pade numerical schemes for the sine gordon equation abstract the sine gordon equation turn up in several problems in science and engineering although it is integrable in practical applications its numerical solution is powerful and versatile four novel implicit finite difference methods based on q s pade approximations with q s th order in space have been developed and analyzed for this equation all share the same treatment for the nonlinearity and integration in time concretely 0 4 2 2 2 4 and 4 4 pade methods additionally the energy conserving strauss vazquez scheme has been considered in a 0 2 pade implementation these methods have been compared among them for both the kink antikink and breather solutions in terms of global error computational cost and energy conservation the 0 4 and 2 4 pade methods are the most cost effective ones for small and large global error respectively our results indicate that spatial order of accuracy is more relevant to effectiveness of a method than energy conservation even in very long time integrations

nonlinear fourier transforms for the sine gordon equation in the quarter plane abstract using the unified transform also known as the fokas method the solution of the sine gordon equation in the quarter plane can be expressed in terms of the solution of a matrix riemann hilbert problem whose definition involves four spectral functions a b a b the functions a k and b k are defined via a nonlinear fourier transform of the initial data whereas a k and b k are defined via a nonlinear fourier transform of the boundary values in this paper we provide an extensive study of these nonlinear fourier transforms and the associated eigenfunctions under weak regularity and decay assumptions on the initial and boundary values the results can be used to determine the long time asymptotics of the sine gordon quarter plane solution via nonlinear steepest descent techniques

mathcal p mathcal t symmetric sine gordon breathers in this work we explore a prototypical example of a genuine continuum breather i e not a standing wave and the conditions under which it can persist in a symmetric medium as our model of interest we will explore the sine gordon equation in the presence of a symmetric perturbation our main finding is that the breather of the sine gordon model will only persist at the interface between gain and loss that symmetry imposes but will not be preserved if centered at the lossy or at the gain side the latter dynamics is found to be interesting in its own right giving rise to kink antikink pairs on the gain side and complete decay of the breather on the lossy side lastly the stability of the breathers centered at the interface is studied as may be anticipated on the basis of their delicate existence properties such breathers are found to be destabilized through a hopf bifurcation in the corresponding floquet analysis

energy conservative solutions to the system of full variational sine gordon equations in a unit sphere we establish the global existence of a conservative weak solution to the cauchy problem for a complete system of variational sine gordon equations which models the motion of long waves on a neutral dipole chain in the continuum limit in a unit sphere although singularities may develop in finite time the energy of the solution is conserved across singular times we also obtain the continuous dependence of solutions on the given initial data

spectral and modulational stability of periodic wavetrains for the nonlinear klein gordon equation abstract this paper is a detailed and self contained study of the stability properties of periodic traveling wave solutions of the nonlinear klein gordon equation u t t u x x v u 0 where u is a scalar valued function of x and t and the potential v u is of class c 2 and periodic stability is considered both from the point of view of spectral analysis of the linearized problem spectral stability analysis and from the point of view of wave modulation theory the strongly nonlinear theory due to whitham as well as the weakly nonlinear theory of wave packets the aim is to develop and present new spectral stability results for periodic traveling waves and to make a solid connection between these results and predictions of the formal modulation theory which has been developed by others but which we review for completeness

kam theorem revisited abstract a combinatorial proof of the kam theorem for the thirring model is presented by using renormalization group techniques usual in the formalism of quantum field theory

quasiperiodic motions in dynamical systems review of a renormalization group approach power series expansions naturally arise whenever solutions of ordinary differential equations are studied in the regime of perturbation theory in the case of quasiperiodic solutions the issue of convergence of the series is bedeviled by the so called small divisor problem in this paper we review a method recently introduced to deal with such a problem based on renormalization group ideas and multiscale techniques applications to both quasi integrable hamiltonian systems kolmogorov arnold moser kam theory and non hamiltonian dissipative systems are discussed the method is also suited to situations in which the perturbation series diverges and a resummation procedure can be envisaged leading to a solution which is not analytic in the perturbation parameter we consider explicitly examples of solutions which are only c in the perturbation parameter or even defined on a cantor set

