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

multiple brake orbits for some classes of singular hamiltonian systems from which one can easily construct a periodic solution of period 2t by reflecting q around t 0 t if q x e e 1 v x hj is homeomorphic to a ball then the existence of one brake orbit is a classical result of seifert 11 generalizations of that result as well as alternative proofs were given in 2 6 141 in the case in which the topology of sz is richer then the existence of multiple solutions has been proved by bolotin and kozlov in 7 see also 8 in this paper we are mainly concerned with the case in which the potential v has singularities inside 2 in the sense that v x co as x x e ai and the topology of q is nontrivial more precisely in section 2 we shall assume that sz has a certain number of holes k and that v behaves like lxi o1 with 0 cx 2 near x 0 and we shall show that 1 3 has at least k classical brake orbits let us point out that the main interest of this result lies in the fact that we are able to prove that the solutions we find are classical ones that is they do not collide with the singularity while most of the papers dealing with existence of fixed energy 9 or fixed period lo solutions for singular potentials only prove existence of generalized solutions that is solutions which can collide with the singularity existence of a noncollision periodic solution for a singular potential

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

sharp upper bound on the blow up rate for the critical nonlinear schrodinger equation we consider the critical nonlinear schrodinger equation iu t delta u u 4 n with initial condition u 0 x u0

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

persistence of solutions to higher order nonlinear schrodinger equation abstract applying an abstract interpolation lemma we showed persistence of solutions of the initial value problem to higher order nonlinear schrodinger equation also called airy schrodinger equation in weighted sobolev spaces x 2 for 0 1

orbital stability of spatially synchronized solitary waves of an m coupled nonlinear schrodinger system in this paper we investigate the orbital stability of solitary wave solutions for an m coupled nonlinear schrodinger system i tuj 2 x2uj i 1mbijui2uj 0 j 1 m where m 2 uj are complex valued functions of x t 2 bjj j 1 2 m and bij i j are positive coupling constants satisfying bij bji it will be shown that spatially synchronized solitary wave solutions of the m coupled nonlinear schrodinger system exist and are orbitally stable here by synchronized solutions we mean solutions in which the components are proportional to one another our results completely settle the question on the existence and stability of synchronized solitary waves for the m coupled system while only partial results were known in the literature for the cases of m 3 heretofore furthermore the conditions imposed on the symmetric matrix b bij satisfied here are both sufficient and necessary for the m coupled nonlinear schrodinger system to admit synchronized ground state solutions

asymptotic stability of small gap solitons in nonlinear dirac equations we prove dispersive decay estimates for the one dimensional dirac operator and use them to prove asymptotic stability of small gap solitons in the nonlinear dirac equations with quintic and higher order nonlinear terms

stability of standing waves for the fractional schrodinger choquard equation abstract in this paper we consider the stability of standing waves for the fractional schrodinger choquard equation with an l 2 critical nonlinearity by using the profile decomposition of bounded sequences in h s and variational methods we prove that the standing waves are orbitally stable we extend the study of bhattarai for a single equation bhattarai 2017 to the l 2 critical case

on the existence of periodic solutions for a class of symetric hamiltonian systems abstract hamiltonian systems are used to model the motion of discrete mechanical systems this paper establishes the existence of periodic solutions for a class of such systems the method developed to prove existence should be useful for other such problems keywords minimax technique ordinary differential equations and variational problem

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

multiple brake orbits on convex hypersurfaces under asymmetric pinch conditions abstract in this paper we give an example of an asymmetric convex compact hypersurface possessing precisely n brake orbits for h c 1 r 2 n r satisfying h p q h p q for all p q r n r n we prove multiplicity result of the brake orbits on h 1 1 pinched by not necessarily symmetric hypersurfaces

the blow up dynamic and upper bound on the blow up rate for critical nonlinear schrodinger equation we consider the critical nonlinear schr odinger equation iut u u 4 n u with initial condition u 0 x u0 in dimension n 1 for u0 h1 local existence in the time of solutions on an interval 0 t is known and there exist finite time blow up solutions that is u0 such that limt t 1

