xi)$ and we have the +eigenvalue problem: +\begin{equation} +\la Y - c \pa_\xi Y + 6 [U_s \pa_\xi Y + Y \pa_\xi U_s ] +\pa_\xi^3 Y = 0 \ . +\label{kdvl} +\end{equation} +One method for studying such eigenvalue problem is carried in two steps \cite{PW92}: +First, one studies the $|\xi| \ra \infty$ limit system +\[ +\la Y - c \pa_\xi Y + \pa_\xi^3 Y = 0 \ , +\] +and finds that it has solutions $Y(\xi)=e^{\mu_j \xi}$ for $j=1,2,3$ where the $\mu_j$ satisfy +\[ +\mbox{Re} \{ \mu_1 \} < 0 < \mbox{Re} \{ \mu_l \} \ \ \ \ \mbox{for} \ +l=2,3. +\] +Thus the equation (\ref{kdvl}) has a one-dimensional subspace of solutions which decay as $x \ra +\infty$, and a two-dimensional subspace of solutions which decay as $x \ra -\infty$. $\la$ will +be an eigenvalue when these subspaces meet nontrivially. In step 2, one measures the angle between +these subspaces by a Wronskian-like analytic function $D(\la)$, named Evans's function. One +interpretation of this function is that it is like a transmission coefficient, in the sense +that for the solution of (\ref{kdvl}) satisfying +\[ +Y(\xi) \sim e^{\mu_1 \xi}\ \ \ \ \mbox{as}\ \xi \ra \infty \ , +\] +we have +\[ +Y(\xi) \sim D(\la) e^{\mu_1 \xi}\ \ \ \ \mbox{as}\ \xi \ra -\infty \ . +\] +In equation (\ref{kdvl}), for $\mbox{Re}\{\la \} > 0$, if $D(\la)$ vanishes, then $\la$ is +an eigenvalue, and conversely. The conclusion for the equation (\ref{kdvl}) is that there +is no eigenvalue with $\mbox{Re}\{\la \} > 0$. In fact, for the KdV equation (\ref{kdv}), +$U_s$ is $H^1$-orbitally stable \cite{Ben72} \cite{Bon75} \cite{Wei86} \cite{BSS87}. + +For generalized soliton equations, linear instability results have been established \cite{PW92}. +Let $f(u)=u^{p+1}/(p+1)$, and +\[ +U_s(\xi)=a\ \mbox{sech}^{2/p}(b\xi) +\] +for appropriate constants $a$ and $b$. For the generalized KdV equation \cite{PW92} +\[ +\pa_t u +\pa_x f(u) +\pa_x^3 u = 0 \ , +\] +if $p>4$, then $U_s$ is linearly unstable for all $c > 0$. If $1 < p< 4$, then $U_s$ is +$H^1$-orbitally stable. For the generalized Benjamin-Bona-Mahoney equation \cite{PW92} +\[ +\pa_t u +\pa_x u +\pa_x f(u) - \pa_t \pa_x^2 u = 0 \ , +\] +if $p> 4$, there exists a positive number $c_0(p)$, such that $U_s$ with $1 < c< c_0(p)$ are +linearly unstable. For the generalized regularized Boussinesq equation \cite{PW92} +\[ +\pa_t^2 u -\pa_x^2 u -\pa_x^2 f(u)-\pa_t^2 \pa_x^2 u = 0 \ , +\] +if $p > 4$, $U_s$ is linearly unstable if $1 < c^2< c^2_0(p)$, where +\[ +c^2_0(p) = 3p /(4+2p)\ . +\] + +\section{Breathers} + +Breathers are periodic solutions of soliton equations under decay boundary conditions. The +importance of breathers with respect to the phase space structures of soliton equations is +not clear yet. B\"{a}cklund-Darboux transformations may play a role for this. For example, +starting from a breather with one frequency, B\"{a}cklund-Darboux transformations can generate +a new breather with two frequencies, i.e. the new breather is a quasiperiodic solution. The +stability of breathers will be a very important subject to study. + +First we study the reciprocal relation between homoclinic solutions generated through +B\"{a}cklund-Darboux transformations for soliton equations under periodic boundary conditions +and breathers for soliton equations under decay boundary conditions. Consider the sine-Gordon +equation +\begin{equation} +u_{tt}-u_{xx}+\sin u = 0 \ , +\label{sG} +\end{equation} +under the periodic boundary condition +\begin{equation} +u(x+L,t)=u(x,t)\ , +\label{pbc} +\end{equation} +and the decay boundary condition +\begin{equation} +u(x,t) \ra 0\ \ \ \ \mbox{as}\ |x| \ra \infty \ . +\label{dbc} +\end{equation} +Starting from the trivial solution $u = \pi$, one can generate the homoclinic solution to the +Cauchy problem (\ref{sG}) and (\ref{pbc}) through B\"{a}cklund-Darboux transformations +\cite{EFM90}, +\begin{equation} +u_H(x,t) = \pi + 4 \tan^{-1} \bigg [ {\tan \nu \cos [ (\cos \nu ) x] \over \cosh [ +(\sin \nu) t ] } \bigg ] \ , +\label{sghorb} +\end{equation} +where $L = 2 \pi /\cos \nu$. Applying the sine-Gordon symmetry +\[ +(x,t,u) \longrightarrow (t,x,u-\pi)\ , +\] +one can immediately generate the breather solution to the Cauchy problem (\ref{sG}) and +(\ref{dbc}) \cite{EFM90}, +\begin{equation} +u_B(x,t) = 4 \tan^{-1} \bigg [ {\tan \nu \cos [ (\cos \nu ) t] \over \cosh [ (\sin \nu) x ] } +\bigg ] \ . +\label{sgbr} +\end{equation} +The period of this breather is $L = 2 \pi /\cos \nu$. +The linear stability can be investigated through studying the Floquet theory for the linear +partial differential equation which is the linearized equation of (\ref{sG}) at $u_B$ +(\ref{sgbr}): +\begin{equation} +u_{tt}-u_{xx}+(\cos u_B)u=0\ , \label{lsG} +\end{equation} +under the decay boundary condition +\begin{equation} +u(x,t) \ra 0 \quad \mbox{as}\ |x| \ra \infty\ . +\label{lsGbc} +\end{equation} +Such studies will be very important in terms of understanding the phase space structures +in the neighborhood of the breather and developing infinite dimensional Floquet theory. +In fact, McLaughlin and Scott \cite{MS78} had studied the equation (\ref{lsG}) through +infinitessimal B\"acklund transformations, which lead to the so-called radiation of +breather phenomenon. In physical variables only, the B\"acklund transformation for the +sine-Gordon equation (\ref{sG}) is given as, +\begin{eqnarray*} +{\pa \over \pa \xi} \bigg [ {u_+ + u_- \over 2} \bigg ] &=& -{i \over \z} \sin +\bigg [ {u_+ - u_- \over 2} \bigg ] \ , \\ +{\pa \over \pa \eta} \bigg [ {u_+ - u_- \over 2} \bigg ] &=& i\z \sin +\bigg [ {u_+ + u_- \over 2} \bigg ] \ , +\end{eqnarray*} +where $\xi = \frac {1}{2} (x+t)$, $\eta = \frac {1}{2} (x-t)$, and $\z$ is a complex +B\"acklund parameter. Through the nonlinear superposition principle \cite{RS82} \cite{AI79}, +\begin{equation} +\tan \bigg [ {u_+ - u_- \over 4} \bigg ] = \frac {\z_1+\z_2}{\z_1-\z_2} +\tan \bigg [ {u_1 - u_2 \over 4} \bigg ] +\label{nsup} +\end{equation} +built upon the Bianchi diagram \cite{RS82} \cite{AI79}. +The breather $u_B$ (\ref{sgbr}) is produced by starting with $u_-=0$, and choosing +$\z_1 = \cos \nu +i \sin \nu$ and $\z_2 = -\cos \nu +i \sin \nu$. Let $\tu_-$ be a +solution to the linearized sine-Gordon equation (\ref{lsG}) with $u_B$ replaced by $u_-=0$, +\[ +\tu_- = e^{i(kt +\sqrt{k^2-1}x)}\ . +\] +Then variation of the nonlinear superposition principle leads to the solution to the +linearized sine-Gordon equation (\ref{lsG}): +\begin{eqnarray*} +& & \tu_+ = \tu_- +\frac {i\al}{\be}\cos^2 (u_B/4) \bigg [ \cos^2 [(u_1-u_2)/4]\bigg +]^{-1}(\tu_1 -\tu_2) \\ +&=& (1+A^2)^{-1}\bigg [ a_1 +a_2 \frac {\sin^2\be t}{\cosh^2 \al x}+\frac {a_3}{2}\frac +{(\cosh^2 \al x)'}{\cosh^2 \al x}+\frac {a_4}{2}\frac {(\sin^2\be t)'}{\cosh^2 \al x} +\bigg ]\tu_-\ , +\end{eqnarray*} +where +\begin{eqnarray*} +& & \al =\sin \nu\ , \ \ \ \be =\cos \nu\ , \ \ \ A= \frac {\al}{\be}\frac {\sin \be t} +{\cosh \al x}\ , \\ +& & a_1 =\frac {1}{2} (k^2-\be^2)-\al^2\ , \ \ \ a_2 =\frac {1}{2} (k^2-\be^2)\frac{\al^2} +{\be^2}+\al^2\ , \\ +& & a_3=i\sqrt{k^2-1}\ , \ \ \ a_4 = ik \frac{\al^2}{\be^2}\ , +\end{eqnarray*} +which represents