breather solutions in periodic media for nonlinear wave equations existence proofs for breathers are very rare in the spatially homogeneous case up to rescaling the sine gordon equation partial 2 t u partial 2 x u sin u is the only nonlinear wave equation which is known to possess breather solutions for nonlinear wave equations in periodic media no examples of breather solutions have been known so far using spatial dynamics center manifold theory and bifurcation theory for periodic systems we construct for the first time such time periodic solutions of finite energy for a nonlinear wave equation s x partial 2 t u x t partial 2 x u x t q x u x t r x u x t 3 with spatially periodic coefficients s q and r on the real axis such breather solutions play an important role in theoretical scenarios where photonic crystals are used as optical storage

two dimensional synchronization patterns for minimum ambiguity a number of closely related combinatorial problems corresponding to specific assumptions about the type of time frequency sequence which may be appropriate in a particular application are formulated in terms of square or rectangular arrays of dots with appropriate constraints on the two dimensional correlation function the current state of knowledge concerning each of these problems is summarized it is hoped that more general constructions may be found leading to larger families of solutions as well as better computational algorithms for finding individual solutions which may lie outside of the general families

the eigenvalue problem for the focusing nonlinear schrodinger equations new solvable cases abstract in this paper we study the semi classical limit of the zakharov shabat eigenvalue problem for the focusing of nls with some specific initial data in all these cases the eigenvalue problem is reduced to connection problems for the hypergeometric equation and for other classical equations the special initial data suppl prog theor phys 55 1974 284 is contained in our family of initial data parameterized by a real parameter as a particular case 0 we find that beyond a certain value of the parameter the pure point spectrum becomes empty and all the scattering information is contained in the reflection coefficient

bryuno function and the standard map for the standard map the homotopically non trivial invariant curves of rotation number satisfying the bryuno condition are shown to be analytic in the perturbative parameter e provided e is small enough the radius of convergence of the lindstedt series sometimes called critical function of the standard map is studied and the relation with the bryuno function b is derived the quantity log 2 b is proved to be bounded uniformly in

2 d quasi m arrays and gold code arrays two dimensional quasi maximal area arrays quasi m arrays are proposed these quasi m arrays are so named because their cyclic correlation properties are very close to those of the m arrays these arrays are generated by modulo 2 addition of maximal length sequences the cyclic autocorrelation of any quasi m array is close to the delta function in addition the cyclic cross correlation is small compared with the cyclic autocorrelation peak two dimensional gold code arrays generated by the same construction method are also studied correlation properties of these gold code arrays are similar to those of the quasi m arrays the gold code arrays provide families of arrays in which any two arrays have the bound on their cyclic cross correlation these arrays are also shown to have the quasi orthogonal property

degenerate elliptic resonances quasi periodic motions on invariant tori of an integrable system of dimension smaller than half the phase space dimension may continue to exist after small perturbations the parametric equations of the invariant tori can often be computed as a formal power series in the perturbation parameter and can be given a meaning via resummations here we prove that for a class of elliptic tori a resummation algorithm can be devised and proved to be convergent thus extending to such lower dimensional invariant tori the methods employed to prove convergence of the lindstedt series either for the maximal i e kam tori or for the hyperbolic lower dimensional invariant tori

on semiclassical zero dispersion limit solutions of the focusing nonlinear schrodinger equation we calculate the leading order term of the solution of the focusing nonlinear cubic schrodinger equation nls in the semiclassical limit for a certain one parameter family of initial conditions this family contains both solitons and pure radiation in the pure radiation case our result is valid for all times t 0 we utilize the riemann hilbert problem formulation of the inverse scattering problem to obtain the leading order term of the solution error estimates are provided 2004 wiley periodicals inc

purely imaginary eigenvalues of zakharov shabat systems zakharov shabat systems with single hump and real but not necessarily symmetric potentials are shown to have purely imaginary eigenvalues only coupled with examples of double hump potentials with nonimaginary eigenvalues this establishes that confinement of zakharov shabat eigenvalues to the imaginary axis is a characteristic of potentials whose energy is concentrated in a single region of the time axis

