squared wavefunctions approach to periodic solutions of vector nonlinear schrodinger equation we develop a method of squared wavefunctions for the vector nonlinear schrodinger equation the squared wavefunctions of the octet representation of su 3 group give periodic solutions in terms of weierstrass elliptic functions specific limits of the obtained solution are the plane wave the soliton and cnoidal waves which were previously obtained using the ansatz of stationary motion

existence and bifurcation of solutions for an elliptic degenerate problem abstract we investigate the existence multiplicity and bifurcation of solutions of a model nonlinear degenerate elliptic differential equation x 2 u u u p 1 u in 0 1 u 0 u 1 0 this model is related to a simplified version of the nonlinear wheeler dewitt equation as it appears in quantum cosmological models we prove the existence of multiple positive solutions more precisely we show that there exists an infinite number of connected branches of solutions which bifurcate from the bottom of the essential spectrum of the corresponding linear operator nous etudions ici l existence multiplicite et proprietes de bifurcation des solutions d un probleme elliptique degenere x 2 u u u p 1 u in 0 1 u 0 u 1 0 ce probleme modele est proche d une version simplifiee et non lineaire de l equation de wheeler dewitt utilisee dans des modeles de cosmologie quantique nous prouvons l existence d une infinite de branches de solutions qui bifurquent a partir de l infimum du spectre continu de l operateur lineaire correspondant

backlund transformation in bilinear form for a higher order nonlinear schrodinger equation backlund transformation in bilinear form is presented for a higher order nonlinear schrodinger equation which describes the propagation of ultrashort light pulses in optical fibers with symbolic computation and starting from the backlund transformation the analytical soliton solution is obtained from a trivial solution and the inverse scattering transform scheme is also derived furthermore the n soliton solution in double wronskian form is given and the value of the arbitrary constant appearing in the backlund transformation is determined for a transformation between the n 1 and n soliton solutions the results obtained from the backlund transformation might be valuable in optical communications

analysis of the linearization around a critical point of an infinite dimensional hamiltonian system partial differential equations that conserve energy can often be written as infinite dimensional hamiltonian systems of the following general form du dt je u where j x x is a symplectic matrix i e jj 1 and e x r is a c 2 functional defined on some hilbert space x

count of eigenvalues in the generalized eigenvalue problem we study isolated and embedded eigenvalues in the generalized eigenvalue problem defined by two self adjoint operators with a positive essential spectrum and a finite number of isolated eigenvalues the generalized eigenvalue problem determines the spectral stability of nonlinear waves in infinite dimensional hamiltonian systems the theory is based on pontryagin s invariant subspace theorem and extends beyond the scope of earlier papers of pontryagin krein grillakis and others our main results are i the number of unstable and potentially unstable eigenvalues equals the number of negative eigenvalues of the self adjoint operators ii the total number of isolated eigenvalues of the generalized eigenvalue problem is bounded from above by the total number of isolated eigenvalues of the self adjoint operators and iii the quadratic forms defined by the two self adjoint operators are strictly positive on the subspace related to the continuous spectrum of the generalized eigenvalue problem applicatio

hamiltonian bifurcations of the spatial structure for coupled nonlinear schro dinger equations abstract a coupled pair of nonlinear schrodinger equations is used as a model to study bifurcations in space of translation invariant infinite dimensional hamiltonian systems it is usual in translation invariant systems to fix the spatial structure for example spatially periodic functions or functions that decay at infinity in this paper the temporal structure relative equilibria or time periodic functions for example is fixed and an evolution equation in space is constructed for the unknown spatial structure the spatial evolution equation is shown to be hamiltonian and its basic properties are related to the fluxes of energy and momentum of the time dependent problem the gauge fluxes are used to reduce the spatial system to a poisson system on a reduced phase space with two casimir functions hamiltonian bifurcation theory is used to analyze the spatial hamiltonian system particular attention is paid to the role of invariants in spatial bifurcations and temporal stability for spatially periodic travelling waves a global result relating the momentum flux to the position of spatial floquet multipliers is proved

on periodic waves for sine and sinh gordon equations abstract results of existence as well as orbital stability instability of periodic waves associated with sine gordon and sinh gordon equations will be treated in this manuscript the sine gordon equation is known to be completely integrable exactly solvable and it has several special solutions as soliton and breather solutions we obtain in both cases a periodic sign changed wave which is smooth when posed on whole real line the main tool used is the grillakis shatah and strauss theory