the radiation of the breather. + +Next we briefly survey some interesting studies on breathers. The rigidity of sine-Gordon +breathers was studied by Birnir, Mckean and Weinstein \cite{BMW94} \cite{Bir94}. Under the scaling +\[ +x \ra x'=(1+\e a)^{-1/2} x\ , \ \ t \ra t'=(1+\e a)^{-1/2}t\ , \ \ u \ra u'=(1+\e b) u +\] +the sine-Gordon equation (\ref{sG}) is transformed into +\[ +u_{tt}-u_{xx}+\sin u +\e (a \sin u +b u \cos u) +O(\e^2) = 0 \ . +\] +The breather (\ref{sgbr}) is transformed into a breather for the new equation. The +remarkable fact is the rigidity of the breather (\ref{sgbr}) as stated in the following theorem. +\begin{theorem}[Birnir, Mckean and Weinstein \cite{BMW94}] +If $f(u)$ vanishes at $u=0$ and is holomorphic in the open strip $|\mbox{Re}\{u\}| < \pi$, +then $u_{tt}-u_{xx}+\sin u +\e f(u) +O(\e^2) = 0$ has breathers $u=u_B +\e \hat{u}_1 + +O(\e^2)$, one to each $\sin \nu$ from an open subinterval of $(0, 1/\sqrt{2}]$ issuing +smoothly from $u=u_B$ at $\e =0$, with $\hat{u}_1$ vanishing at $x = \pm \infty$, +if and only if $f(u)$ is a linear combination of $\sin u$, $u\cos u$, and/or $1 +3 \cos u +-4 \cos (u/2) +4 \cos u \ln \cos (u/4)$. +\end{theorem} +In fact, it is proved that sine-Gordon equation (\ref{sG}) is the only nonlinear wave +equation possessing small analytic breathers \cite{Kic91}. These results lend color to +the conjecture that, for the general nonlinear wave equation, breathing is an +``arithmetical'' phenomenon in the sense that it takes place only for isolated +nonlinearities, scaling and other trivialities aside. + + +\clearpage{} +\clearpage{}\chapter{Lax Pairs of Euler Equations of Inviscid Fluids} + +The governing equations for the incompressible viscous fluid flow are the +Navier-Stokes equations. Turbulence occurs in the regime of high Reynolds +number. By formally setting the Reynolds number equal to infinity, the +Navier-Stokes equations reduce to the Euler equations of incompressible +inviscid fluid flow. One may view the Navier-Stokes equations with large +Reynolds number as a singular perturbation of the Euler equations. + +Results of T. Kato show that 2D Navier-Stokes equations are globally +well-posed in $C^0([0, \infty); H^s(R^2)), \ s>2$, and for any +$0 < T < \infty$, the mild solutions of the 2D Navier-Stokes equations +approach those of the 2D Euler equations in $C^0([0, T]; H^s(R^2))$ +\cite{Kat86}. 3D Navier-Stokes equations are locally well-posed in +$C^0([0, \tau]; H^s(R^3)), \ s>5/2$, and the mild solutions of the 3D +Navier-Stokes equations approach those of the 3D Euler equations in +$C^0([0, \tau]; H^s(R^3))$, where $\tau$ depends on the norms of the initial +data and the external force \cite{Kat72} \cite{Kat75}. Extensive studies on +the inviscid limit have been carried by J. Wu et al. \cite{Wu96} +\cite{CW96} \cite{Wu98} \cite{BW99}. There is no doubt that mathematical +study on Navier-Stokes (Euler) equations is one of the most important +mathematical problems. In fact, Clay Mathematics Institute has posted the +global well-posedness of 3D Navier-Stokes equations as one of the one +million dollars problems. + +V. Arnold \cite{Arn66} realized that 2D Euler equations are a Hamiltonian +system. Extensive studies on the symplectic structures of 2D Euler equations +have been carried by J. Marsden, T. Ratiu et al. \cite{Mar92}. + +Recently, the author found Lax pair structures for Euler +equations \cite{Li01a} \cite{LY01} \cite{Li02e} \cite{Li02f}. +Understanding the structures of solutions to Euler equations is of +fundamental interest. Of particular interest is the question on the +global well-posedness of 3D Navier-Stokes and Euler equations. Our number +one hope is that the Lax pair structures can be useful in investigating +the global well-posedness. Our secondary hope is that the Darboux +transformation \cite{LY01} associated with the Lax pair can generate +explicit representation of homoclinic structures \cite{Li00a}. + +The philosophical significance of the existence of Lax pairs for Euler +equations is even more important. If one defines integrability of an equation +by the existence of a Lax pair, then both 2D and 3D Euler equations +are integrable. More importantly, both 2D and 3D Navier-Stokes equations +at high Reynolds numbers are singularly perturbed integrable systems. +Such a point of view changes our old ideology on Euler and Navier-Stokes +equations. + +\section{A Lax Pair for 2D Euler Equation} + +The 2D Euler equation can be written in the vorticity form, +\begin{equation} +\pa_t \Om + \{ \Psi, \Om \} = 0 \ , +\label{euler} +\end{equation} +where the bracket $\{\ ,\ \}$ is defined as +\[ +\{ f, g\} = (\pa_x f) (\pa_y g) - (\pa_y f) (\pa_x g) \ , +\] +$\Om$ is the vorticity, and $\Psi$ is the stream function given by, +\[ +u=- \pa_y \Psi \ ,\ \ \ v=\pa_x \Psi \ , +\] +and the relation between vorticity $\Om$ and stream +function $\Psi$ is, +\[ +\Om =\pa_x v - \pa_y u =\Dl \Psi \ . +\] +\begin{theorem}[Li, \cite{Li01a}] +The Lax pair of the 2D Euler equation (\ref{euler}) is given as +\begin{equation} +\left \{ \begin{array}{l} +L \varphi = \la \varphi \ , +\\ +\pa_t \varphi + A \varphi = 0 \ , +\end{array} \right. +\label{laxpair} +\end{equation} +where +\[ +L \varphi = \{ \Om, \varphi \}\ , \ \ \ A \varphi = \{ \Psi, \varphi \}\ , +\] +and $\la$ is a complex constant, and $\varphi$ is a complex-valued function. +\label{2dlp} +\end{theorem} + +\section{A Darboux Transformation for 2D Euler Equation} + +Consider the Lax pair (\ref{laxpair}) at $\la =0$, i.e. +\begin{eqnarray} +& & \{ \Om, p \} = 0 \ , \label{d1} \\ +& & \pa_t p + \{ \Psi, p \} = 0 \ , \label{d2} +\end{eqnarray} +where we replaced the notation $\varphi$ by $p$. +\begin{theorem}[\cite{LY01}] +Let $f = f(t,x,y)$ be any fixed solution to the system +(\ref{d1}, \ref{d2}), we define the Gauge transform $G_f$: +\begin{equation} +\tilde{p} = G_f p = \frac {1}{\Om_x}[p_x -(\pa_x \ln f)p]\ , +\label{gauge} +\end{equation} +and the transforms of the potentials $\Om$ and $\Psi$: +\begin{equation} +\tilde{\Psi} = \Psi + F\ , \ \ \ \tilde{\Om} = \Om + \Dl F \ , +\label{ptl} +\end{equation} +where $F$ is subject to the constraints +\begin{equation} +\{ \Om, \Dl F \} = 0 \ , \ \ \ \{ \Dl F, F \} = 0\ . +\label{constraint} +\end{equation} +Then $\tilde{p}$ solves the system (\ref{d1}, \ref{d2}) at +$(\tilde{\Om}, \tilde{\Psi})$. Thus (\ref{gauge}) and +(\ref{ptl}) form the Darboux transformation for the 2D +Euler equation (\ref{euler}) and its Lax pair (\ref{d1}, \ref{d2}). +\label{dt} +\end{theorem} +\begin{remark} +For KdV equation and many other soliton equations, the +Gauge transform is of the form \cite{MS91}, +\[ +\tilde{p} = p_x -(\pa_x \ln f)p \ . +\] +In general, Gauge transform does not involve potentials. +For 2D Euler equation, a potential factor $\frac {1}{\Om_x}$ +is needed. From (\ref{d1}), one has +\[ +\frac{p_x}{\Om_x} = \frac{p_y}{\Om_y} \ . +\] +The Gauge transform (\ref{gauge}) can be rewritten as +\[ +\tilde{p} = \frac{p_x}{\Om_x} - \frac{f_x}{\Om_x} \frac{p}{f} +=\frac{p_y}{\Om_y} - \frac{f_y}{\Om_y} \frac{p}{f}\ . +\] +The Lax pair (\ref{d1}, \ref{d2}) has a