the n soliton of the focusing nonlinear schrodinger equation for n large we present a detailed analysis of the solution of the focusing nonlinear schrodinger equation with initial condition x 0 n sech x in the limit n 1 we begin by presenting new and more accurate numerical reconstructions of the n soliton by inverse scattering numerical linear algebra for n 5 10 20 and 40 we then recast the inverse scattering problem as a riemann hilbert problem and provide a rigorous asymptotic analysis of this problem in the large n limit for those x t where results have been obtained by other authors we improve the error estimates from o n 1 3 to o n 1 we also analyze the fourier power spectrum in this regime and relate the results to the optical phenomenon of supercontinuum generation we then study the n soliton for values of x t where analysis has not been carried out before and we uncover new phenomena the main discovery of this paper is the mathematical mechanism for a secondary caustic phase transition which turns out to differ from the mechanism that generates the primary caustic the mechanism for the generation of the secondary caustic depends essentially on the discrete nature of the spectrum for the n soliton and more significantly cannot be recovered from an analysis of an ostensibly similar riemann hilbert problem in the conditions of which a certain formal continuum limit is taken on an ad hoc basis

image coding using robust quantization for noisy digital transmission a robust quantizer is developed for encoding memoryless sources and transmission over the binary symmetric channel bsc the system combines channel optimized scalar quantization cosq with all pass filtering the latter performed using a binary phase scrambling descrambling method applied to a broad class of sources the robust quantizer achieves the same performance as the gaussian cosq for the memoryless gaussian source this quantizer is used in image coding for transmission over a bsc the peak signal to noise ratio psnr performance degrades gracefully as the channel bit error rate increases

on the variational structure of breather solutions i sine gordon equation abstract in this paper we describe stability properties of the sine gordon breather solution these properties are first described by suitable variational elliptic equations which also implies that the stability problem reduces in some sense to i the study of the spectrum of explicit linear systems and ii the understanding of how bad directions if any can be controlled using low regularity conservation laws then we discuss spectral properties of a fourth order linear matrix system using numerical methods we confirm that all spectral assumptions leading to the h 2 h 1 stability of sg breathers are numerically satisfied even in the ultra relativistic singular regime

summation of divergent series and borel summability for strongly dissipative differential equations with periodic or quasiperiodic forcing terms we consider a class of second order ordinary differential equations describing one dimensional systems with a quasiperiodic analytic forcing term and in the presence of damping as a physical application one can think of a resistor inductor varactor circuit with a periodic or quasiperiodic forcing function even if the range of applicability of the theory is much wider in the limit of large damping we look for quasiperiodic solutions which have the same frequency vector of the forcing term and we study their analyticity properties in the inverse of the damping coefficient we find that even the case of periodic forcing terms is nontrivial as the solution is not analytic in a neighborhood of the origin it turns out to be borel summable in the case of quasiperiodic forcing terms we need renormalization group techniques in order to control the small divisors arising in the perturbation series we show the existence of a summation criterion of the series in this case also however this cannot be interpreted as borel summability

resummation of perturbation series and reducibility for bryuno skew product flows we consider skew product systems on d l sl 2 for bryuno base flows close to constant coefficients depending on a parameter in any dimension d and we prove reducibility for a large measure set of values of the parameter the proof is based on a resummation procedure of the formal power series for the conjugation and uses techniques of renormalisation group in quantum field theory

krein signatures for the faddeev takhtajan eigenvalue problem one of the difficulties in analyzing eigenvalue problems that arise in connection with integrable systems is that they are frequently non self adjoint making it difficult to determine where the spectrum lies in this paper we consider the problem of locating and counting the discrete eigenvalues associated with the faddeev takhtajan eigenvalue problem for which the sine gordon equation is the isospectral flow in particular we show that for potentials having either zero topological charge or topological charge 1 and satisfying certain monotonicity conditions the point spectrum lies on the unit circle and is simple furthermore we give an exact count of the number of eigenvalues this result is an analog of that of klaus and shaw for the zakharov shabat eigenvalue problem we also relate our results as well as those of klaus and shaw to the krein stability theory for j unitary matrices in particular we show that the eigenvalue problem associated to the sine gordon equation has a j unitary structure and under the above conditions the point eigenvalues have a definite krein signature and are thus simple and lie on the unit circle

