数理解析研究所講究録
900 巻 1995 年 42-46

42

Qualitative behavior of solutions to Ginzburg-Landau type systems
Yoshihisa Morita
Department of Applied Mathematics and Informatics
Ryukoku University
Seta Ohtsu 520-21
森田善久

龍谷大学理工学部
数理情報学科

1Introduction

We are concerned with the time-dependent Ginzburg-Landau (G-L) equation on a circle:$U_{t}=U+\lambda(xx-1|U|2)U$,$X\in^{s^{1}=R}/2\pi Z$

(1)

where

$U=$ ,and$\lambda$$|U|=(u+v^{2})^{1/}22$,,$U_{xx}= \frac{\partial^{2}}{\partial x^{2}}U$$U_{t}= \frac{\partial}{\partial t}U$

is a positive parameter. This equation has a Lyapunov function:$\mathcal{E}(U):=\int_{S}1\frac{\lambda}{2}\{|U_{x}|^{2}+(1-|U|2)^{2}\}dx$(2)generated by solutions admits a global attractor $A$ , that is,and the semiflow inthe maximal compact invariant set. Considering this fact and that the semiflow is analytic,
we see from [5] that every solution converges to an equilibrium solution, namely a solution
to$x\in S^{1}$(3)$U_{xx}+\lambda(1-|U|^{2})U=0$ ,.
$H^{1}(S^{1}; R^{2})$Hence if we could obtain all the solution to (3) and Morse index of them, we would be able
to discuss the existence of connecting orbits between equilibrium solutions by applying theConley index theory (see [3]). As seen below, however, it is not so easy to determine theMorse index of the solutions because Equation (3) has many secondary bifurcating solutions
for large . In addition when we consider a little generalized system written as$\lambda$$U_{t}= \frac{1}{a(x)}(a(X)Ux)_{x}+\lambda(1-|U|^{2})U$,(4)$a(\cdot)\in C^{22}(S^{1};R)$, $a(x)>0$ ,such an approach is not useful for studying of the dynamical structure.

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Here we take account of some qualitaive behaivior of the solution to Equation (1) (or (4))
and give an insight into the structure of the attractor. Actually we propose a non-trivial
Morse decompotion of the attractor, where the Morse decomposition means a decompositionof the attractor into a finite number of disjoint compact invariant sets. We believe that thisdecomposition will be useful when we discuss the existence of connecting orbits between theequilibrium solutions.

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Equilibrium Solutions

We state a structure of equilibrium solutions to (3); all the solutions can be classified asfollows:i) constant solutions:$U=$ ( is any constant);$U=0$ ,$\xi$ii) collinear solutions:$U=w(x)$ ,$\xi$

is any given number and $w(x)$ is a non-constant solution to$w_{xx}+\lambda(1-w2)w=0$ ,$x\in S^{1}$(5).

In this case the projection of solution $U(x)$ into $(u, v)$ -plane,$C(U):=\{(u(X), v(_{X)})\in R^{2} :x\in S^{1}\}$

make a line segment.
iii) winding solutions: $C(U)$ for the solution $U(x)$ winds around $\mathcal{O}=(0,0)$ clockwise oranti-clockwise, where the winding number of $C(U)$ is defined as$W(C(U)):= \frac{\theta(2\pi)-\theta(0)}{2\pi}$where$\theta(x)$is the angle of the vector $(u(x), v(x))$ .

Remark 2.1 We easily see that for$\lambda>m^{2}$Equation (3) has solutions:$U(x)=a_{m}(\cos(_{X}+\sin(_{X+\xi)}\xi)),$$a_{m}=\sqrt{1-m^{2}/\lambda}$$U(x)=(\cos(\sin(\xi)\xi)\mathrm{I}^{\phi_{\lambda}(X})$

(6)

,

(7)

,

as. Inis a solution of (5) which bifurcated from origin atwhereplacetake$(\lambda=1,2,\cdots,2m-1)$
secondary,frombifurcations
passes throughisit
Unfortunately
appear.
solutionswinding
type
difficultandthe solution (6)ofdifferent
to solve the linearized eigenvalue problem of the secondary bifurcating solutions.
$\lambda=m^{2}$$\phi_{\lambda}(x)$$3m^{2}-\lambda^{2}/2$$additi_{\mathit{0}}n_{f}$$\lambda$