symmetry, i.e. it is +invariant under the transform $(t,x,y) \ra (-t,y,x)$. The form +of the Gauge transform (\ref{gauge}) resulted from the inclusion +of the potential factor $\frac {1}{\Om_x}$, is consistent with +this symmetry. +\end{remark} +Our hope is to use the Darboux transformation to generate homoclinic +structures for 2D Euler equation \cite{Li00a}. + + +\section{A Lax Pair for Rossby Wave Equation} + +The Rossby wave equation is +\[ +\pa_t \Om + \{ \Psi , \Om \} + \be \pa_x \Psi = 0 \ , +\] +where $\Om = \Om (t,x,y)$ is the vorticity, +$\{ \Psi , \Om \} = \Psi_x \Om_y - \Psi_y \Om_x $, +and $\Psi = \Dl^{-1} \Om$ +is the stream function. Its Lax pair can be obtained +by formally conducting the transformation, $\Om = \tilde{\Om} +\be y$, +to the 2D Euler equation \cite{Li01a}, +\[ +\{ \Om , \varphi \} - \be \pa_x \varphi = \la \varphi \ , +\quad \pa_t \varphi + \{ \Psi , \varphi \} = 0 \ , +\] +where $\varphi$ is a complex-valued function, and $\la$ is +a complex parameter. + +\section{Lax Pairs for 3D Euler Equation} + +The 3D Euler equation can be written in vorticity form, +\begin{equation} +\pa_t \Om + (u \cdot \na) \Om - (\Om \cdot \na) u = 0 \ , +\label{3deuler} +\end{equation} +where $u = (u_1, u_2, u_3)$ is the velocity, $\Om = (\Om_1, \Om_2, \Om_3)$ +is the vorticity, $\na = (\pa_x, \pa_y, \pa_z)$, +$\Om = \na \times u$, and $\na \cdot u = 0$. $u$ can be +represented by $\Om$ for example through Biot-Savart law. +\begin{theorem} +The Lax pair of the 3D Euler equation (\ref{3deuler}) is given as +\begin{equation} +\left \{ \begin{array}{l} +L \phi = \la \phi \ , +\\ +\pa_t \phi + A \phi = 0 \ , +\end{array} \right. +\label{alaxpair} +\end{equation} +where +\[ +L \phi = (\Om \cdot \na )\phi \ , +\ \ \ A \varphi = (u \cdot \na )\phi \ , +\] +$\la$ is a complex constant, and $\phi$ is a complex scalar-valued function. +\end{theorem} +\begin{theorem}[\cite{Chi00}] +Another Lax pair of the 3D Euler equation (\ref{3deuler}) is given as +\begin{equation} +\left \{ \begin{array}{l} +L \varphi = \la \varphi \ , +\\ +\pa_t \varphi + A \varphi = 0 \ , +\end{array} \right. +\label{3dlaxpair} +\end{equation} +where +\[ +L \varphi = (\Om \cdot \na )\varphi - (\varphi \cdot \na )\Om \ , +\ \ \ A \varphi = (u \cdot \na )\varphi - (\varphi \cdot \na ) u \ , +\] +$\la$ is a complex constant, and $\varphi = (\varphi_1, \varphi_2, +\varphi_3)$ is a complex 3-vector valued function. +\end{theorem} +Our hope is that the infinitely many conservation laws generated by $\la +\in C$ can provide a priori estimates for the global well-posedness of +3D Navier-Stokes equations, or better understanding on the global +well-posedness. For more informations on the topics, +see \cite{LY01}. + + + + + + + + + + + + + +\clearpage{} +\clearpage{}\chapter{Linearized 2D Euler Equation at a Fixed Point} + +To begin an infinite dimensional dynamical system study, we investigate +the linearized 2D Euler equation at a fixed point \cite{Li00}. We consider the +2D Euler equation (\ref{euler}) under periodic boundary condition in +both $x$ and $y$ directions with period $2\pi$. Expanding $\Om$ into +Fourier series, +\[ +\Om =\sum_{k\in \Z} \om_k \ e^{ik\cdot X}\ , +\] +where $\om_{-k}=\overline{\om_k}\ $, $k=(k_1,k_2)^T$, +and $X=(x,y)^T$. The 2D Euler equation +can be rewritten as +\begin{equation} +\dot{\om}_k = \sum_{k=p+q} A(p,q) \ \om_p \om_q \ , +\label{Keuler} +\end{equation} +where $A(p,q)$ is given by, +\begin{eqnarray} +A(p,q)&=& {1\over 2}[|q|^{-2}-|p|^{-2}](p_1 q_2 -p_2 q_1) \nonumber \\ +\label{Af} \\ + &=& {1\over 2}[|q|^{-2}-|p|^{-2}]\left | \begin{array}{lr} +p_1 & q_1 \\ p_2 & q_2 \\ \end{array} \right | \ , \nonumber +\end{eqnarray} +where $|q|^2 =q_1^2 +q_2^2$ for $q=(q_1,q_2)^T$, similarly for $p$. +The 2D Euler equation (\ref{Keuler}) has huge dimensional equilibrium +manifolds. +\begin{proposition} +For any $k \in Z^2/\{0\}$, the infinite dimensional space +\[ +E^1_k \equiv \bigg \{ \{ \om_{k'}\} \ \bigg | \ \om_{k'}=0, +\ \mbox{if}\ k' \neq rk,\ \forall r \in R \bigg \} \ , +\] +and the finite dimensional space +\[ +E^2_k \equiv \bigg \{ \{ \om_{k'}\} \ \bigg | \ \om_{k'}=0, +\ \mbox{if}\ |k'| \neq |k| \bigg \} \ , +\] +entirely consist of fixed points of the system (\ref{Keuler}). +\label{eman} +\end{proposition} +Fig.\ref{eulman} shows an example on the locations of the modes +($k'=rk$) and ($|k'|=|k|$) in the definitions of $E_k^1$ and $E_k^2$ +(Proposition \ref{eman}). +\begin{figure}[ht] + \begin{center} + \leavevmode + \setlength{\unitlength}{2ex} + \begin{picture}(36,27.8)(-18,-12) + \thinlines +\multiput(-12,-11.5)(2,0){13}{\line(0,1){23}} +\multiput(-16,-10)(0,2){11}{\line(1,0){32}} + \thicklines +\put(0,-14){\vector(0,1){28}} +\put(-18,0){\vector(1,0){36}} +\put(0,15){\makebox(0,0){$k_2$}} +\put(18.5,0){\makebox(0,0)[l]{$k_1$}} +\qbezier(-5.5,0)(-5.275,5.275)(0,5.5) +\qbezier(0,5.5)(5.275,5.275)(5.5,0) +\qbezier(5.5,0)(5.275,-5.275)(0,-5.5) +\qbezier(0,-5.5)(-5.275,-5.275)(-5.5,0) + \thinlines +\put(4,4){\circle*{0.5}} +\put(0,0){\vector(1,1){3.7}} +\put(4,-4){\circle*{0.5}} +\put(-4,4){\circle*{0.5}} +\put(2,2){\circle*{0.5}} +\put(6,6){\circle*{0.5}} +\put(8,8){\circle*{0.5}} +\put(10,10){\circle*{0.5}} +\put(-2,-2){\circle*{0.5}} +\put(-4,-4){\circle*{0.5}} +\put(-6,-6){\circle*{0.5}} +\put(-8,-8){\circle*{0.5}} +\put(-10,-10){\circle*{0.5}} +\put(-10.6,-10.6){\line(1,1){21.5}} +\put(4.35,4.65){$k$} +\put(12.5,-12.5){\makebox(0,0)[t]{$|k'|=|k|$}} +\put(12.4,-12.4){\vector(-1,1){8.2}} +\put(14.5,-2.5){\makebox(0,0)[t]{$k'=rk$}} +\put(14.4,-2.4){\vector(-1,1){8.2}} + \end{picture} + \end{center} +\caption{An illustration on the locations of the modes ($k'=rk$) and +($|k'|=|k|$) in the definitions of $E^1_k$ and $E^2_k$ +(Proposition \ref{eman}).} +\label{eulman} +\end{figure} + +\section{Hamiltonian Structure of 2D Euler Equation} + +For any two functionals $F_1$ and $F_2$ of $\{ \om_k \}$, +define their Lie-Poisson bracket: +\begin{equation} +\{ F_1,F_2 \} = \sum_{k+p+q=0} \left | \begin{array}{lr} +q_1 & p_1 \\ q_2 & p_2 \\ \end{array} \right | \ \om_k \ +{\pa F_1 \over \pa \overline{\om_p}} \ {\pa F_2 \over \pa \overline{\om_q}}\ . +\label{Liebr} +\end{equation} +Then the 2D Euler equation (\ref{Keuler}) is a Hamiltonian system \cite{Arn66}, +\begin{equation} +\dot{\om}_k = \{ \om_k, H\}, \label{hEft} +\end{equation} +where the Hamiltonian $H$ is the kinetic energy, +\begin{equation} +H= {1\over 2} \sum_{k \in Z^2/\{0\}} |k|^{-2} |\om_k |^2. +\label{Ih} +\end{equation} +Following are Casimirs (i.e. invariants that Poisson commute with +any functional) of the Hamiltonian system (\ref{hEft}): +\begin{equation} +J_n = \sum_{k_1 + \cdot \cdot \cdot +k_n =0} \om_{k_1} +\cdot \cdot \cdot \om_{k_n}. \label{Ic} +\end{equation} + +\section{Linearized 2D Euler Equation at a Unimodal Fixed Point} + +Denote $\{ \om_k \}_{k\in \Z}$ by $\om$. We consider the simple fixed point +$\om^*$ \cite{Li00}: +\begin{equation} +\om^*_p = \Ga,\ \ \ \om^*_k = 0 ,\ \mbox{if} \ k \neq p \ \mbox{or}\ -p, +\label{fixpt} +\end{equation} +of the 2D Euler equation (\ref{Keuler}), where +$\Ga$ is an arbitrary complex constant. +The {\em{linearized two-dimensional Euler equation}} at $\om^*$ is given by, +\begin{equation} +\dot{\om}_k = A(p,k-p)\ \Ga \ \om_{k-p} + A(-p,k+p)\ \bar{\Ga}\ \om_{k+p}\ . +\label{LE} +\end{equation} +\begin{definition}[Classes] +For any $\hk \in \Z$, we define the class $\Sg_{\hk}$ to be the subset of +$\Z$: +\[ +\Sg_{\hk} = \bigg \{ \hk + n p \in \Z \ \bigg | \ n \in Z, \ +\ p \ \mbox{is specified in (\ref{fixpt})} \bigg \}. +\] +\label{classify} +\end{definition} +\nid +See Fig.