kam theory in configuration space and cancellations in the lindstedt series the kam theorem for analytic quasi integrable anisochronous hamiltonian systems yields that the perturbation expansion lindstedt series for any quasi periodic solution with diophantine frequency vector converges if one studies the lindstedt series by following a perturbation theory approach one finds that convergence is ultimately related to the presence of cancellations between contributions of the same perturbation order in turn this is due to symmetries in the problem such symmetries are easily visualised in action angle coordinates where the kam theorem is usually formulated by exploiting the analogy between lindstedt series and perturbation expansions in quantum field theory and in particular the possibility of expressing the solutions in terms of tree graphs which are the analogue of feynman diagrams if the unperturbed system is isochronous moser s modifying terms theorem ensures that an analytic quasi periodic solution with the same diophantine frequency vector as the unperturbed hamiltonian exists for the system obtained by adding a suitable constant counterterm to the vector field also in this case one can follow the alternative approach of studying the perturbation expansion for both the solution and the counterterm and again convergence of the two series is obtained as a consequence of deep cancellations between contributions of the same order in this paper we revisit moser s theorem by studying the perturbation expansion one obtains by working in cartesian coordinates we investigate the symmetries giving rise to the cancellations which makes possible the convergence of the series we find that the cancellation mechanism works in a completely different way in cartesian coordinates and the interpretation of the underlying symmetries in terms of tree graphs is much more subtle than in the case of action angle coordinates

breathers as homoclinic geometric wave maps a classical breather solution of the sine gordon equation can be viewed as a homoclinic wave map from periodic minkowski space mp2 into s2 this wave map is an unstable orbit of a simple steady state wave map homoclinic wave maps are also constructed into non round spheres

inverse scattering transform for systems with rational spectral dependence abstract we characterize a class of first order systems for which the inverse scattering problems can be formulated as matrix riemann hilbert problems for this class complete direct and inverse scattering results are obtained

decay of a square pulse to sine gordon breathers we investigate numerically and analytically the existence of thresholds for the production of sine gordon single and multiple breathers from simple initial pulses

kam theorem and quantum field theory we give a new proof of the kam theorem for analytic hamiltonians the proof is inspired by a quantum field theory formulation of the problem and is based on a renormalization group argument treating the small denominators inductively scale by scale the crucial cancellations of resonances are shown to follow from the ward identities expressing the translation invariance of the corresponding field theory

periodic nonlinear waves on a half line nontrivial solutions of the equationutt uxx g u which are 2 periodic int and which decay asx are shown to exist ifg a 0 andg 0 1 breather like solutions which also decay asx can be interpreted as homoclinic solutions in thex dynamics their existence is still in question for generalg

quasiperiodic attractors borel summability and the bryuno condition for strongly dissipative systems we consider a class of ordinary differential equations describing one dimensional analytic systems with a quasiperiodic forcing term and in the presence of damping in the limit of large damping under some generic nondegeneracy condition on the force there are quasiperiodic solutions which have the same frequency vector as the forcing term we prove that such solutions are borel summable at the origin when the frequency vector is either any one dimensional number or a two dimensional vector such that the ratio of its components is an irrational number of constant type in the first case the proof given simplifies that provided in a previous work of ours we also show that in any dimension d for the existence of a quasiperiodic solution with the same frequency vector as the forcing term the standard diophantine condition can be weakened into the bryuno condition in all cases under a suitable positivity condition the quasiperiodic solution is proved to describe a local attractor

modulating pulse solutions to quadratic quasilinear wave equations over exponentially long length scales this paper presents an existence proof for modulating pulse solutions to a wide class of quadratic quasilinear klein gordon equations of the form partial t 2 u partial x 2 u u f 1 u partial x u partial t u partial x 2 u f 2 u partial x u partial t u modulating pulse solutions consist of a pulse like envelope advancing in the laboratory frame and modulating an underlying wave train they are also referred to as moving breathers since they are time periodic in a moving frame of reference the problem is formulated as an infinite dimensional dynamical system with three stable three unstable and infinitely many neutral directions by transforming part of the equation into a normal form with an exponentially small remainder term and using a generalisation of local invariant manifold theory to the quasilinear setting we prove the existence of small amplitude modulating pulses on domains in space whose length is exponentially large compared to the magnitude of the pulse

