conditional stability of solitary waves propagating in elastic rods abstract the problem of dynamic stability of solitary wave solutions of the equations describing bending oscillations of inextensible elastic rods is treated the governing equations describe sufficiently large displacements though we are restricted to small strains we show that under the condition of well posedness of the initial value problem in some specific sense the family of solitary wave solutions of the mentioned equations is stable for the planar elastic rod the framework of the analysis is largely based on the spectral properties of the linearized hamiltonian h we show that for planar oscillations h is positively semidefinite subject to a certain constraint which implies the stability

variational properties and orbital stability of standing waves for nls equation on a star graph we study standing waves for a model of nonlinear schr odinger equation on a graph the graph is obtained joining n halflines at a vertex i e it is a star graph at the vertex an interaction occurs due to a boundary condition of delta type with strength alpha leqslant 0 which includes the free or kirchhoff case alpha 0 there imposed the nonlinearity is of focusing power type the dynamics is given by an equation of the form i frac d dt psi t h psi t psi t 2 mu psi t where h is the s a operator which generates the linear schr odinger dynamics on the graph in the hilbert space l 2 gg we show the existence of several families of standing waves solutions of the form phi t e i omega t psi omega where psi omega is a vector amplitude parametrized by a frequency omega they exist for every sign of the coupling at the vertex attractive alpha 0 and for every omega frac alpha 2 n 2 the number of the families depends on the number n of the edges of the graph we moreover determine the ground state which is the standing wave the amplitude of which minimizes the action on the natural nehari constraint and order the various families of standing waves according to their increasing action thus determining a nonlinear spectrum of the problem finally we show that the ground state is orbitally stable for every allowed omega if the nonlinearity is subcritical or critical for supercritical nonlinearities there exists an omega frac alpha 2 n 2 such that the ground state is orbitally stable for frac alpha 2 n 2 omega omega action minimization requires technical subtleties in particular an adaptation of decreasing rearrangements on a star graph is developed with a generalization of the p olya szeg h o inequality

stability and instability of periodic travelling wave solutions for the critical korteweg de vries and nonlinear schrodinger equations abstract in this paper we establish new results about the existence stability and instability of periodic travelling wave solutions related to the critical korteweg de vries equation u t 5 u 4 u x u x x x 0 and the critical nonlinear schrodinger equation i v t v x x v 4 v 0 the periodic travelling wave solutions obtained in our study tend to the classical solitary wave solutions in the infinite wavelength scenario the stability approach is based on the theory developed by angulo natali in j angulo f natali positivity properties of the fourier transform and the stability of periodic travelling wave solutions siam j math anal 40 2008 1123 1151 for positive periodic travelling wave solutions associated to dispersive evolution equations of korteweg de vries type the instability approach is based on an extension to the periodic setting of arguments found in bona souganidis strauss j l bona p e souganidis w a strauss stability and instability of solitary waves of korteweg de vries type proc roy soc london ser a 411 1987 395 412 regarding the critical schrodinger equation stability instability theories similar to the critical korteweg de vries equation are obtained by using the classical grillakis shatah strauss theory in m grillakis j shatah w strauss stability theory of solitary waves in the presence of symmetry ii j funct anal 94 1990 308 348 m grillakis j shatah w strauss stability theory of solitary waves in the presence of symmetry i j funct anal 74 1987 160 197 the arguments presented in this investigation have prospects for the study of the stability of periodic travelling wave solutions of other nonlinear evolution equations

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

instability of standing waves for klein gordon zakharov equations with different propagation speeds in three space dimensions abstract this paper is concerned with the standing wave for klein gordon zakharov equations with different propagation speeds in three space dimensions the existence of standing wave with the ground state is established by applying an intricate variational argument and the instability of the standing wave is shown by applying pagne and sattinger s potential well argument and levine s concavity method

stability and asymptotic stability in the energy space of the sum of n solitons for subcritical gkdv equations we prove in this paper the stability and asymptotic stability in h1 of a decoupled sum of n solitons for the subcritical generalized kdv equations the proof of the stability result is based on energy arguments and monotonicity of the local l2 norm note that the result is new even for p 2 the kdv equation the asymptotic stability result then follows directly from a rigidity theorem in 16