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3Main Theorem

Before stating main results obtained in [4], we introduce notations. We$\backslash \mathrm{w}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{e}$$U^{\gamma}(t, x)=:=U(t, x)$ ,and define$Z_{N}(U(t, \cdot);\gamma):=$provided that$u^{\gamma}(t, x)${the number of zeros of $x-u^{\gamma}(t,$ }$x)$

is not identically zero. Denote an alpha and omefa-limit sets by,$\alpha(U_{0})=\mathrm{n}_{\tau\leq 0}c\ell(\bigcup_{\mathcal{T}\geq^{U(t}}t, \cdot;U_{0}))$$\omega(U_{0})=\bigcap_{\tau\geq 0}C\ell(\bigcup_{\mathcal{T}\leq^{U(t}}t, \cdot;U_{0}))$,where $U(t, \cdot;U0)$ is a solution of (1) with $U(0, \cdot)=U_{0}$ . Set$M_{2k}=$$\{U_{0}\in A:Z_{N}(U(t, \cdot;U_{0);\gamma})=2k(-\infty<t<\infty)$

for any$M_{2k}^{\mathrm{c}}=$$M_{2k}$$0\not\in\alpha(U_{0})\}$,$\{U\in A:U=\phi_{k}(x)(0\leq\gamma<2\pi)$ ,$\phi_{k}(x)$Note thatand$\gamma$

is a bifurcating solution from zero$\mathrm{a}\mathrm{t}\lambda=k^{2}$

contains the winding solutions with winding number}.

$k\mathrm{o}\mathrm{r}-k$.

, then $M(2k):=M_{2k}\cup M_{2k}^{\mathrm{c}}$ is non-empty and compact invariant set
Lemma 3.1 If
in A. Moreover $M(2k)\cap M(2j)=\emptyset$ for $k\neq j,$ $0<k^{2},j^{2}<\lambda$ .
$k^{2}<\lambda$

This lemma leads us to a Morse decompotion as follows:

Theorem 3.2 Given any$\lambda$in $(k^{2}, (k+1)^{2})$ , set $\Lambda=\{0,2,4, \cdots, 2k, 2k+1\}$ and$M(0)=\{U= :0\leq\gamma<2\pi\}$ ,, where
Then $A$ is decomposed as$U_{0}\inC$orbitsj hence forthere exists $K,$$M(L)$ .
$A=\mathcal{M}\cup C$$M(2k+1)=\{0\}$ .

and$K>L$ such that$\mathcal{M}=\bigcup_{K\in\Lambda}M(K)$$L\in\Lambda,$$C$

is the set of connecting$\alpha(U_{0})\subset M(K),\omega(U_{0})\subset$

Remark 3.3 With a little modification of the definition of the Morse sets $M(K)$ , we also
obtain quite a similar Morse decompotion of the global attractor for (4).
be a family of parametrized domain as defined byLet$\Omega(\epsilon)$$\Omega(\epsilon)=\{x=(r\cos\theta, r\sin\theta)\in R^{2} :and consider the G-L equation in$\Omega(\epsilon)$1<|x|<1+\epsilon a(\theta), 0\leq\theta<2\pi\}$,with Neumann boundary condition:$U_{t}=\triangle U+\lambda(1-|U|^{2})U$in$\Omega(\epsilon)$,$\frac{\partial u}{\partial n}=0$on$\partial\Omega(\epsilon)$.

(8). Indeed theThen Equation (4) can be regarded as a limit equation of (8) when
dynamical system (of finite dimension) on the global attractor
for (8) is a nice perturbed
system of the one for (4) in some sense (see [1]). Hence the Morse decompotion obtained
above persists under the perturbation caused by this deformation of the domain.
$\epsilonarrow 0$$A_{\epsilon}$

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References[1] J. K. Hale and G. Raugel, Reaction-Diffusion Equation on Thin Domains, J. Math.
Pures Appl. Vol.71, 1992, pp.33-95.
[2] S. Jimbo and Y. Morita, Stability of Non-constant Steady State Solutions to a GinzburgLandau Equation in Higher Space Dimensions, Nonlinear Analysis: T.M.A., Vol.22,
1994, pp.753-770.

[3] K. Mischaikow and Y. Morita, Dynamics on the Global Attractor of a Gradient Flow
Arising from the Ginzburg-Landau Equation, Japan Journal of Industorial and AppliedMathematics, Vol.11, 1994, pp.185-202.
[4] K. Mischaikow and Y. Morita, Dynamical Structure of Ginzburg-Landau Type Systems
on , in preparation.
$S^{1}$[5] L. Simon, Asymptotics for a class of non-linear evolution equations, with applications
to geometric problems, Annals of Mathematics, Vol.118, 1983, pp.525-571.