resonances radiation damping and instabilitym in hamiltonian nonlinear wave equations we consider a class of nonlinear klein gordon equations which are hamiltonian and are perturbations of linear dispersive equations the unperturbed dynamical system has a bound state a spatially localized and time periodic solution we show that for ge neric nonlinear hamiltonian perturbations all small amplitude so lutions decay to zero as time tends to infinity at an anomalously slow rate in particular spatially localized and time periodic solutions of the linear problem are destroyed by generic nonlinear hamiltonian perturbations via slow radiation of energy to infinity these solutions can therefore be thought of as metastable states the main mecha nism is a nonlinear resonant interaction of bound states eigenfunc tions and radiation continuous spectral modes leading to energy transfer from the discrete to continuum modes this is in contrast to the kam theory in which appropriate nonresonance conditions im ply the persistence of invariant tori a hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the fermi golden rule arising in the theory of resonances in quantum mechanics the techniques used involve i a time dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues ii a generalization of the hamiltonian normal form appropriate for infinite dimensional dis persive systems and iii ideas from scattering theory the arguments are quite general and we expect them to apply to a large class of

small generalized breathers with exponentially small tails for klein gordon equations abstract we consider a class of nonlinear klein gordon equation u t t u x x u f u and show that generically there exist small breathers with exponentially small tails

conservation of resonant periodic solutions for the one dimensional nonlinear schrodinger equation we consider the one dimensional nonlinear schrodinger equation with dirichlet boundary conditions in the fully resonant case absence of the mass term we investigate conservation of small amplitude periodic solutions for a large measure set of frequencies in particular we show that there are infinitely many periodic solutions which continue the linear ones involving an arbitrary number of resonant modes provided the corresponding frequencies are large enough say greater than a certain threshold value depending on the number of resonant modes if the frequencies of the latter are close enough to such a threshold then they can not be too distant from each other hence we can interpret such solutions as perturbations of wave packets with large wave number

birkhoff kolmogorov arnold moser tori in convex hamiltonian systems for the hamiltonian systems of kam type it is proved that some lower dimensional invariant tori always exist in the resonance gaps although those maximum tori can not survive small perturbations in the generic case

exact solutions of semiclassical non characteristic cauchy problems for the sine gordon equation abstract the use of the sine gordon equation as a model of magnetic flux propagation in josephson junctions motivates studying the initial value problem for this equation in the semiclassical limit in which the dispersion parameter e tends to zero assuming natural initial data having the profile of a moving 2 kink at time zero we analytically calculate the scattering data of this completely integrable cauchy problem for all e 0 sufficiently small and further we invert the scattering transform to calculate the solution for a sequence of arbitrarily small e this sequence of exact solutions is analogous to that of the well known n soliton or higher order soliton solutions of the focusing nonlinear schrodinger equation we then use plots obtained from a careful numerical implementation of the inverse scattering algorithm for reflectionless potentials to study the asymptotic behavior of solutions in the semiclassical limit in the limit e 0 one observes the appearance of nonlinear caustics i e curves in space time that are independent of e but vary with the initial data and that separate regions in which the solution is expected to have different numbers of nonlinear phases in the appendices we give a self contained account of the cauchy problem from the perspectives of both inverse scattering and classical analysis picard iteration specifically appendix a contains a complete formulation of the inverse scattering method for generic l 1 sobolev initial data and appendix b establishes the well posedness for l p sobolev initial data which in particular completely justifies the inverse scattering analysis in appendix a

on the long time limit of semiclassical zero dispersion limit solutions of the focusing nonlinear schrodinger equation pure radiation case in a previous paper 13 we calculated the leading order term q0 x t e of the solution ofq x t e the focusing nonlinear cubic schrodinger nls equation in the semiclassical limit e 0 for a certain one parameter family of initial conditions this family contains both solitons and pure radiation in the pure radiation case our result is valid for all times t 0 the aim of the present paper is to calculate the long term behavior of the semiclassical solution q x t e in the pure radiation case as before our main tool is the riemann hilbert problem rhp formulation of the inverse scattering problem and the corresponding system of moment and integral conditions known also as a system of modulation equations c 2006 wiley periodicals inc

conservative solutions to a nonlinear variational sine gordon equation abstract this paper is concerned with a nonlinear variational sine gordon equation u t t c u c u u x x 2 2 s i n 2 u 0 which describes the motion of long waves on a neutral dipole chain in the continuum limit and a few other physical phenomena we establish the global existence of an energy conservative weak solution to its cauchy problem for initial data of finite energy to deal with the possible blowup of solutions we introduce a new set of variables depending on the energy whereby all singularities are resolved