stability and instability of solitary waves of korteweg de vries type considered herein are the stability and instability properties of solitary wave solutions of a general class of equations that arise as mathematical models for the unidirectional propagation of weakly nonlinear dispersive long waves special cases for which our analysis is decisive include equations of the korteweg de vries and benjamin ono type necessary and sufficient conditions are formulated in terms of the linearized dispersion relation and the nonlinearity for the solitary waves to be stable

domain walls in the coupled gross pitaevskii equations a thorough study of domain wall solutions in coupled gross pitaevskii equations on the real line is carried out including existence of these solutions their spectral and nonlinear stability their persistence and stability under a small localized potential the proof of existence is variational and is presented in a general framework we show that the domain wall solutions are energy minimizers within a class of vector valued functions with nontrivial conditions at infinity the admissible energy functionals include those corresponding to coupled gross pitaevskii equations arising in modeling of bose einstein condensates the results on spectral and nonlinear stability follow from properties of the linearized operator about the domain wall the methods apply to many systems of interest and integrability is not germane to our analysis finally sufficient conditions for persistence and stability of domain wall solutions are obtained to show that stable pinning occurs near maxima of the potential thus giving rigorous justification to earlier results in the physics literature

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

solitary wave dynamics in an external potential we study the behavior of solitary wave solutions of some generalized nonlinear schrodinger equations with an external potential the equations have the feature that in the absence of the external potential they have solutions describing inertial motions of stable solitary waves we consider solutions of the equations with a non vanishing external potential corresponding to initial conditions close to one of these solitary wave solutions and show that over a large interval of time they describe a solitary wave whose center of mass motion is a solution of newton s equations of motion for a point particle in the given external potential up to small corrections corresponding to radiation damping

the lyapunov stability of the n soliton solutions in the lax hierarchy of the benjamin ono equation the lyapunov stability is established for the n soliton solutions in the lax hierarchy of the benjamin ono bo equation we characterize the n soliton profiles as critical points of certain lyapunov functional by using several results derived by the inverse scattering transform of the bo equation we demonstrate the convexity of the lyapunov functional when evaluated at the n soliton profiles from this fact we deduce that the n soliton solutions are energetically stable

linear stability of perturbed hamiltonian systems theory and a case example in this paper we present a general theory based on a lyapunov schmidt reduction for the linearized stability of a perturbed hamiltonian system with a number of symmetries the reduction leads to an eigenvalue problem in a small subspace of the original system whose eigenvalues and eigenvectors can be used to derive the ones of the perturbed original system the krein signature of the eigenvalues can also be obtained in this framework the method is applied to the case example of the coupled nonlinear schr dinger equations giving excellent agreement with the full numerical linear stability results

on asymptotic stability of stationary solutions to nonlinear wave and klein gordon equations we consider nonlinear wave and klein gordon equations with general nonlinear terms localized in space conditions are found which provide asymptotic stability of stationary solutions in local energy norms these conditions are formulated in terms of spectral properties of the schrodinger operator corresponding to the linearized problem they are natural extensions to partial differential equations of the known lyapunov condition for the nonlinear wave equation in three dimensional space we find asymptotic expansions as t of the solutions which are close enough to a stationary asymptotically stable solution

on the turing patterns in one dimensional gradient skew gradient dissipative systems in this article fundamental properties concerning the turing patterns are considered in one dimensional dissipative systems with gradient skew gradient structure introduced in m kuwamura and e yanagida phys d 175 2003 pp 185 195 it is a natural extension of free energy which covers reaction diffusion systems of activator inhibitor type the theory based on this concept provides a new perspective on a fundamental problem of what unique turing pattern is to be selected among many