on instability of excited states of the nonlinear schrodinger equation we introduce a new notion of linear stability for standing waves of the nonlinear schrodinger equation nls which requires not only that the spectrum of the linearization be real but also that the generalized kernel be not degenerate and that the signature of all the positive eigenvalues be positive we prove that excited states of the nls are not linearly stable in this more restrictive sense we then give a partial proof that this more restrictive notion of linear stability is a necessary condition to have orbital stability

stability of standing waves for nls with perturbed lame potential abstract we perturb a linear schrodinger equation with lame potential with a small positive or negative potential the new perturbed operator has one or more eigenvalues at most one in each spectral gap we then add a nonlinear term and study the stability of the corresponding nonlinear stationary waves

the sub harmonic bifurcation of stokes waves steady periodic water waves on the free surface of an infinitely deep irrotational flow under gravity without surface tension stokes waves can be described in terms of solutions of a quasi linear equation which involves the hilbert transform and which is the euler lagrange equation of a simple functional the unknowns are a 2 periodic function w which gives the wave profile and the froude number a dimensionless parameter reflecting the wavelength when the wave speed is fixed and vice versa

graphical krein signature theory and evans krein functions two concepts evidently very different in nature have proved to be useful in analytical and numerical studies of spectral stability in nonlinear wave theory i the krein signature of an eigenvalue a quantity usually defined in terms of the relative orientation of certain subspaces that is capable of detecting the structural instability of imaginary eigenvalues and hence their potential for moving into the right half plane leading to dynamical instability under perturbation of the system and ii the evans function an analytic function detecting the location of eigenvalues one might expect these two concepts to be related but unfortunately examples demonstrate that there is no way in general to deduce the krein signature of an eigenvalue from the evans function for example by studying derivatives of the latter the purpose of this paper is to recall and popularize a simple graphical interpretation of the krein signature well known in the spectral theory of polynomial operator pencils once esta

computing periodic orbits with arbitrary precision this paper deals with the computation of periodic orbits of dynamical systems up to any arbitrary precision these very high requirements are useful for example in the studies of complex pole location in many physical systems the algorithm is based on an optimized shooting method combined with a numerical ordinary differential equation ode solver tides that uses a taylor series method nowadays this methodology is the only one capable of reaching precision up to thousands of digits for odes the method is shown to be quadratically convergent some numerical tests for the paradigmatic lorenz model and the henon heiles hamiltonian are presented giving periodic orbits up to 1000 digits

spectral stability and time evolution of n solitons in the kdv hierarchy this paper concerns spectral stability and time evolution of n solitons in the korteweg de vries kdv hierarchy with mixed commuting time flows spectral stability problem is analysed by using a pair of self adjoint operators with finite numbers of negative eigenvalues we show that the absence of unstable eigenvalues in the stability problem is related to the absence of negative eigenvalues of these operators in the constrained function spaces time evolution of n solitons is uniquely characterized from the inverse scattering transform technique

stability properties of steady water waves with vorticity we present two stability analyses for exact periodic traveling water waves with vorticity the first approach leads in particular to linear stability properties of water waves for which the vorticity decreases with depth the second approach leads to a formal stability property for long water waves that have small vorticity and amplitude although we do not use a small amplitude or long wave approximation 2006 wiley periodicals inc

spatial hamiltonian structure energy flux and the water wave problem travelling waves and relative equilibrium states of infinite dimensional hamiltonian systems are considered with particular attention to the water wave problem relative equilibria correspond to solutions that are stationary in a moving frame of reference the governing equations for relative equilibria are recast as a hamiltonian evolution equation in space the mass momentum and energy flux densities play a fundamental role in the spatial hamiltonian structure the energy flux generates the basic symplectic operator the momentum flux is the spatial hamiltonian and the mass flux is a second independent spatial integral a complete theory is given for the water wave problem with surface tension the hamiltonian structure is used to show the existence of dual variational principles for travelling waves and to give an analysis of a singularity in the dispersion relation for capillary gravity waves from a spatial hamiltonian systems viewpoint

an instability mechanism for radially symmetric standing waves of a nonlinear schrodinger equation abstract a condition is proved for the spectrum of nonlinear schrodinger equations linearised at a standing wave to have a positive eigenvalue the standing waves considered are radially symmetric ones in higher space dimensions the instability result is applied to show that if there are multiple asymptotically positive nondegenerate waves with a fixed number of zeroes then there is an unstable one the techniques used are dynamical systems arguments and involve a shooting argument in the space of lagrangian planes