\ref{class} for an illustration of the classes. +According to the classification +defined in Definition \ref{classify}, the linearized two-dimensional Euler +equation (\ref{LE}) decouples into infinitely many {\em{invariant subsystems}}: +\begin{eqnarray} +\dot{\omega}_{\hat{k} + np} &=& A(p, \hat{k} + (n-1) p) + \ \Gamma \ \omega_{\hat{k} + (n-1) p} \nonumber \\ +& & + \ A(-p, \hat{k} + (n+1)p)\ + \bar{\Gamma} \ \omega_{\hat{k} +(n+1)p}\ . \label{CLE} +\end{eqnarray} +\begin{figure}[ht] + \begin{center} + \leavevmode + \setlength{\unitlength}{2ex} + \begin{picture}(36,27.8)(-18,-12) +\thinlines +\multiput(-12,-11.5)(2,0){13}{\line(0,1){23}} +\multiput(-16,-10)(0,2){11}{\line(1,0){32}} +\thicklines +\put(0,-14){\vector(0,1){28}} +\put(-18,0){\vector(1,0){36}} +\put(0,15){\makebox(0,0){$k_2$}} +\put(18.5,0){\makebox(0,0)[l]{$k_1$}} +\qbezier(-5.5,0)(-5.275,5.275)(0,5.5) +\qbezier(0,5.5)(5.275,5.275)(5.5,0) +\qbezier(5.5,0)(5.275,-5.275)(0,-5.5) +\qbezier(0,-5.5)(-5.275,-5.275)(-5.5,0) +\thinlines +\put(4,4){\circle*{0.5}} +\put(0,0){\vector(1,1){3.7}} +\put(4.35,4.35){$p$} +\put(4,-4){\circle*{0.5}} +\put(8,0){\circle*{0.5}} +\put(-8,0){\circle*{0.5}} +\put(-8,-2){\circle*{0.5}} +\put(-12,-4){\circle*{0.5}} +\put(-12,-6){\circle*{0.5}} +\put(-4,2){\circle*{0.5}} +\put(-4,4){\circle*{0.5}} +\put(0,6){\circle*{0.5}} +\put(0,8){\circle*{0.5}} +\put(4,10){\circle*{0.5}} +\put(12,4){\circle*{0.5}} +\put(0,-8){\circle*{0.5}} +\put(-4,-12){\line(1,1){17.5}} +\put(-13.5,-7.5){\line(1,1){19.5}} +\put(-13.5,-5.5){\line(1,1){17.5}} +\put(-3.6,1.3){$\hat{k}$} +\put(-7,12.1){\makebox(0,0)[b]{$(-p_2, p_1)^T$}} +\put(-6.7,12){\vector(1,-3){2.55}} +\put(6.5,13.6){\makebox(0,0)[l]{$\Sg_{\hat{k}}$}} +\put(6.4,13.5){\vector(-2,-3){2.0}} +\put(7,-12.1){\makebox(0,0)[t]{$(p_2, -p_1)^T$}} +\put(6.7,-12.25){\vector(-1,3){2.62}} +\put(-4.4,-13.6){\makebox(0,0)[r]{$\bar{D}_{|p|}$}} +\put(-4.85,-12.55){\vector(1,3){2.45}} +\end{picture} +\end{center} +\caption{An illustration of the classes $\Sg_{\hk}$ and the disk +$\bar{D}_{|p|}$.} +\label{class} +\end{figure} + +\subsection{Linear Hamiltonian Systems} + +Each invariant subsystem (\ref{CLE}) can be rewritten as a linear +Hamiltonian system as shown below. +\begin{definition}[The Quadratic Hamiltonian] +The quadratic Hamiltonian $\HH_{\hat{k}}$ is defined as: +\begin{eqnarray} +\HH_{\hat{k}} &=& -2 \ \im \bigg \{ \sum_{n \in Z} \rho_n \ \Gamma +\ A(p, \hat{k} + (n-1)p)\ \omega_{\hat{k} + (n-1)p} \ +\bar{\omega}_{\hat{k} +np} \bigg \} \nonumber \\ +\label{CHAM} \\ +&=& - \left| + \begin{array}{cc} + p_1 & \hat{k}_1\\ + p_2 & \hat{k}_2 + \end{array} + \right| \ \im + \bigg \{ \sum_{n \in Z} \Gamma \ \rho_n \ \rho_{n-1} + \ \omega_{\hat{k} + (n-1)p} + \ \bar{\omega}_{\hat{k} + np} + \bigg \}, \nonumber +\end{eqnarray} +where $\rho_n = [ |\hat{k} + np |^{-2}-|p|^{-2}]$, +``\ $\im$\ '' denotes `` imaginary part ''. +\end{definition} +Then the invariant subsystem (\ref{CLE}) can be rewritten as a linear +Hamiltonian system \cite{Li00}, +\begin{equation} +i \ \dot{\omega}_{\hat{k} +n p} = \rho^{-1}_n \ \frac{\pa \HH_{\hat{k}}} + {\partial \bar{\omega}_{\hat{k} + np}}\ . +\label{CHAF} +\end{equation} +For a finite dimensional linear Hamiltonian system, it is well-known +that the eigenvalues are of +four types: real pairs ($c, -c$), purely imaginary pairs ($id, -id$), +quadruples ($\pm c \pm id$), and zero eigenvalues \cite{Poi99} \cite{Lia49} +\cite{Arn80}. There is also a complete theorem on the normal forms of such +Hamiltonians \cite{Arn80}. For the above infinite dimensional system +(\ref{CLE}), the classical proofs can not be applied. Nevertheless, the +conclusion is still true with different proof \cite{Li00} \cite{Li02m}. +Let $\LL_{\hk}$ be the linear operator defined by the right hand side of +(\ref{CLE}), and $H^s$ be the Sobolev space where $s \geq 0$ is an integer and +$H^0=\ell_2$. +\begin{theorem}[\cite{Li00} \cite{Li02m}] +The eigenvalues of the linear operator $\LL_{\hk}$ in $H^s$ are of +four types: real pairs ($c, -c$), purely imaginary pairs ($id, -id$), +quadruples ($\pm c \pm id$), and zero eigenvalues. +\end{theorem} + +\subsection{Liapunov Stability} + +\begin{definition}[An Important Functional] +For each invariant subsystem (\ref{CLE}), we define the functional +$I_{\hat{k}}$, +\begin{eqnarray} + I_{\hat{k}} &=& I_{(\hbox{\textup{\small restricted to }} + \Sg_{\hat{k}})} \nonumber \\ +\label{invIk} \\ +&=& \sum_{n \in Z} \{ | \hat{k} + np |^{-2} + - | p |^{-2} \} \left| + \omega_{\hat{k} + np} \right|^2 \, . \nonumber +\end{eqnarray} +\end{definition} +\begin{lemma} +$I_{\hat{k}}$ is a constant of motion for the system (\ref{CHAF}). +\label{const} +\end{lemma} +\begin{definition}[The Disk] +The disk of radius $| p |$ in $\Z$, denoted by +$\bar{D}_{| p |}$, is defined as +\[ + \bar{D}_{| p |} = \bigg \{ k \in \Z \ \bigg| + \ | k | \leq | p | \bigg \} \, . +\] +\end{definition} +\nid +See Fig.\ref{class} for an illustration. +\begin{theorem}[Unstable Disk Theorem, \cite{Li00}] +If $\ \Sigma_{\hat{k}} \cap \bar{D}_{\left| p \right|} = \emptyset,\ $ +then the invariant subsystem (\ref{CLE}) is Liapunov stable +for all $t \in R$, in fact, +\[ + \sum_{n \in Z} \left| + \omega_{\hat{k}+np}(t) \right|^2 \leq \sigma \ \sum_{n \in Z} \left| + \omega_{\hat{k}+np}(0) \right|^2 \, , \quad \quad \forall t \in R \, , +\] +where +\[ + \sigma = \left[ \max_{n \in Z} + \left\{ - \rho_n \right\} + \right] \, + \left[ \min_{n \in Z} + \left\{ -\rho_n \right\} + \right]^{-1} \, , \quad 0< \sigma < \infty \, . +\] +\label{UDT} +\end{theorem} + +\subsection{Spectral Theorems} + +Again denote by $\LL_{\hk}$ the linear operator defined by the right hand +side of (\ref{CLE}). +\begin{theorem}[The Spectral Theorem, \cite{Li00} \cite{Li02m}] We have +the following claims on the spectrum of the linear operator $\LL_{\hk}$: +\begin{enumerate} +\item If $\Sg_{\hat{k}} \cap \bar{D}_{|p|} = \emptyset$, then the entire +$H^s$ spectrum of the linear operator $\LL_{\hk}$ +is its continuous spectrum. See Figure \ref{splb}, where +$b= - \frac{1}{2}|\Gamma | |p|^{-2} +\left| + \begin{array}{cc} +p_1 & \hat{k}_1 \\ +p_2 & \hat{k}_2 + \end{array} +\right| \ .