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

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

a sharp threshold of blow up for coupled nonlinear schrodinger equations this paper is concerned with the coupled supercritical nonlinear schrodinger equations which have applications in many physical problems especially in nonlinear optics two types of new invariant evolution flows are established a sharp threshold of blow up and global existence of solutions for the equations is derived it is shown that the main result obtained includes parts of those presented by fanelli and montefusco 2007 j phys a math theor 40 14139 50x

stable standing waves of nonlinear klein gordon equations in this paper we give sufficient conditions for the stability of the standing waves of least energy for nonlinear klein gordon equations

stability for line solitary waves of zakharov kuznetsov equation abstract in this paper we consider the stability for line solitary waves of the two dimensional zakharov kuznetsov equation on r t l which is one of a high dimensional generalization of korteweg de vries equation where t l is the torus with the 2 l period the orbital and asymptotic stability of the one soliton of korteweg de vries equation on the energy space was proved by benjamin 2 pego and weinstein 41 and martel and merle 30 we regard the one soliton of korteweg de vries equation as a line solitary wave of zakharov kuznetsov equation on r t l we prove the stability and the transverse instability of the line solitary waves of zakharov kuznetsov equation by applying the method of evans function and the argument of rousset and tzvetkov 44 moreover we prove the asymptotic stability for orbitally stable line solitary waves of zakharov kuznetsov equation by using the argument of martel and merle 30 31 32 and a liouville type theorem if l is the critical period with respect to a line solitary wave the line solitary wave is orbitally stable however since this line solitary wave is a bifurcation point of the stationary equation the linearized operator of the stationary equation is degenerate because of the degeneracy of the linearized operator we can not show the liouville type theorem for the line solitary wave by using the usual virial type estimate to show the liouville type theorem for the line solitary wave we modify a virial type estimate

on the low regularity solutions and wave breaking for an equation modeling shallow water waves of moderate amplitude abstract in this paper we consider a nonlinear evolution equation which models the propagation of surface waves of moderate amplitude in shallow water regime we first prove the existence of the low regularity solutions for the cauchy problem of this equation then we discuss the initial profiles under which the solutions of this equation blow up in finite time

on the dynamics of solitons in the nonlinear schrodinger equation we study the behavior of the soliton solutions of the equation i frac partial psi partial t frac 1 2m delta psi frac 1 2 w varepsilon prime psi v x psi where w varepsilon prime is a suitable nonlinear term which is singular for varepsilon 0 we use the strong nonlinearity to obtain results on existence shape stability and dynamics of the soliton the main result of this paper theorem 1 shows that for varepsilon rightarrow 0 the orbit of our soliton approaches the orbit of a classical particle in a potential v x

stability and folds it is known that when one branch of a simple fold in a bifurcation diagram represents linearly stable solutions the other branch represents unstable solutions the theory developed here can predict instability of some branches close to folds without knowledge of stability of the adjacent branch provided that the underlying problem has a variational structure first one particular bifurcation diagram is identified as playing a special role the relevant diagram being specified by the choice of functional plotted as ordinate the results are then stated in terms of the shape of the solution branch in this distinguished bifurcation diagram in many problems arising in elasticity the preferred bifurcation diagram is the loaddisplacement graph the theory is particularly useful in applications where a solution branch has a succession of folds

internal modes of envelope solitons abstract we extend the concept of internal mode to envelope solitons and show that this mode is responsible for long lived weakly damped periodic oscillations of the soliton amplitude observed in numerical simulations we present analytical and numerical results for solitons of the generalized nonlinear schrodinger equation and analyze the example of the cubic quintic nonlinearity in more detail we obtain also analytical criteria for the existence and bifurcations of the soliton internal mode and calculate the rate of the radiation induced damping of the soliton oscillations induced by excitation of the internal mode

stability in h 1 of the sum of k solitary waves for some nonlinear schrodinger equations in this article we consider nonlinear schrodinger nls equations in rd for d 1 2 and 3 we consider nonlinearities satisfying a flatness condition at zero and such that solitary waves are stable let rk t x be k solitary wave solutions of the equation with different speeds v1 v2 vk provided that the relative speeds of the solitary waves vk vk 1 are large enough and that no interaction of two solitary waves takes place for positive time we prove that the sum of the rk t is stable for t 0 in some suitable sense in h1 to prove this result we use an energy method and a new monotonicity property on quantities related to momentum for solutions of the nonlinear schrodinger equation this property is similar to the l2 monotonicity property that has been proved by martel and merle for the generalized korteweg de vries gkdv equations see 12 lem 16 proof of prop 6 and that was used to prove the stability of the sum of k solitons of the gkdv equations by the authors of the present article see 15 th 1 i