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

large time asymptotics for the fourth order nonlinear schrodinger equation abstract we consider the cauchy problem for the fourth order nonlinear schrodinger equation in the super critical case i t u 1 4 x 4 u x u 1 u where 4 c we prove the global existence and the large time asymptotics of solutions for small initial data u 0 h 1 h 1 2 1 when the order of the nonlinearity 4

stable solutions of the logarithmic schrodinger equation on considere l equation de schrodinger suivante u t u ulog u 2 0 dans r r n on donne des resultats sur le comportement asymptotique des solutions

a solution to the focusing 3d nls that blows up on a contracting sphere we rigorously construct radial h 1 solutions to the 3d cubic focusing nls equation i t 2j j 2 0 that blow up along a contracting sphere with blow up time set to t 0 the solutions concentrate on a sphere at radius t 1 3 but focus towards this sphere at the faster rate t 2 3 such dynamics were originally proposed heuristically by degtyarev zakharov rudakov 2 in 1975 and independently later in holmer roudenko 5 in 2007 where it was demonstrated to be consistent with all conservation laws of this equation in the latter paper it was proposed as a solution that would yield divergence of the l 3 norm within the wide radius kr u t k 1 2 l2 but not within the tight radius kr u t k 2 l2

sharp thresholds of global existence and blowup for a system of schrodinger equations with combined power type nonlinearities in this paper we consider the cauchy problem of a nonlinear schrodinger system through establishing a sharp weighted vector valued gagliardo nirenberg s inequality we find that the best constant in this inequality can be regarded as the criterion of blowup and global existence of the solutions when p 4 n and we prove that the solutions of this system will always exist globally if p 4 n the sharp thresholds for blowup and global existence are also obtained when 4 n p 4 n 2

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

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

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

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

on a theorem by ekeland and lasry concerning the number of periodic hamiltonian trajectories this note deals with the existence of iz distinct modes of vibration for hamihonian systems with n degrees of freedom on a prescribed energy surface this problem has been studied in a recent remarkable paper by ekeland and lasry 5 who have given an important contribution to the complete extension in the large of the local results of liapunov ri weinstein 111 and moser s the purpose of the present paper is to give a new proof of the result of 5 jointly with a slight improvement

self focusing on bounded domains the critical nonlinear schrodinger equation nls on bounded domains models the propagation of cw laser beams in hollow core fibers unlike the nls on unbounded domains which models propagation in bulk media the ground state waveguide solutions are stable and the condition of critical power for singularity formation is generically sharp

multiple brake orbits in bounded convex symmetric domains abstract in this paper we prove that there exist at least two geometrically distinct brake orbits in every bounded convex symmetric domain in r n for n 2

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

stability of the solitary wave solutions to a coupled bbm system abstract in this work we present a stability criteria for the solitary wave solutions to a bbm system that contains coupled nonlinear terms using the idea by bona chen and karakashian 5 and exploiting the accurate point spectrum information of the associated schrodinger operator we improve the stability results previously gotten by pereira 15

stability of the solitary manifold of the perturbed sine gordon equation abstract we study the perturbed sine gordon equation t t x x sin f e x where f is of differentiability class c n in e and the first k derivatives vanish at e 0 i e e l f 0 0 for 0 l k we construct implicitly a virtual solitary manifold by deformation of the classical solitary manifold in n iteration steps our main result establishes that the initial value problem with an appropriate initial state e n close to the virtual solitary manifold has a unique solution which follows up to time 1 c e k 1 2 and errors of order e n a trajectory on the virtual solitary manifold the trajectory on the virtual solitary manifold is described by two parameters which satisfy a system of odes in contrast to previous works our stability result yields arbitrarily high accuracy as long as the perturbation f is sufficiently often differentiable

wellposedness for the fourth order nonlinear schrodinger equations abstract we study the local smoothing effects and wellposedness of cauchy problem for the fourth order nonlinear schrodinger equations in 1d i t u x 4 u p x u 2 x u 2 t x r where p is a polynomial excluding constant or linear terms