$ +That is, both the residual and the point spectra of $\LL_{\hk}$ are empty. +\item If $\Sg_{\hat{k}} \cap \bar{D}_{|p|} \neq \emptyset$, then the entire +essential $H^s$ spectrum of the linear operator $\LL_{\hk}$ is its +continuous spectrum. +That is, the residual +spectrum of $\LL_{\hk}$ is empty. The point +spectrum of $\LL_{\hk}$ is symmetric with respect to both real and +imaginary axes. +See Figure \ref{spla2}. +\end{enumerate} +\label{SST} +\end{theorem} +\begin{figure}[ht] + \begin{center} + \leavevmode + \setlength{\unitlength}{2ex} + \begin{picture}(36,27.8)(-18,-12) + \thicklines +\put(0,-14){\vector(0,1){28}} +\put(-18,0){\vector(1,0){36}} +\put(0,15){\makebox(0,0){$\Im \{ \la \}$}} +\put(18.5,0){\makebox(0,0)[l]{$\Re \{ \la \}$}} +\put(0.1,-7){\line(0,1){14}} +\put(.2,-.2){\makebox(0,0)[tl]{$0$}} +\put(-0.2,-7){\line(1,0){0.4}} +\put(-0.2,7){\line(1,0){0.4}} +\put(2.0,-6.4){\makebox(0,0)[t]{$-i2|b|$}} +\put(2.0,7.6){\makebox(0,0)[t]{$i2|b|$}} +\end{picture} + \end{center} +\caption{The spectrum of $\LL_{\hk}$ in case (1).} +\label{splb} +\end{figure} +\begin{figure}[ht] + \begin{center} + \leavevmode + \setlength{\unitlength}{2ex} + \begin{picture}(36,27.8)(-18,-12) + \thicklines +\put(0,-14){\vector(0,1){28}} +\put(-18,0){\vector(1,0){36}} +\put(0,15){\makebox(0,0){$\Im \{ \la \}$}} +\put(18.5,0){\makebox(0,0)[l]{$\Re \{ \la \}$}} +\put(0.1,-7){\line(0,1){14}} +\put(.2,-.2){\makebox(0,0)[tl]{$0$}} +\put(-0.2,-7){\line(1,0){0.4}} +\put(-0.2,7){\line(1,0){0.4}} +\put(2.0,-6.4){\makebox(0,0)[t]{$-i2|b|$}} +\put(2.0,7.6){\makebox(0,0)[t]{$i2|b|$}} +\put(2.4,3.5){\circle*{0.5}} +\put(-2.4,3.5){\circle*{0.5}} +\put(2.4,-3.5){\circle*{0.5}} +\put(-2.4,-3.5){\circle*{0.5}} +\put(5,4){\circle*{0.5}} +\put(-5,4){\circle*{0.5}} +\put(5,-4){\circle*{0.5}} +\put(-5,-4){\circle*{0.5}} +\put(8,6){\circle*{0.5}} +\put(-8,6){\circle*{0.5}} +\put(8,-6){\circle*{0.5}} +\put(-8,-6){\circle*{0.5}} +\end{picture} + \end{center} +\caption{The spectrum of $\LL_{\hk}$ in case (2).} +\label{spla2} +\end{figure} +For a detailed proof of this theorem, see \cite{Li00} \cite{Li02m}. +Denote by $L$ the right hand side of (\ref{LE}), i.e. the whole +linearized 2D Euler operator, the spectral mapping theorem +holds. +\begin{theorem}[\cite{LLM01}] +$$\sigma(e^{tL})=e^{t\sigma(L)}, t\neq 0.$$ +\end{theorem} + +\subsection{A Continued Fraction Calculation of Eigenvalues} + +Since the introduction of continued fractions for calculating +the eigenvalues of steady fluid flow, by Meshalkin and Sinai \cite{MS61}, +this topics had been extensively explored \cite{Yud65} \cite{Liu92a} +\cite{Liu92b} \cite{Liu93} \cite{Liu94a} \cite{Liu95} \cite{BFY99} +\cite{Li00}. Rigorous justification on the continued fraction calculation +was given in \cite{Li00} \cite{Liu95}. + +Rewrite the equation (\ref{CLE}) as follows, +\begin{equation} +\rho^{-1}_n \dot{\tz}_n = a\ \bigg [ \tz_{n+1} - \tz_{n-1} \bigg ]\ , +\label{cfr1} +\end{equation} +where $\tz_n = \rho_n e^{in (\th +\pi /2)} \om_{\hat{k}+np}$, $\th +\ga +=\pi /2$, $\Ga = |\Ga| e^{i\ga}$, $a = {1 \over 2} |\Ga| \left | \begin{array} +{lr} p_1 & \hat{k}_1 \\ p_2 & \hat{k}_2 \\ \end{array} \right |$, +$\rho_n = | \hat{k}+np|^{-2} - |p|^{-2}$. Let $\tz_n = e^{\la t} z_n$, where +$\la \in C$; then $z_n$ satisfies +\begin{equation} +a_n z_n +z_{n-1} - z_{n+1} = 0 \ , +\label{cfr2} +\end{equation} +where $a_n = \la (a \rho_n)^{-1}$. Let $w_n = z_n / z_{n-1}$ \cite{MS61}; +then $w_n$ satisfies +\begin{equation} +a_n + {1 \over w_n} = w_{n+1}\ . +\label{cfr3} +\end{equation} +Iteration of (\ref{cfr3}) leads to the continued fraction solution \cite{MS61}, +\begin{equation} +w_n^{(1)}=a_{n-1} +{1 \over a_{n-2} + {1 \over a_{n-3}+_{\ \ddots}}}\ \ . +\label{cfr4} +\end{equation} +Rewrite (\ref{cfr3}) as follows, +\begin{equation} +w_n = {1 \over -a_n +w_{n+1}}\ . +\label{cfr5} +\end{equation} +Iteration of (\ref{cfr5}) leads to the continued fraction solution +\cite{MS61}, +\begin{equation} +w_n^{(2)}=-{1 \over a_{n} + {1 \over a_{n+1}+{1 \over a_{n+2} ++_{\ \ddots}}}}\ \ . +\label{cfr6} +\end{equation} +The eigenvalues are given by the condition $w_1^{(1)}=w_1^{(2)}$, +i.e. +\begin{equation} +f= a_0 + \bigg ( {1 \over a_{-1} + {1 \over a_{-2} +{1 \over a_{-3} ++_{\ \ddots}}}} \bigg ) + \bigg ( {1 \over a_{1} + {1 \over a_{2} + +{1 \over a_{3}+_{\ \ddots}}}} \bigg ) = 0 \ , +\label{cfr17} +\end{equation} +where $f = f(\tla,\hat{k},p)$, $\tla = \la /a$. + +As an example, we take $p=(1,1)^T$. When $\Ga \neq 0$, the fixed point +has $4$ eigenvalues which form a +quadruple. These four eigenvalues appear in the invariant +linear subsystem labeled by $\hk = (-3,-2)^T$. One of them is \cite{Li00}: +\begin{equation} +\tla=2 \lambda / | \Gamma | = 0.24822302478255 \ + \ i \ 0.35172076526520\ . +\label{evun} +\end{equation} +See Figure \ref{figev} for an illustration. The essential spectrum +(= continuous spectrum) of $\LL_{\hk}$ with $\hk = (-3,-2)^T$ is the segment +on the imaginary axis shown in Figure \ref{figev}, where +$b = -\frac{1}{4} \Ga$. The essential spectrum +(= continuous spectrum) of the linear 2D Euler operator at this fixed point +is the entire imaginary axis. +\begin{figure}[ht] + \begin{center} + \leavevmode + \setlength{\unitlength}{2ex} + \begin{picture}(36,27.8)(-18,-12) + \thicklines +\put(0,-14){\vector(0,1){28}} +\put(-18,0){\vector(1,0){36}} +\put(0,15){\makebox(0,0){$\Im \{ \la \}$}} +\put(18.5,0){\makebox(0,0)[l]{$\Re \{ \la \}$}} +\put(2.4,3.5){\circle*{0.5}} +\put(-2.4,3.5){\circle*{0.5}} +\put(2.4,-3.5){\circle*{0.5}} +\put(-2.4,-3.5){\circle*{0.5}} +\put(0.1,-10){\line(0,1){20}} +\put(.2,-.2){\makebox(0,0)[tl]{$0$}} +\put(-0.2,-10){\line(1,0){0.4}} +\put(-0.2,10){\line(1,0){0.4}} +\put(2.0,-9.4){\makebox(0,0)[t]{$-i2|b|$}} +\put(2.0,10.6){\makebox(0,0)[t]{$i2|b|$}} +\end{picture} + \end{center} +\caption{The spectrum of $\LL_{\hk}$ with $\hk = (-3,-2)^T$, when $p=(1,1)^T$.} +\label{figev} +\end{figure} +Denote by $L$ the right hand side of (\ref{LE}), i.e. the whole +linearized 2D Euler operator. Let $\z$ denote the number of points $q \in +\Z$ that belong to the open disk of radius $|p|$, +and such that $q$ is not parallel to $p$. +\begin{theorem}[\cite{LLM01}] +The number of nonimaginary eigenvalues of $L$ (counting the multiplicities) +does not exceed $2\z$. +\end{theorem} +Another interesting discussion upon the discrete spectrum can be found in +\cite{Fad71}. + +A rather well-known open problem is proving the existence of unstable, stable, +and center manifolds. The main difficulty comes from the fact that the +nonlinear term is non-Lipschitzian. + + + + + + +\clearpage{} +\clearpage{}\chapter{Arnold's Liapunov Stability Theory} + +\section{A Brief Summary} + +Let $D$ be a region on the ($x,y$)-plane bounded +by the curves $\Ga_i$ ($i=1,2$), an ideal fluid flow in $D$ is governed +by the 2D Euler equation written in the stream-function form: +\begin{equation} +{\pa \over \pa t} \Dl \psi = [ \na \psi, \na \Dl \psi ]\ , \label{sfef} +\end{equation} +where +\[ +[ \na \psi, \na \Dl \psi ] = {\pa \psi \over \pa x} +{\pa \Dl \psi \over \pa y} +- {\pa \psi \over \pa y}{\pa \Dl \psi \over \pa x}\ , +\] +with the boundary conditions, +\[ +\psi|_{\Ga_i}=c_i(t)\ ,\ \ c_1 \equiv 0\ ,\ \ {d \over dt} \oint_{\Ga_i} +{\pa \psi \over \pa n} ds = 0\ . +\] +For every function $f(z)$, the functional +\begin{equation} +F= \int\int_{D} f(\Dl \psi) \ dxdy \label{brs1} +\end{equation} +is a constant of motion (a Casimir) for (\ref{sfef}). The conditional +extremum of the kinetic energy +\begin{equation} +E={1 \over 2}\int\int_{D}\na \psi \cdot \na \psi \ dxdy \label{brs2} +\end{equation} +for fixed $F$ is given