joint modeling and clustering paired generalized longitudinal trajectories with application to cocaine abuse treatment data in a cocaine dependence treatment study we have paired binary longitudinal trajectories that record the cocaine use patterns of each patient before and after a treatment to better understand the drug using behaviors among the patients we propose a general framework based on functional data analysis to jointly model and cluster these paired non gaussian longitudinal trajectories our approach assumes that the response variables follow distributions from the exponential family with the canonical parameters determined by some latent gaussian processes to reduce the dimensionality of the latent processes we express them by a truncated karhunen loeve kl expansion allowing the mean and covariance functions to be different across clusters we further represent the mean and eigenfunctions functions by flexible spline bases and determine the orders of the truncated kl expansions using data driven methods by treating the cluster membership as a missing value we cluster the cocaine use trajectories by a l

stability of solitary waves for a nonlinear dispersive system in a critical case we study a nonlinear dispersive system with critical power nonlinearities building on the works of weinstein and angulo bona linares and scialom a conditional stability modulo the group of translations in space and dilations is proved for a solitary wave solution

local well posedness and wave breaking results for periodic solutions of a shallow water equation for waves of moderate amplitude abstract we study the local well posedness of a periodic nonlinear equation for surface waves of moderate amplitude in shallow water we use an approach proposed by kato which is based on semigroup theory for quasi linear equations we also show that singularities for the model equation can occur only in the form of wave breaking in particular surging breakers

relaxation of solitons in nonlinear schrodinger equations with potential abstract in this paper we study dynamics of solitons in the generalized nonlinear schrodinger equation nls with an external potential in all dimensions except for 2 for a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space centered near different critical points of the potential we call those solutions which are centered near the minima of the potential and which minimize energy restricted to l 2 unit sphere trapped solitons or just solitons in this paper we prove under certain conditions on the potentials and initial conditions that trapped solitons are asymptotically stable moreover if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position the dynamical law of motion of the soliton i e effective equations of motion for the soliton s center and momentum is close to newton s equation but with a dissipative term due to radiation of the energy to infinity

description of the inelastic collision of two solitary waves for the bbm equation we prove that the collision of two solitary waves of the bbm equation is inelastic but almost elastic in the case where one solitary wave is small in the energy space we show precise estimates of the nonzero residue due to the collision moreover we give a precise description of the collision phenomenon change of size of the solitary waves and shifts in their trajectories to prove these results we extend the method introduced in martel and merle description of two soliton collision for the quartic gkdv equation submitted preprint http arxiv org abs 0709 2672 commun math phys 286 39 79 2009 for the generalized kdv equation in particular in the quartic case the main argument is the construction of an explicit approximate solution in the collision region

instability of nonlinear bound states we establish a sharp instability theorem for the bound states of lowest energy of the nonlinear klein gordon equation utt u f u 0 and the nonlinear schrodinger equation iut u f u 0

orbital stability of double solitons for the benjamin ono equation in this paper we prove the orbital stability of double solitons for the benjamin ono equation in the case of the kdv equation this stability has been proved in 17 parts of the proof given there rely on the fact that the euler lagrange equations for the conserved quantities of the kdv equation are ordinary differential equations since this is not the case for the benjamin ono equation new methods are required our approach consists in using a new invariant for multi solitons and certain new identities motivated by the sylvester law of inertia

unstable ground state of nonlinear klein gordon equations on demontre l instabilite de l etat fondamental des equations de klein gordon non lineaires a n dimensions d espace n 3

a centre stable manifold for the focussing cubic nls in mathbb r 1 3 consider the focussing cubic nonlinear schrodinger equation in mathbb r 3 i psi t delta psi psi 2 psi quad 0 1 it admits special solutions of the form eit where phi in mathcal s mathbb r 3 is a positive 0 solution of delta phi alpha phi phi 3 quad 0 2 the space of all such solutions together with those obtained from them by rescaling and applying phase and galilean coordinate changes called standing waves is the 8 dimensional manifold that consists of functions of the form e i v cdot gamma phi cdot y alpha we prove that any solution starting sufficiently close to a standing wave in the sigma w 1 2 mathbb r 3 cap x 1 l 2 mathbb r 3 norm and situated on a certain codimension one local lipschitz manifold exists globally in time and converges to a point on the manifold of standing waves furthermore we show that mathcal n is invariant under the hamiltonian flow locally in time and is a centre stable manifold in the sense of bates jones batjon the proof is based on the modulation method introduced by soffer and weinstein for the l2 subcritical case and adapted by schlag to the l2 supercritical case an important part of the proof is the keel tao endpoint strichartz estimate in mathbb r 3 for the nonselfadjoint schrodinger operator obtained by linearizing 0 1 around a standing wave solution all results in this paper depend on the standard spectral assumption that the hamiltonian mathcal h left begin array cc delta 2 phi cdot alpha 2 alpha quad phi cdot alpha 2 phi cdot alpha 2 quad delta 2 phi cdot alpha 2 alpha end array right quad 0 3 has no embedded eigenvalues in the interior of its essential spectrum infty alpha cup alpha infty