multi symplectic lagrangian one dimensional gas dynamics the equations of lagrangian ideal one dimensional compressible gas dynamics are written in a multi symplectic form using the lagrangian mass coordinate m and time t as independent variables and in which the eulerian position of the fluid element x x m t is one of the dependent variables this approach differs from the eulerian multi symplectic approach using clebsch variables lagrangian constraints are used to specify equations for xm xt and st consistent with the lagrangian map where s is the entropy of the gas we require st 0 corresponding to advection of the entropy s with the flow we show that the lagrangian hamiltonian equations are related to the de donder weyl multi momentum formulation the pullback conservation laws and the symplecticity conservation laws are discussed the pullback conservation laws correspond to invariance of the action with respect to translations in time energy conservation and translations in m in noether s theorem the conservation law due to m translatio

on the behavior of the lorenz equation backward in time abstract the sets of solutions to the lorenz equations that exist backward in time and are bounded at an exponential rate determined by the eigenvalues of the linear part of the equation are examined the set associated with the middle eigenvalue is shown to project surjectively onto a plane thereby providing a lower estimate for its dimension specific bounds are also found for a cone containing this set

spectral stability analysis for special solutions of second order in time pdes the higher dimensional case abstract we develop a general theory to treat the linear stability of certain special solutions of second order in time evolutionary pde we apply these results to standing waves of the following problems the klein gordon equation for which we consider both ground states and certain excited states the klein gordon zakharov system and the beam equation we also discuss applications to excited states for the klein gordon model as well as multidimensional traveling waves not necessarily homoclinic to zero for general second order equations of this type in all cases our abstract results provide a complete characterization of the linear stability of such solutions

eigenvalues of zero energy in the linearized nls problem we study a pair of neutrally stable eigenvalues of zero energy in the linearized nls equation we prove that the pair of isolated eigenvalues where each eigenvalue has geometric multiplicity one and algebraic multiplicity n is associated with 2p negative eigenvalues of the energy operator where p n 2 if n is even and p n 1 2 or p n 1 2 if n is odd when the potential of the linearized nls problem is perturbed due to parameter continuations we compute the exact number of unstable eigenvalues that bifurcate from the neutrally stable eigenvalues of zero energy

benjamin feir type instability of sine gordon equation and spectrum of lame equation the benjamin feir type instability of the nonhomoclinic wavetrain solution of the sine gordon equation against the spatially periodic small perturbation is observed by relating the fourier coefficients of the first order approximate solution of fourier type to the band structure of the spectrum of the lame equation

structural stability of non ground state traveling waves of coupled nonlinear schrodinger equations abstract we consider the linear stability and structural stability of non ground state traveling waves of a pair of coupled nonlinear schrodinger equations cnls which describe the evolution of co propagating polarized pulses in the presence of birefringence viewing the cnls equations as a hamiltonian perturbation of the manakov equations we find parameter regimes in which there are two stable families of traveling waves the usual variational methods for stability analysis of ground states do not apply instead we employ a liapunov schmidt type reduction to detect eigenvalues bifurcating from the imaginary axis we also demonstrate the instability of a family of vector solitons

benjamin feir instability in nonlinear dispersive waves in this paper the authors extended the derivation to the nonlinear schrodinger equation in two dimensions modified by the effect of non uniformity the authors derived several classes of soliton solutions in 2 1 dimensions when the solution is assumed to depend on space and time only through a single argument of the function they showed that the two dimensional nonlinear schrodinger equation is reduced either to the sine gordon for the hyperbolic case or sinh gordon equations for the elliptic case moreover the authors extended this method to obtain analytical solutions to the nonlinear schrodinger equation in two space dimensions plus time this contains some interesting solutions such as the plane solitons the n multiple solitons the propagating breathers and quadratic solitons the authors displayed graphically the obtained solutions by using the software mathematica 5