by the Lagrange's formula \cite{Arn65}, +\begin{equation} +\dl H = \dl (E+\la F) =0\ ,\ \ \ \ \Rightarrow \ \ \psi_0 = +\la f'(\Dl \psi_0)\ . +\label{brs3} +\end{equation} +where $\la$ is the Lagrange multiplier. Thus, $\psi_0$ is the stream +function of a stationary flow, which satisfies +\begin{equation} +\psi_0 = \Phi(\Dl \psi_0)\ , \label{brs4} +\end{equation} +where $\Phi = \la f'$. The second variation is given by \cite{Arn65}, +\begin{equation} +\dl^2 H = {1 \over 2} \int\int_{D} \bigg \{ \na \phi \cdot \na \phi + +\Phi'(\Dl \psi_0) \ (\Dl \phi)^2 \bigg \} dxdy\ . \label{brs5} +\end{equation} +Let $\psi = \psi_0 + \varphi$ be a solution to the 2D Euler equation +(\ref{sfef}), Arnold proved the estimates \cite{Arn69}: (a). when $c \leq +\Phi'(\Dl \psi_0) \leq C$, $0 < c \leq C <\infty$, +\[ +\int\int_{D} \bigg \{ \na \varphi(t) \cdot \na \varphi(t) + c +[\Dl \varphi(t)]^2 \bigg \} \ dxdy +\leq \int\int_{D} \bigg \{ \na \varphi(0) \cdot \na \varphi(0) + C +[\Dl \varphi(0)]^2 \bigg \} \ dxdy, +\] +for all $t \in (-\infty, +\infty)$, (b). when $c \leq +-\Phi'(\Dl \psi_0) \leq C$, $0 < c < C <\infty$, +\[ +\int\int_{D} \bigg \{ c [\Dl \varphi(t)]^2 - \na \varphi(t) \cdot +\na \varphi(t)\bigg \} \ dxdy +\leq \int\int_{D} \bigg \{ C [\Dl \varphi(0)]^2 - \na \varphi(0) +\cdot \na \varphi(0) \bigg \} \ dxdy, +\] +for all $t \in (-\infty, +\infty)$. Therefore, when the second variation +(\ref{brs5}) is positive definite, or when +\[ +\int\int_{D} \bigg \{ \na \phi \cdot \na \phi + +[\max \Phi'(\Dl \psi_0)] \ (\Dl \phi)^2 \bigg \} \ dxdy +\] +is negative definite, the stationary flow (\ref{brs4}) is nonlinearly stable +(Liapunov stable). + +Arnold's Liapunov stability theory had been extensively explored, see e.g. +\cite{HMRW85} \cite{Mar92} \cite{WS00}. + +\section{Miscellaneous Remarks} + +Establishing Liapunov instability +along the line of the above, has not been successful. Some rather technical, +with no clear physical meaning as above, argument showing nonlinear +instability starting from linear instability, has been established +by Guo et al. \cite{Guo96} \cite{ABG97} \cite{FSV97}. + +Yudovich \cite{Yud00} had been promoting the importance of the +so-called ``slow collapse'', that is, not finite time blowup, rather +growing to infinity in time. The main thought is that if derivatives +growing to infinity in time, the function itself should gain randomness. +Yudovich \cite{Yud65} had been studying bifurcations of fluid flows. + +There are also interests \cite{GMW01} in studying structures of divergent free +2D vector fields on 2-tori. + + + + + + + + + + + +\clearpage{} +\clearpage{}\chapter{Miscellaneous Topics} + +This chapter serves as a guide to other interesting topics. +Some topics are already well-developed. Others are poorly developed +in terms of partial differential equations. + +\section{KAM Theory} + +KAM (Kolmogorov-Arnold-Moser) theory in finite dimensions has +been a well-known topic \cite{Arn63}. It dealt with the persistence +of Liouville tori in integrable Hamiltonian systems under Hamiltonian +perturbations. A natural idea of constructing such tori in perturbed +systems is conducting canonical transformations which lead to small +divisor problem. To overcome such difficulties, Kolmogorov introduced +the Newton's method to speed up the rate of convergence of the +canonical transformation series. Under certain non-resonance condition +and non-degeneracy condition of certain Hessian, Arnold completed +the proof of a rather general theorem \cite{Arn63}. Arnold proved the +case that the Hamiltonian is an analytic function. Moser was able to +prove the theorem for the case that the Hamiltonian is 333-times +differentiable \cite{Mos66a} \cite{Mos66b}, with the help of Nash +implicit function theorem. Another related topic is the Arnold theorem +on circle map \cite{Arn61}. It answers the question when a circle map +is equivalent to a rotation. Yoccoz \cite{Yoc92} was able to prove +an if and only if condition for such equivalence, using Brjuno number. + +KAM theory for partial differential equations has also been studied +\cite{Way84} \cite{Kuk93} \cite{Kuk98} \cite{Bou96} \cite{Bou98}. +For partial differential equations, KAM theory is studied on a case +by case base. There is no general theorem. So far the common types of +equations studied are nonlinear wave equations as perturbations of certain +linear wave equations, and solition equations under Hamiltonian perturbations. +Often the persistent Liouville tori are limited to finite dimensional, +sometimes, even one dimensional, i.e. periodic solutions \cite{CW93}. + +\section{Gibbs Measure} + +Gibbs measure is one of the important concepts in thermodynamic and +statistical mechanics. In an effort to understand the statistical +mechanics of nonlinear wave equations, Gibbs measure was introduced +\cite{MV94} \cite{Bou94} \cite{Bou96b}, which is built upon the Hamiltonians +of such systems. The calculation of a Gibbs measure is +similar to that in quantum field theory. In terms of classical analysis, +the Gibbs measure is not well-defined. One of the central questions is +which space such Gibbs measure is supported upon. Bourgain \cite{Bou94} +\cite{Bou96b} was able to give a brilliant answer. For example, +for periodic nonlinear Schr\"odinger equation, it is supported on $H^{-1/2}$ +\cite{Bou94}. For the thermodynamic formalism of infinite dynamical +systems, Gibbs measure should be an important concept in the future. + +\section{Inertial Manifolds and Global Attractors} + +Global attractor is a concept for dissipative systems, and inertial manifold +is a concept for strongly dissipative systems. A global attractor is a set +in the phase space, that attracts all the big balls to it as time approaches +infinity. An inertial manifold is an invariant manifold that attracts its +neighborhood +exponentially. A global attractor can be just a point, and often it is +just a set. There is no manifold structure with it. Based upon the idea +of reducing the complex infinite dimensional flows, like Navier-Stokes flow, +to finite dimensional flows, the concept of inertial manifold is introduced. +Often inertial manifolds are finite dimensional, have manifold structures, +and most importantly attract their neighborhoods exponentially. Therefore, one +hopes that the complex infinite dimensional dynamics is slaved by the +finite dimensional dynamics on the inertial manifolds. More ambitiously, +one hopes that a finite system of ordinary differential equations can be +derived to govern the dynamics on the inertial manifold. Under certain +spectral gap conditions, inertial manifolds can be obtained for many +evolution equations \cite{Con89} \cite{CFNT89}. Usually, global attractors +can be established rather easily. Unfortunately, inertial manifolds for +either 2D or 3D Navier-Stokes equations have not been established. + +\section{Zero-Dispersion Limit} + +Take the KdV equation as an example +\[ +u_t-6uu_x +\e^2 u_{xxx}=0\ , +\] +Lax \cite{LL83} asked the question: what happens to the dynamics as +$\e \ra 0$, i.e. what is the +zero-dispersion limit ? One can view the KdV