stability and instability of the kdv solitary wave under the kp i flow we consider the kp i and gkp i equations in mathbb r times mathbb r 2 pi mathbb z we prove that the kdv soliton with subcritical speed 0 c in the spirit of the work by duyckaerts and merle gafa 18 1787 1840 2009 we sharpen our previous instability result and construct a global solution which is different from the solitary wave and its translates and which converges to the solitary wave as time goes to infinity this last result also holds for the gkp i equation

scaling properties of functionals and existence of constrained minimizers in this paper we develop a new method to prove the existence of minimizers for a class of constrained minimization problems on hilbert spaces that are invariant under translations our method permits to exclude the dichotomy of the minimizing sequences for a large class of functionals we introduce family of maps called scaling paths that permits to show the strong subadditivity inequality as byproduct the strong convergence of the minimizing sequences up to translations is proved we give an application to the energy functional i associated to the schrodinger poisson equation in r3 i t x 1 2 p 2 0 when 2 0 in this way we recover the case studied in sanchez and soler 2004 20 for p 8 3 and we complete the case studied by the authors for 3 p 10 3 in bellazzini and siciliano 2011 4

solitary waves of nonlinear dispersive evolution equations with critical power nonlinearities we study the nonlinear schrodinger nls generalized korteweg de vries gkdv and generalized benjamin ono equations gbo with critical power nonlinearities the ground state solitary wave is conditionally stable modulo the group of translations in space and phase and dilations this is a natural extension of the result for subcritical nonlinearities where solitary waves are stable modulo translations alone to a case where solutions may blow up we also present results on global existence for gbo and the structure of singularities for certain solutions of gbo which are analogues of previous results by the author for nls and gkdv

on spectral stability of the nonlinear dirac equation abstract we study the point spectrum of the nonlinear dirac equation in any spatial dimension linearized at one of the solitary wave solutions we prove that in any dimension the linearized equation has no embedded eigenvalues in the part of the essential spectrum beyond the embedded thresholds we then prove that the birth of point eigenvalues with nonzero real part the ones which lead to linear instability from the essential spectrum is only possible from the embedded eigenvalues or thresholds and therefore can not take place beyond the embedded thresholds we also prove that in the nonrelativistic limit m the point eigenvalues can only accumulate to 0 and 2 m i

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

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

global existence and blow up of solutions for the inhomogeneous nonlinear schrodinger equation in r2 abstract this paper discusses a class of inhomogeneous nonlinear schrodinger equation i t u t x u t x v x u t x p 1 u t x u 0 x u 0 x where t x r r 2 v x satisfies some assumptions by a constrained variational problem we firstly define some cross constrained invariant sets for the inhomogeneous nonlinear schrodinger equation then we obtain some sharp conditions for global existence and blow up of solutions as a consequence it is shown that the solution is globally well posed in h r 1 r 2 with the h 1 norm of the initial data u 0 which is dominated by the minimal value of the constrained variational problem

long time dynamics of variable coefficient modified korteweg de vries solitary waves we study the long time behavior of solutions to the korteweg de vries type equation partial derivative t u partial derivative x partial derivative 2 x u f u b t x u with initial conditions cl

on the asymptotic stability of solitary waves in the discrete schrodinger equation coupled to a nonlinear oscillator abstract the long time asymptotics is analyzed for finite energy solutions of the 1d discrete schrodinger equation coupled to a nonlinear oscillator the coupled system is invariant with respect to the phase rotation group u 1 for initial states close to a solitary wave the solution converges to a sum of another solitary wave and a dispersive wave which is a solution to the free schrodinger equation the proofs use the strategy of buslaev perelman v s buslaev g s perelman on the stability of solitary waves for nonlinear schrodinger equations amer math soc trans 2 164 1995 75 98 the linearization of the dynamics on the solitary manifold the symplectic orthogonal projection method of majorants etc