the fractal property of the lorenz attractor in a 1963 paper lorenz inferred that the lorenz attractor must be an infinite complex of surfaces we investigate this fractal property of the lorenz attractor in two ways firstly we obtain explicit plots of the fractal structure of the lorenz attractor using symbolic dynamics and multiple precision computations of periodic orbits the method we derive for multiple precision computation is based on iterative refinement and can compute even highly unstable periodic orbits with long symbol sequences with as many as 100 digits of accuracy ordinary numerical integrations are much too crude to show even the coarsest splitting of surfaces and there appear to be no other explicit computations of the fractal structure in the extensive literature about the lorenz attractor secondly we apply a well known formula that gives the hausdorff dimension of the lorenz attractor in terms of the characteristic multipliers of its unstable periodic orbits the formula converges impressively and the hausdorff dimension of the lorenz attractor appears to be 2 0627160 we use comparison with explicit computations of the fractal structure and discuss the accuracy of this formula and its applicability to the lorenz equations additionally we apply periodic orbit theory to the lorenz attractor and exhibit its spectral determinant and compute its lyapunov exponent

on the spectra of periodic waves for infinite dimensional hamiltonian systems abstract we consider the problem of determining the spectrum for the linearization of an infinite dimensional hamiltonian system about a spatially periodic traveling wave by using a bloch wave decomposition we recast the problem as determining the point spectra for a family of operators j l where j is skew symmetric with bounded inverse and l is symmetric with compact inverse our main result relates the number of unstable eigenvalues of the operator j l to the number of negative eigenvalues of the symmetric operator l the compactness of the resolvent operators allows us to greatly simplify the proofs as compared to those where similar results are obtained for linearizations about localized waves the theoretical results are general and apply to a larger class of problems than those considered herein the theory is applied to a study of the spectra associated with periodic and quasi periodic solutions to the nonlinear schrodinger equation as well as periodic solutions to the generalized korteweg de vries equation with power nonlinearity

linearized instability for nonlinear schrodinger and klein gordon equations in this paper i am introducing a new technique for proving instability of bound states for hamiltonian systems there are already two disparate types of instability results in the literature the approach developed by strauss shatah 20 gave an instability criterion coming from the variational structure of the problem on the other hand jones approach 11 produced a complementary criterion related to the difference between the number of negative eigenvalues of two selfadjoint operators using quite different techniques it turns out that with the methods developed in this paper these two criteria can be derived within a single framework that also leads to a generalization of the previous results finally in order to demonstrate how this method works i apply the instability criterion in some specific examples

instability and stability of rolls in the swift hohenberg equation we develop a method for the stability analysis of bifurcating spatially periodic patterns under general nonperiodic perturbations in particular it enables us to detect sideband instabilities we treat in all detail the stability question of roll solutions in the two dimensional swift hohenberg equation and derive a condition on the amplitude and the wave number of the rolls which is necessary and sufficent for stability moreover we characterize the set of those wave vectors which give rise to unstable perturbations

conservation laws in higher order nonlinear schrodinger equations conservation laws of the nonlinear schr odinger equation are studied in the presence of higher order optical effects including the third order dispersion and the self steepening in a context of group theory we derive general expressions for infinitely many conserved currents and charges of a coupled higher order nonlinear schr odinger equation the first few currents and associated charges are also presented explicitly due to the higher order effects the conservation laws of the nonlinear schr odinger equation are violated in general the differences between the types of the conserved currents for the hirota and the sasa satsuma equations imply that the higher order terms determine the inherent types of conserved quantities for each integrable case of the higher order nonlinear schr odinger equation

stability of periodic solutions of conservation laws with viscosity analysis of the evans function nonclassical conservation laws with viscosity arising in multiphase fluid and solid mechanics exhibit a rich variety of traveling wave phenomena including homoclinic pulse type and periodic solutions along with the standard heteroclinic shock or front type solutions here we investigate stability of periodic traveling waves within the abstract evans function framework established by r a gardner our main result is to derive a useful stability index analogous to that developed by gardner and zumbrun in the traveling front or pulse context giving necessary conditions for stability with respect to initial perturbations that are periodic on the same period t as the traveling wave moreover we show that the periodic stability index has an interpretation analogous to that of the traveling front or pulse index in terms of well posedness of an associated riemann problem for an inviscid medium now to be interpreted as allowing a wider class of measure valued solutions or alternatively in terms of existence and nonsingularity of a local mass map from perturbation mass to potential time asymptotic t periodic states a closely related calculation yields also a complementary long wave stability criterion necessary for stability with respect to periodic perturbations of arbitrarily large period nt n we augment these analytical results with numerical investigations analogous to those carried out by brin in the traveling front or pulse case approximating the spectrum of the linearized operator about the wave