equation as a singular +perturbation of the corresponding inviscid Burgers equation, +do the solutions of the KdV equation converge strongly, or weakly, or +not at all to those of the Burgers equation ? In a series of three +papers \cite{LL83}, Lax and Levermore investigated these questions. +It turns out that in the zero-dispersion limit, fast oscillations +are generated instead of shocks or multi-valuedness. Certain +weak convergences can also established. + +In comparison with the singular perturbation studies of soliton +equations in previous sections, our interests are focused upon +dynamical systems objects like invariant manifolds and homoclinic +orbits. In the zero-parameter limit, regularity of invariant manifolds +changes \cite{Li01b}. + +\section{Zero-Viscosity Limit} + +Take the viscous Burgers equation as an example +\[ +u_t+uu_x = \mu u_{xx}\ , +\] +through the Cole-Hopf transformation \cite{Hop50} \cite{Col51}, +this equation can be transformed into the heat equation which leads +to an explicit expression of the solution to the Burgers equation. +Hopf \cite{Hop50}, Cole \cite{Col51}, and Whitham \cite{Whi74} asked the +question on the zero-viscosity limit. It turns out that the limit +of a solution to the viscous Burgers equation can form shocks instead +of the multi-valuedness of the solution to the inviscid Burgers equation +\[ +u_t+uu_x = 0\ . +\] +That is, strong convergence does not happen. Nevertheless, the location +of the shock is determined by the multi-valued portion of the solution to +the inviscid Burgers equation. + +One can also ask the zero-viscosity limit question for Navier-Stokes equations. +In fact, this question is the core of the studies on fully developed +turbulence. In most of Kato's papers on fluids \cite{Kat72} \cite{Kat75} +\cite{Kat86}, he studied the zero-viscosity limits of the solutions to +Navier-Stokes equations in finite (or small) time interval. It turns +out that strong convergence can be established for 2D in finite time interval +\cite{Kat86}, and 3D in small time interval \cite{Kat75}. Order $\sqrt{\nu}$ +rate of convergence was obtained \cite{Kat72}. More recently, Constantin +and Wu \cite{CW96} had investigated the zero-viscosity limit problem for vortex +patches. + +\section{Finite Time Blowup} + +Finite time blowup and its general negative Hamiltonian criterion +for nonlinear Schr\"odinger equations have been well-known \cite{SS99}. +The criterion was obtained from a variance relation found by Zakharov +\cite{SS99}. Extension of such criterion to nonlinear Schr\"odinger equations +under periodic boundary condition, is also obtained \cite{Kav87}. +Extension of such criterion to Davey-Stewartson type equations +is also obtained \cite{GS90}. + +An explicit finite time blowup solution to the integrable Davey-Stewartson +II equation, was obtained by Ozawa \cite{Oza92} by inventing an extra +conservation law due to a symmetry. This shows that integrability +and finite time blowup are compatible. The explicit solution $q(t)$ has +the property that +\[ +\| q(t) \|_{L^2} = 2 \sqrt{\pi}\ ,\ \ \forall t\ , +\ \ \ \ q(t) \not\in H^1(R^2)\ ,\ \ \forall t\ . +\] +\[ +q(t) \in H^s(R^2)\ ,\ \ s \in (0,1)\ , \ \ t\in [0,T]\ , +\] +for some $T > 0$. When $t \ra T$, +\[ +\| q(t) \|_{H^s} \geq C |T-t|^{-s} \ra \infty\ , \ \ s \in (0,1)\ . +\] +The question of global well-posedness of Davey-Stewartson +II equation in $H^s(R^2)$ $(s >1)$ is open. + +Unlike the success in nonlinear wave equations, the search for +finite time blowup solutions for 3D Euler equations and other +equations of fluids has not been successful. The well-known result is the +Beale-Kato-Majda necessary condition \cite{BKM84}. There are results +on non-existence of finite time blowup \cite{CF01} \cite{CF02}. + +\section{Slow Collapse} + +Since 1960's, Yudovich \cite{Yud00} had been promoting the idea of +slow collapse. That is, although there is no finite time blowup, if +the function's derivative grows to infinity in norm as time approaches +infinity, then the function itself should gain randomness in space. + +The recent works of Fefferman and Cordoba \cite{CF01} \cite{CF02} +are in resonance with the slow collapse idea. Indeed, they found +the temporal growth can be as fast as $e^{e^t}$ and beyond. + +\section{Burgers Equation} + +Burgers equation +\[ +u_t+uu_x = \mu u_{xx} + f(t,x)\ , +\] +was introduced by J. M. Burgers \cite{Bur39} as a simple model of +turbulence. By 1950, E. Hopf \cite{Hop50} had conducted serious mathematical +study on the Burgers equation +\[ +u_t+uu_x = \mu u_{xx} \ , +\] +and found interesting mathematical structures of this equation including +the so-called Cole-Hopf transformation \cite{Hop50} \cite{Col51} and +Legendre transform. Moreover, +Hopf investigated the limits $\mu \ra 0$ and $t\ra \infty$. It turns out +that the order of taking the two limits is important. Especially with the +works of Lax \cite{Lax53} \cite{Lax54}, this led to a huge interests of +studies on conservation laws for many years. Since 1992, Sinai \cite{Sin92} +had led a study on Burgers equation with random data which can be random +initial data or random forcing \cite{EKMS00}. + +Another old model introduced by E. Hopf \cite{Hop48} did not catch +too much attention. As a turbulence model, the main drawback of Burgers +equation is lack of +incompressibility condition. In fact, there are studies focusing upon +only incompressibility \cite{GMW01}. To remedy this drawback, other models are +necessary. + +\section{Other Model Equations} + +A model to describe the hyperbolic structures in a neighborhood of +a fixed point of 2D Euler equation under periodic boundary condition, +was introduced in \cite{Li02l} \cite{Li02m} \cite{Li02e} \cite{Li02f}. +It is a good model in terms of capturing the linear instability. +At special value of a parameter, the explicit expression for the +hyperbolic structure can be calculated \cite{Li02l} \cite{Li02e}. +One fixed point's unstable manifold is the stable manifold of another +fixed point, and vice versa. All are 2D ellipsoidal surfaces. Together, +they form a lip shape hyperbolic structure. + +Another model is the so-called shell model \cite{KLWB95} \cite{SKL95} +\cite{KLS97} which model the energy transfer in the spectral space to +understand for example Kolmogorov spectra. + +\section{Kolmogorov Spectra and An Old Theory of Hopf} + +The famous Kolmogorov $-5/3$ law of homogeneous isotropic turbulence +\cite{Kol41a} \cite{Kol41b} \cite{Kol41c} still fascinates a lot of +researchers even nowadays. On the contrary, a statistical theory of +Hopf on turbulence \cite{Hop52} \cite{HT53} \cite{Hop57} \cite{Hop62} +\cite{VF86} has almost been forgotten. By introducing initial probability +in a function space, and realizing the conservation of probability under +the Navier-Stokes +flow, Hopf derived a functional equation for the characteristic functional +of the probability, around 1940, published in \cite{Hop52}. Initially, +Hopf tried to find some near Gaussian solution \cite{HT53}, then he +knew the work of Kolmogorov on the $-5/3$ law which is far from Gaussian +probability, and