the symplectic evans matrix and the instability of solitary waves and fronts hamiltonian evolution equations which are equivariant with respect to the action of a lie group are models for physical phenomena such as oceanographic flows optical fibres and atmospheric flows and such systems often have a wide variety of solitary wave or front solutions in this paper we present a new symplectic framework for analyzing the spectral problem associated with the linearization about such solitary waves and fronts at the heart of the analysis is a multi symplectic formulation of hamiltonian partial differential equations where a distinct symplectic structure is assigned for the time and space directions with a third symplectic structure with two form denoted by omega associated with a coordinate frame moving at the speed of the wave this leads to a geometric decomposition and symplectification of the evans function formulation for the linearization about solitary waves and fronts we introduce the concept of the symplectic evans matrix a matrix consisting of restricted omega symplectic forms by applying hodge duality to the exterior algebra formulation of the evans function we find that the zeros of the evans function correspond to zeros of the determinant of the symplectic evans matrix based on this formulation we prove several new properties of the evans function restricting the spectral parameter lambda to the real axis we obtain rigorous results on the derivatives of the evans function near the origin based solely on the abstract geometry of the equations and results for the large lambda behaviour which use primarily the symplectic structure but also extend to the non symplectic case the lie group symmetry affects the evans function by generating zero eigenvalues of large multiplicity in the so called systems at infinity we present a new geometric theory which describes precisely how these zero eigenvalues behave under perturbation by combining all these results a new rigorous sufficient condition for instability of solitary waves and fronts is obtained the theory applies to a large class of solitary waves and fronts including waves which are biasymptotic to a nonconstant manifold of states as x tends to infinity to illustrate the theory it is applied to three examples a boussinesq model from oceanography a class of nonlinear schrodinger equations from optics and a nonlinear klein gordon equation from atmospheric dynamics

asymptotic stability of ground states in 2d nonlinear schrodinger equation including subcritical cases abstract we consider a class of nonlinear schrodinger equations in two space dimensions with an attractive potential the nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical in l 2 nonlinearities we study the asymptotic stability of the nonlinear bound states i e periodic in time localized in space solutions our result shows that all solutions with small initial data converge to a nonlinear bound state therefore the nonlinear bound states are asymptotically stable the proof hinges on dispersive estimates that we obtain for the time dependent hamiltonian linearized dynamics around a carefully chosen one parameter family of bound states that shadows the nonlinear evolution of the system due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations

strong instability of solitary wave solutions of a generalized boussinesq equation abstract in this paper we prove the strong instability of solitary wave solutions with small speed of propagation for a generalized boussinesq equation

asymptotic stability of ground states in 3d nonlinear schrodinger equation including subcritical cases we consider a class of nonlinear schrodinger equation in three space dimensions with an attractive potential the nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical in l2 nonlinearities we study the asymptotic stability of the nonlinear bound states i e periodic in time localized in space solutions our result shows that all solutions with small initial data converge to a nonlinear bound state therefore the nonlinear bound states are asymptotically stable the proof hinges on dispersive estimates that we obtain for the time dependent hamiltonian linearized dynamics around a careful chosen one parameter family of bound states that shadows the nonlinear evolution of the system due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations

on weak interaction between a ground state and a non trapping potential abstract we show that ground states of the nls moving at nonzero speed are asymptotically stable if they either stay far from the potential or the potential is small or the ground state has large speed we search an effective hamiltonian using the birkhoff normal forms argument in 11 treating the potential as a perturbation the so called fermi golden rule which is used to describe the decay to 0 of the internal discrete modes of the ground state is similar to that in 12 the continuous modes dispersion requires the theory in 40 on charge transfer models

the regularized benjamin ono and bbm equations well posedness and nonlinear stability abstract nonlinear stability of nonlinear periodic solutions of the regularized benjamin ono equation and the benjamin bona mahony equation with respect to perturbations of the same wavelength is analytically studied these perturbations are shown to be stable we also develop a global well posedness theory for the regularized benjamin ono equation in the periodic and in the line setting in particular we show that the cauchy problem in both periodic and nonperiodic case cannot be solved by an iteration scheme based on the duhamel formula for negative sobolev indices