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

unique continuation properties of the higher order nonlinear schrodinger equations in one dimension abstract in this paper we study the unique continuation properties of the higher order nonlinear schrodinger equations in one dimension and show exponential decay weighted estimates as well as a l p type carleman estimate based on the littlewood paley theory as a consequence we obtain that if u is a solution of the schrodinger equation such that there exists x 0 with supp u 0 supp u 1 x 0 or x 0 then u 0 in r 0 1

an index theory and existence of multiple brake orbits for star shaped hamiltonian systems soit h r 2n r une fonction continument differentiable et j o i ou i est la matrice identite n n on considere le systeme hamiltonien x jh x on etudie l existence de solutions periodiques sur une hypersurface compacte donnee h 1 1

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

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

existence and orbital stability of standing waves for nonlinear schrodinger systems abstract in this paper we investigate the existence of solutions in h 1 r n h 1 r n for nonlinear schrodinger systems of the form u 1 1 u 1 1 u 1 p 1 2 u 1 r 1 u 1 r 1 2 u 1 u 2 r 2 u 2 2 u 2 2 u 2 p 2 2 u 2 r 2 u 1 r 1 u 2 r 2 2 u 2 under the constraints r n u 1 2 d x a 1 0 r n u 2 2 d x a 2 0 here n 1 0 i 0 r i 1 2 p i 2 4 n for i 1 2 and r 1 r 2 2 4 n this problem is motivated by the search of standing waves for an evolution problem appearing in several physical models our solutions are obtained as constrained global minimizers of an associated functional note that in the system 1 and 2 are unknown and will correspond to the lagrange multipliers our main result is the precompactness of the minimizing sequences up to translation assuming the local well posedness of the associated evolution problem we then obtain the orbital stability of the standing waves associated to the set of minimizers

global existence for a coupled system of schrodinger equations with power type nonlinearities in this manuscript we consider the cauchy problem for a schrodinger system with power type nonlinearitiesi tuj uj k 1majk uk p uj p 2uj 0 uj x 0 j0 x where uj rn r c j0 rn c for j 1 2 m and ajk akj are positive real numbers global existence for the cauchy problem is established for a certain range of p a sharp form of a vector valued gagliardo nirenberg inequality is deduced which yields the minimal embedding constant for the inequality using this minimal embedding constant global existence for small initial data is shown for the critical case p 1 2 n finite time blow up as well as stability of solutions in the critical case is discussed

stability and instability of standing waves to a system of schrodinger equations with combined power type nonlinearities abstract we obtain the existence of standing wave solutions to a coupled nonlinear schrodinger system i u 1 t u 1 u 1 p k 1 m k u 1 p k 2 u 2 q k u 1 in r n 0 t i u 2 t u 2 u 2 p k 1 m k u 1 p k u 2 q k 2 u 2 in r n 0 t u 1 x 0 u 10 x u 2 x 0 u 20 x in r n where 0 p 4 n 2 and k k p k q k are real constants with 1 p k p 1 1 q k p 1 p k q k p 2 and k q k k p k k 1 m we also discuss the stability and instability of the standing wave solutions

existence of multiple normal mode trajectories on convex energy surfaces of even classical hamiltonian systems hamiltonian systems of n degrees of freedom for which the hamiltonian is a function that is even both in its joint n coordinate variables as well as in its joint n momentum variables are discussed for such systems the number of distinct trajectories which correspond to particular periodic solutions normal modes with the same energy is investigated to that end a constrained dual action principle is introduced applying min max methods to this variational problem several results are obtained among which the existence of at least n distinct trajectories if specific conditions are satisfied

nonlinear fourth order schrodinger equations with radial data abstract we prove the small data scattering and large data local well posedness for the nonlinear fourth order schrodinger equation with radial data in critical h s c space and in particular for some s c 0 the new ingredient is the improved strichartz estimates for fourth order schrodinger equations with radial initial data

mapping between generalized nonlinear schrodinger equations and neutral scalar field theories and new solutions of the cubic quintic nls equation we highlight an interesting mapping between the moving breather solutions of the generalized nonlinear schrodinger nls equations and the static solutions of neutral scalar field theories using this connection we then obtain several new moving breather solutions of the cubic quintic nls equation both with and without uniform phase in space the stability of some stationary solutions is investigated numerically and the results are confirmed via dynamical evolution