counting eigenvalues via the krein signature in infinite dimensional hamiltonian systems spectra of nonlinear waves in infinite dimensional hamiltonian systems are investigated we establish a connection via the krein signature between the number of negative directions of the second variation of the energy and the number of potentially unstable eigenvalues of the linearization about a nonlinear wave we apply our results to determine the effect of symmetry breaking on the spectral stability of nonlinear waves in weakly coupled nonlinear schrodinger equations

integrating the nonintegrable analytic structure of the lorenz system revisited abstract a study of the complex time analytic structure of the lorenz system in nonintegrable parameter regimes reveals the special sets of parameter values for which one time dependent integral of motion exists furthermore the analysis yields the exact form of the part of the integral with the highest homogeneous weight and a method to construct the rest of the integral recursive clustering of singularities in the chaotic regimes of the system is observed in computer studies and explained by a simple analytic argument the analytic techniques used in these studies a systematic resummation of a logarithmic psi series appears to be quite general and can provide explicit representations of a solution even in the chaotic regimes in the neighborhood of a given movable singularity furthermore we suggest that this technique provides a type of renormalization program to study a wide class of nonintegrable systems

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

generalized heisenberg ferromagnet models via hermitian symmetric spaces we study the generalized heisenberg ferromagnet model using a duality with the hermitian symmetric space nonlinear schrodinger equation we explain the mathematical structure of the hermiticity of the space in connection with the soliton surface theory and obtain various spin configurations in n dimension using this correspondence

normal form for the symmetry breaking bifurcation in the nonlinear schrodinger equation abstract we derive and justify a normal form reduction of the nonlinear schrodinger equation for a general pitchfork bifurcation of the symmetric bound state that occurs in a double well symmetric potential we prove persistence of normal form dynamics for both supercritical and subcritical pitchfork bifurcations in the time dependent solutions of the nonlinear schrodinger equation over long but finite time intervals

complex singularities and the lorenz attractor the lorenz attractor is one of the best known examples of applied mathematics however much of what is known about it is a result of numerical calculations and not of mathemat ical analysis as a step toward mathematical analysis we allow the time variable in the three dimensional lorenz system to be complex hoping that solutions that have resisted analysis on the real line will give up their secrets in the complex plane knowledge of singularities being fundamental to any investigation in the complex plane we build upon earlier work and give a complete and consistent formal development of complex singular ities of the lorenz system using the psi series the psi series contain two undetermined constants in addition the location of the singularity is undetermined as a consequence of the autonomous nature of the lorenz system we prove that the psi series converge using a technique that is simpler and more powerful than that of hille thus implying a two parameter family of singular solutions of the lorenz system we pose three ques tions answers to which may bring us closer to understanding the connection of complex singularities to lorenz dynamics

bifurcations from the endpoints of the essential spectrum in the linearized nonlinear schrodinger problem we study bifurcations of eigenvalues from the endpoints of the essential spectrum in the linearized nonlinear schrodinger problem in three dimensions we show that a resonance and an eigenvalue of positive energy at the endpoint may bifurcate only to a real eigenvalue of positive energy while an eigenvalue of negative energy at the endpoint may also bifurcate to complex eigenvalues

the krein matrix general theory and concrete applications in atomic bose einstein condensates when finding the nonzero eigenvalues for hamiltonian eigenvalue problems it is especially important to locate not only the unstable eigenvalues i e those with positive real part but also those which are purely imaginary but have negative krein signature these latter eigenvalues have the property that they can become unstable upon collision with other purely imaginary eigenvalues i e they are a necessary building block in the mechanism leading to the so called hamiltonian hopf bifurcation in this paper we review a general theory for constructing a meromorphic matrix valued function the so called krein matrix which has the property of not only locating the unstable eigenvalues but also those with negative krein signature these eigenvalues are realized as zeros of the determinant the resulting finite dimensional problem obtained by setting the determinant of the krein matrix to zero presents a valuable simplification in this paper the usefulness of the technique is illustrated through prototypi