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

instability of travelling waves for weakly coupled kdv systems abstract in this paper we consider the weakly coupled kdv system 0 1 u t u x x f u f u u v x v t v x x g v f v u v x where is a small parameter assuming that for 0 the decoupled system has a travelling wave we first give a sufficient condition for the persistence of such a wave secondly assuming that each wave for the unperturbed system is stable and under a generic condition we show that at least for a one sided interval for the parameter the persistent wave is unstable

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

long time dynamics of kdv solitary waves over a variable bottom we study the variable bottom generalized korteweg de vries bkdv equation tu x u f u b t x u where f is a nonlinearity and b is a small bounded and slowly varying function related to the varying depth of a channel of water many variable coefficient kdv type equations including the variable coefficient variable bottom kdv equation can be rescaled into the bkdv we study the long time behavior of solutions with initial conditions close to a stable b 0 solitary wave we prove that for long time intervals such solutions have the form of the solitary wave whose center and scale evolve according to a certain dynamical law involving the function b t x plus an h1 small fluctuation 2005 wiley periodicals inc

the hamiltonian structure of the nonlinear schrodinger equation and the asymptotic stability of its ground states in this paper we prove that ground states of the nls which satisfy the sufficient conditions for orbital stability of m weinstein are also asymptotically stable for seemingly generic equations the key issue is to prove that a certain coefficient is non negative because is a square power we assume that the nls has a smooth short range nonlinearity we assume also the presence of a very short range and smooth linear potential to avoid translation invariance the basic idea is to perform a birkhoff normal form argument on the hamiltonian as in a paper by bambusi and cuccagna on the stability of the 0 solution for nlkg but in our case the natural coordinates arising from the linearization are not canonical so we need also to apply the darboux theorem with some care though in order not to destroy some nice features of the initial hamiltonian

orbital stability of periodic standing waves for the logarithmic klein gordon equation abstract the main goal of this paper is to present orbital stability results of periodic standing waves for the one dimensional logarithmic klein gordon equation to do so we first use compactness arguments and a non standard analysis to obtain the existence and uniqueness of weak solutions for the associated cauchy problem in the energy space second we prove the orbital stability of standing waves using a stablity analysis of conservative systems

on the bound states of the nonlinear schrodinger equation with a linear potential abstract we study the nonlinear schrodinger equation with an attractive linear potential i t v x 2 0 n 2 nls which arises in the mathematical description of phenomena in nonlinear optics and plasma physics nonlinear bound states are finite energy localized solutions which if dynamically stable play an important role in the structure of general solutions of nls we discuss the existence and nonlinear orbital stability of nonlinear ground states of nls in particular if v supports a linear bound state then nls has stable nonlinear ground states in the supercritical and critical cases 2 n where blow up self focusing or collapse can occur this is a phenomenon not present in the case where v 0 in addition islands of stability in the regime of large h1 norm exist

nonlinear stability of gardner breathers abstract we show that breather solutions of the gardner equation a natural generalization of the kdv and mkdv equations are h 2 r stable through a variational approach we characterize gardner breathers as minimizers of a new lyapunov functional and we study the associated spectral problem through i the analysis of the spectrum of explicit linear systems spectral stability and i i controlling degenerated directions by using low regularity conservation laws

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

sharp threshold of global existence and instability of standing wave for a davey stewartson system this paper concerns the sharp threshold of blowup and global existence of the solution as well as the strong instability of standing wave phi t x e i omega t u x for the system i phi t delta phi a phi p 1 phi b e 1 phi 2 phi 0 quad t geq 0 quad x in bf r n quad quad quad rm ds where a 0 b 0 1 0 by combining the former results

long time oscillation of solutions of nonlinear schrodinger equations near minimal mass ground state abstract in this paper we consider the long time dynamics of radially symmetric solutions of nonlinear schrodinger equations nls having a minimal mass ground state in particular we show that there exist solutions with initial data near the minimal mass ground state that oscillate for long time more precisely we introduce a coordinate defined near the minimal mass ground state which consists of finite and infinite dimensional part associated to the discrete and continuous part of the linearized operator then we show that the finite dimensional 2 d part approximately obeys newton s equation of motion for a particle in an anharmonic potential well showing that the infinite dimensional part is well separated from the finite dimensional part we will have long time oscillation