verified by experiments. So the topic was not pursued much +further. + +\section{Onsager Conjecture} + +In 1949, L. Onsager \cite{Ons49} conjectured that solutions of +the incompressible Euler equation with H\"older continuous velocity of +order $\nu > 1/3$ conserves the energy, but not necessarily if +$\nu \leq 1/3$. In terms of Besov spaces, if the weak solution of +Euler equation has certain regularity, it can be proved that energy +indeed conserves \cite{Eyi94} \cite{CET94}. + +\section{Weak Turbulence} + +Motivated by a study on 2D Euler equation \cite{Zak90}, Zakharov +led a study on infinte dimensional Hamiltonian systems in the +spectral space \cite{Zak98}. Under near Gaussian and small amplitude +assumptions, +Zakharov heuristically gave a closure relation for the averaged equation. +He also heuristically found some stationary solution to the averaged equation +which leads to power-type energy spectra all of which he called Kolmogorov +spectra. One of the mathematical manipulations he often used is the +canonical transformation, based upon which he classified the kinetic +Hamiltonian systems into 3-wave and 4-wave resonant systems. + +\section{Renormalization Idea} + +Renormalization group approach has been very successful in proving +universal property, especially Feigenbaum constants, of one dimensional +maps \cite{CEL80}. Renormalization-group-type idea has also been +applied to turbulence \cite{OY87}, although not very successful. + +\section{Random Forcing} + +As mentioned above, there are studies on Burgers equation under random +forcing \cite{EKMS00}, and studies on the structures of incompressible +vector fields \cite{GMW01}. Take any steady solution of 2D Euler equation, +it defines a Hamiltonian system with the stream function being the Hamiltonian. +There have been studies on such systems under random forcings \cite{FP94} +\cite{FK02}. + +\section{Strange Attractors and SBR Invariant Measure} + +Roughly speaking, strange attractors are attractors in which dynamics has +sensitive dependence upon initial data. If the Cantor sets proved in +previous sections are also attractors, then they will be strange attractors. +In view of the obvious fact that often the attractor may contain many +different objects rather than only the Cantor set, global +strange attractors are the ideal cases. For one dimensional logistic-type +maps, and two dimensional H\'enon map in the small parameter range such that +the 2D map can viewed as a perturbation of a 1D logistic map, there have been +proofs on the existence of strange attractors and SBR (Sinai-Bowen-Ruelle) +invariant measures \cite{BC91} \cite{BY92} \cite{BY93}. + +\section{Arnold Diffusions} + +The term Arnold diffusion started from the paper \cite{Arn64}. This paper +is right after Arnold +completed the proof on the persistence of KAM tori in \cite{Arn63}. If +the degree of freedom is 2, the persistent KAM tori are 2 dimensional, +and the level set of the perturbed Hamiltonian is 3 dimensional, then +the 2 dimensional KAM tori will isolate the level set, and diffusion is +impossible. In \cite{Arn64}, Arnold gave an explicit example to show that +diffusion is possible when the degree of freedom is more than 2. Another +interesting point is that in \cite{Arn64} Arnold also derived an integral +expression for a distance measurement, which is the so-called Melnikov +integral \cite{Mel63}. Arnold's derivation was quite unique. There have +been a lot of studies upon Arnold diffusions \cite{Loc99}, unfortunately, +doable examples are not much beyond that of Arnold \cite{Arn64}. +Sometimes, the diffusion can be very slow as the famous Nekhoroshev's theorem +shows \cite{Nek77} \cite{Nek79}. + +\section{Averaging Technique} + +For systems with fast small oscillations, long time dynamics are +governed by their averaged systems. There are quite good estimates +for long time deviation of solutions of the averaged systems from those +of the original systems \cite{Arn65b} \cite{Arn80}. Whether or not +the averaging method can be useful in chaos in partial differential +equations is still to be seen. + + + + + + + + + + + + + + + + + + + + +\clearpage{} + +\backmatter + +\bibliographystyle{amsalpha} +\begin{thebibliography}{A} + +\bibitem{AC91} +M.~J. Ablowitz and P.~A. Clarkson. +\newblock {\em Solitons, {N}onlinear {E}volution {E}quations and {I}nverse + {S}cattering}. +\newblock London Math. Soc. Lect. Note Ser. 149, Cambridge Univ. Press, 1991. + +\bibitem{AL76} +M.~J. Ablowitz and J.~F. Ladik. +\newblock A {N}onlinear {D}ifference {S}cheme and {I}nverse {S}cattering. +\newblock {\em Stud. Appl. Math.}, 55:213, 1976. + +\bibitem{AOT99} +M.~J. Ablowitz, Y.~Ohta, and A.~D. Trubatch. +\newblock On {D}iscretizations of the {V}ector {N}onlinear {S}chr{\"{o}}dinger + {E}quation. +\newblock {\em Phys. Lett. A}, 253, no.5-6:287--304, 1999. + +\bibitem{AOT00} +M.~J. Ablowitz, Y.~Ohta, and A.~D. 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Appl.}, 216:466--480, 1997. + +\end{thebibliography} + + +\clearpage{} + +\begin{theindex} + +\item Asymptotic phase shift +\subitem for DSII, 25 +\subitem for NLS, 14 + +\indexspace + +\item B\"acklund-Darboux transformation +\subitem for discrete NLS, 18 +\subitem for DSII, 21 +\subitem for NLS, 13 +\subitem for 2D Euler equation, 92 +\item Breather solution, 3 + +\indexspace + +\item Casimir, 96 +\item Chaos theorem, 76 +\item Conley-Moser conditions, 74 +\item Continued fraction, 100 +\item Counting lemma, 29 + +\indexspace + +\item Equivariant smooth linearization, 71 +\item Evolution operator, 2 + +\indexspace + +\item Floquet discriminant +\subitem for discrete NLS, 17 +\subitem for NLS, 12 +\item Fiber theorem +\subitem for perturbed DSII, 53 +\subitem for regularly perturbed NLS, 40 +\subitem for singularly perturbed NLS, 41 + +\indexspace + +\item Hadamard's method, 39 +\item Heteroclinic orbit, 2 +\item Homoclinic orbit, 2 + +\indexspace + +\item Inflowing invariance, 3 + +\indexspace + +\item $\la$-lemma, 83 +\item Lax pair +\subitem for discrete NLS, 16 +\subitem for DSII, 20 +\subitem for NLS, 11 +\subitem for Rossby wave equation, 93 +\subitem for 3D Euler equation, 93 +\subitem for 2D Euelr equation, 92 +\item Lie-Poisson bracket, 95 +\item Local invariance, 3 + +\indexspace + +\item Overflowing invariance, 3 + +\indexspace + +\item Perron's method, 39 +\item Persistence theorem +\subitem for perturbed DSII, 53 +\subitem for regularly perturbed NLS, 40 +\subitem for singularly perturbed NLS, 42 +\item Pseudo-orbit, 79 + +\indexspace + +\item Quadratic products of eigenfunctions +\subitem for DSII, 34 +\subitem for NLS, 16 + +\indexspace + +\item Shadowing lemma, 80 +\item Shift automorphism, 75 +\item Silnikov homoclinic orbit +\subitem for discrete NLS under perturbations, 65 +\subitem for regularly perturbed NLS, 55 +\subitem for singularly perturbed NLS, 57 +\subitem for vector NLS under perturbations, 65 +\item Spectral mapping theorem, 99 + +\indexspace + +\item Transversal homoclinic orbit +\subitem for a derivative NLS, 69 +\subitem for periodically perturbed SG, 69 + +\end{theindex} + +\clearpage{} +\end{document}