variational necessary and sufficient stability conditions for inviscid shear flow a necessary and sufficient condition for linear stability of inviscid parallel shear flow is formulated by developing a novel variational principle where the velocity profile is assumed to be monotonic and analytic it is shown that unstable eigenvalues of rayleigh s equation which is a non self adjoint eigenvalue problem can be associated with positive eigenvalues of a certain self adjoint operator the stability is therefore determined by maximizing a quadratic form which is theoretically and numerically more tractable than directly solving rayleigh s equation this variational stability criterion is based on the understanding of kre n signature for continuous spectra and is applicable to other stability problems of infinite dimensional hamiltonian systems

the algebraic approximation of attractors the finite dimensional case abstract in this article we present a constructive method for approximating attractors the approximate sets are parts of algebraic sets i e the zeros of high degree polynomial functions and they can approximate the attractor at an arbitrary high level of accuracy

a proof of the benjamin feir instability the existence and linear stability problem for the stokes periodic wavetrain on fluids of finite depth is formulated in terms of the spatial and temporal hamiltonian structure of the water wave problem a proof within the hamiltonian framework of instability of the stokes periodic wavetrain is presented a hamiltonian center manifold analysis reduces the linear stability problem to an ordinary differential eigenvalue problem on 4 a projection of the reduced stability problem onto the tangent space of the 2 manifold of periodic stokes waves is used to prove the existence of a dispersion relation i1 i2 0 where e is the stability exponent for the stokes wave with amplitude i1 and mass flux i2 and is the sideband or spatial exponent a rigorous analysis of the dispersion relation proves the result first discovered in the 1960 s that the stokes gravity wavetrain of sufficiently small amplitude is unstable for f e 0 f0 where f0 0 8 and f is the froude number

stable manifolds for all monic supercritical focusing nonlinear schrodinger equations in one dimension keywords critical schrodinger equation stable manifolds modulation theory spectral theory klein gordon equations non linear schrodinger blow up rate standing waves nonintegrable equations invariant manifolds stability theory solitary waves ground states instability reference epfl article 155183doi 10 1090 s0894 0347 06 00524 8 record created on 2010 11 19 modified on 2017 05 12

dispersive analysis of charge transfer models we prove dispersive estimates for the time dependent schrodinger equation with a charge transfer hamiltonian as a by product we also obtain another proof of asymptotic completeness of the wave operators for a charge transfer model established earlier by k yajima and j m graf we also consider a more general matrix non self adjoint charge transfer problem this model appears naturally in the study of nonlinear multisoliton systems and is specifically motivated by the problem of asymptotic stability of multisoliton states of a nonlinear schrodinger equation 2004 wiley periodicals inc

a symplectic perspective on constrained eigenvalue problems abstract the maslov index is a powerful tool for computing spectra of selfadjoint elliptic boundary value problems this is done by counting intersections of a fixed lagrangian subspace which designates the boundary conditions with the set of cauchy data for the differential operator we apply this methodology to constrained eigenvalue problems in which the operator is restricted to a not necessarily invariant subspace the maslov index is defined and used to compute the morse index of the constrained operator we then prove a constrained morse index theorem which says that the morse index of the constrained problem equals the number of constrained conjugate points counted with multiplicity and give an application to the nonlinear schrodinger equation

spectra of positive and negative energies in the linearized nls problem we study the spectrum of the linearized nls equation in three dimensions in association with the energy spectrum we prove that unstable eigenvalues of the linearized nls problem are related to negative eigenvalues of the energy spectrum while neutrally stable eigenvalues may have both positive and negative energies the nonsingular part of the neutrally stable essential spectrum is always related to the positive energy spectrum we derive bounds on the number of unstable eigenvalues of the linearized nls problem and study bifurcations of embedded eigenvalues of positive and negative energies we develop the l 2 scattering theory for the linearized nls operators and recover results of grillakis 5 with a fermi golden rule c

asymptotic dynamics of nonlinear schrodinger equations with many bound states abstract we consider a nonlinear schrodinger equation with a bounded localized potential in r 3 the linear hamiltonian is assumed to have three or more bound states with the eigenvalues satisfying some resonance conditions suppose that the initial data is localized and small of order n in h1 and that its ground state component is larger than n3 e with e 0 small we prove that the solution will converge locally to a nonlinear ground state as the time tends to infinity

the deift zhou steepest descent method to long time asymptotics for the sasa satsuma equation abstract the initial value problem for the sasa satsuma equation is transformed to a 3 3 matrix riemann hilbert problem with the help of the corresponding lax pair two distinct factorizations of the jump matrix and a decomposition of the vector valued function k are given from which the long time asymptotics for the sasa satsuma equation with decaying initial data is obtained by using the nonlinear steepest descent method