on the dispersion decay for crystals in the linearized schrodinger poisson model abstract the schrodinger poisson newton equations for crystals with a cubic lattice and one ion per cell are considered the ion charge density is assumed i to satisfy the wiener and jellium conditions introduced in our previous paper 25 and ii to be exponentially decaying at infinity the corresponding examples are given we study the linearized dynamics at the ground state the dispersion relations are introduced via spectral resolution for the non selfadjoint hamilton generator using the positivity of the energy established in 25 our main result is the dispersion decay in the weighted sobolev norms for solutions with initial states from the space of continuous spectrum of the hamilton generator we also prove the absence of singular spectrum and limiting absorption principle the multiplicity of every eigenvalue is shown to be infinite the proofs rely on novel exact bounds and compactness for the inversion of the bloch generators and on uniform asymptotics for the dispersion relations we derive the bounds by the energy positivity from 25 we also use the theory of analytic sets

periodic finite genus solutions of the kdv equation are orbitally stable abstract the stability of periodic solutions of partial differential equations has been an area of increasing interest in the last decade the kdv equation is known to have large families of periodic solutions that are parameterized by hyperelliptic riemann surfaces they are generalizations of the famous multi soliton solutions we show that all such periodic solutions are orbitally stable with respect to subharmonic perturbations perturbations that are periodic with period equal to an integer multiple of the period of the underlying solution

spectral stability analysis for periodic traveling wave solutions of nls and cgl perturbations abstract we consider cnoidal traveling wave solutions to the focusing nonlinear schrodinger equation nls that have been shown to persist when the nls is perturbed to the complex ginzburg landau equation cgl we show that while these periodic traveling waves are spectrally stable solutions of nls with respect to periodic perturbations they are unstable with respect to bounded perturbations furthermore we use an argument based on the fredholm alternative to find an instability criterion for the persisting solutions to cgl

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

on the standing waves of the nls log equation with a point interaction on a star graph abstract we study the nonlinear schrodinger equation with logarithmic nonlinearity on a star graph g at the vertex an interaction occurs described by a boundary condition of delta type with strength r we investigate the orbital stability and the spectral instability of the standing wave solutions e i t x to the equation when the profile x has mixed structure i e has bumps and tails in our approach we essentially use the extension theory of symmetric operators by krein von neumann and the analytic perturbations theory

lower bounds for local approximation in the study of deterministic distributed algorithms it is commonly assumed that each node has a unique o log n bit identifier we prove that for a general class of graph problems local algorithms constant time distributed algorithms do not need such identifiers a port numbering and orientation is sufficient our result holds for so called simple po checkable graph optimisation problems this includes many classical packing and covering problems such as vertex covers edge covers matchings independent sets dominating sets and edge dominating sets we focus on the case of bounded degree graphs and show that if a local algorithm finds a constant factor approximation of a simple po checkable graph problem with the help of unique identifiers then the same approximation ratio can be achieved on anonymous networks as a corollary of our result we derive a tight lower bound on the local approximability of the minimum edge dominating set problem by prior work there is a deterministic local algorithm that achieves the approximation factor of 4 1 2 in graphs of maximum degree this approximation ratio is known to be optimal in the port numbering model our main theorem implies that it is optimal also in the standard model in which each node has a unique identifier our main technical tool is an algebraic construction of homogeneously ordered graphs we say that a graph is r homogeneous if its nodes are linearly ordered so that an fraction of nodes have pairwise isomorphic radius r neighbourhoods we show that there exists a finite r homogeneous 2k regular graph of girth at least g for any

on the solitary wave dynamics under slowly varying medium for nonlinear schrodinger equations we consider the problem of a solitary wave propagation in a slowly varying medium for a variable coefficients nonlinear schrodinger equation we prove global existence and uniqueness of solitary wave solutions for a large class of slowly varying media moreover we describe for all time the behavior of these solutions which include refracted and reflected solitary waves depending on the initial energy

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