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

inertia law for spectral stability of solitary waves in coupled nonlinear schrodinger equations spectral stability analysis for solitary waves is developed in the context of the hamiltonian system of coupled nonlinear schrodinger equations the linear eigen value problem for a non self adjoint operator is studied with two self adjoint matrix schrodinger operators sharp bounds on the number and type of unstable eigen values in the spectral problem are found from the inertia law for quadratic forms associated with the two self adjoint operators symmetry breaking stability analysis is also developed with the same method

standing waves in a non linear 1d lattice floquet multipliers krein signatures and stability abstract we construct a class of exact commensurate and incommensurate standing wave sw solutions in a piecewise smooth analogue of the discrete non linear schrodinger dnls model and present their linear stability analysis in the case of the commensurate sw solutions the analysis reduces to the eigenvalue problem of a transfer matrix depending parametrically on the eigenfrequency the spectrum of eigenfrequencies and the corresponding eigenmodes can thereby be determined exactly the spatial periodicity of a commensurate sw implies that the eigenmodes are of the bloch form characterised by an even number of floquet multipliers the spectrum is made up of bands that in general include a number of transition points corresponding to changes in the disposition of the floquet multipliers the latter characterise the different band segments an alternative characterisation of the segments is in terms of the krein signatures associated with the eigenfrequencies when one or more parameters characterising the sw solution is made to vary one occasionally encounters collisions between the band edges or the intra band transition points and depending on the the krein signatures of the colliding bands or segments the spectrum may stretch out in the complex plane leading to the onset of instability we elucidate the correlation between the disposition of floquet multipliers and the krein signatures presenting two specific examples where the sw possesses a definite window of stability as distinct from the sw s obtained close to the anti continuous and linear limits of the dnls model

a borel transform method for locating singularities of taylor and fourier series given a taylor series with a finite radius of convergence its borel transform defines an entire function a theorem of polya relates the large distance behavior of the borel transform in different directions to singularities of the original function with the help of the new asymptotic interpolation method of van der hoeven we show that from the knowledge of a large number of taylor coefficients we can identify precisely the location of such singularities as well as their type when they are isolated there is no risk of getting artefacts with this method which also gives us access to some of the singularities beyond the convergence disk the method can also be applied to fourier series of analytic periodic functions and is here tested on various instances constructed from solutions to the burgers equation large precision on scaling exponents up to twenty accurate digits can be achieved

instability of spatially quasi periodic states of the ginzburg landau equation the ginzburg landau gl equation with real coefficients is a model equation appearing in superconductor physics and near critical hydrodynamic stability problems the stationary gl equation has a two parameter i 1 i 2 family of spatially quasi periodic qp states with frequencies omega 1 omega 2 and frequency map with determinant delta k partial omega 1 omega 2 partial i 1 i 2 in this paper the linear stability of these qp states is studied and an expression for the stability exponent is obtained which has a novel geometric interpretation in terms of delta k when delta k 0 is a necessary but not sufficient condition for linear stability there is an interesting relation between delta k and the kam persistence theorem for invariant toroids

an instability index theory for quadratic pencils and applications primarily motivated by the stability analysis of nonlinear waves in second order in time hamiltonian systems in this paper we develop an instability index theory for quadratic operator pencils acting on a hilbert space in an extension of the known theory for linear pencils explicit connections are made between the number of eigenvalues of a given quadratic operator pencil with positive real parts to spectral information about the individual operators comprising the coefficients of the spectral parameter in the pencil as an application we apply the general theory developed here to yield spectral and nonlinear stability instability results for abstract second order in time wave equations more specifically we consider the problem of the existence and stability of spatially periodic waves for the good boussinesq equation in the analysis our instability index theory provides an explicit and somewhat surprising connection between the stability of a given periodic traveling wave solution of the good boussinesq equation and the stability of the same periodic profile but with different wavespeed in the nonlinear dynamics of a related generalized korteweg de vries equation

stability properties of periodic waves for the klein gordon equation with quintic nonlinearity in this paper we prove results of existence and orbital instability of periodic waves for the klein gordon equation with quintic nonlinearity the main tool used here is a recent computational approach in order to decide about the nonpositive spectrum of the linearized operator combined with the grillakis shatah and strauss stability theory