stability of solitary waves for the vector nonlinear schrodinger equation in higher order sobolev spaces abstract in this manuscript a sharp form of orbital stability for the schrodinger system also known as the vector nls i t u j 2 x x u j 2 i 1 m u i 2 u j 0 where u j are complex valued functions of x t r 2 j 1 2 m is established in l 2 based sobolev classes of arbitrarily high order the result means practically that not only does the bulk of what emanates from the perturbed solitary wave stay close in shape and propagation and phase speeds to the original solitary wave but emerging residual oscillations must also be very small and not only in the energy norm

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

on the covering of a hill s region by solutions in systems with gyroscopic forces abstract consider a lagrangian system with the lagrangian containing terms linear in velocity by analogy with the systems in celestial mechanics we call a bounded connected component of the possible motion area of such a system a hill s region suppose that the energy level is fixed and the corresponding hill s region is compact we present sufficient conditions under which any point in the hill s region can be connected with its boundary by a solution with the given energy the result is illustrated by examples from mechanics

erratum to sharp thresholds of global existence and blowup for a system of schrodinger equations with combined power type nonlinearities j math phys 51 033509 2010

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

orbital stability of solitary waves of a 3 coupled nonlinear schrodinger system abstract in this paper consideration is given to the 3 coupled nonlinear schrodinger system i t u j 2 x 2 u j i 1 3 b i j u i 2 u j 0 where u j are complex valued functions of x t r 2 j 1 2 3 and b i j are positive constants satisfying b i j b j i it will be shown first that if the symmetric matrix b b i j satisfies certain conditions then ground state solutions of the 3 coupled nonlinear schrodinger system exist and moreover they are orbitally stable the theory is then extended to include solitary waves as well in particular it will be shown that when a solitary wave is perturbed the perturbed solution must stay close to a solitary wave profile in which the translation and phase parameters are prescribed functions of time properties of these functions are then studied this is a continuous work of our previous paper where the 2 coupled nonlinear schrodinger system was considered

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

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

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

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

global solution for a sixth order nonlinear schrodinger equation abstract in this paper we are concerned with a sixth order nonlinear schrodinger equation by applying the factorization formula and an analytical method we prove the existence and uniqueness of global solution for the cauchy problem of the sixth order nonlinear schrodinger equation

the gardner equation and the l 2 stability of the n soliton solution of the korteweg de vries equation multi soliton solutions of the korteweg de vries equation kdv are shown to be globally l2 stable and asymptotically stable in the sense of martel and merle the proof is surprisingly simple and combines the gardner transform which links the gardner and kdv equations together with the martel merle tsai and martel merle recent results on stability and asymptotic stability in the energy space applied this time to the gardner equation as a by product the results of maddocks sachs and merle vega are improved in several directions

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

well posedness of higher order nonlinear schrodinger equations in sobolev spaces hs rn and applications in this paper we establish local well posedness in the sobolev space hs rn with s s0 for a general class of nonlinear dispersive equations of the type tu ip dx u f u where p dx is an elliptic differential operator on rn with a real symbol f u is a nonlinear function which behaves like u u for some constant 0 and s0 is a critical index suggested by a standard scaling argument by using such local result and conservation laws we improve the known and obtain some new global well posedness results for the fourth order nonlinear schrodinger equation i tu a u b 2u c u u

seifert conjecture in the even convex case in this paper we prove that there exist at least n geometrically distinct brake orbits on every c2 compact convex symmetric hypersurface in 2n satisfying the reversible condition n with n diag in in as a consequence we show that if the hamiltonian function is convex and even then seifert conjecture of 1948 on the multiplicity of brake orbits holds for any positive integern 2014 wiley periodicals inc

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

iteration theory of l index and multiplicity of brake orbits abstract in this paper we first establish the bott type iteration formulas and some abstract precise iteration formulas of the maslov type index theory associated with a lagrangian subspace for symplectic paths as an application we prove that there exist at least n 2 1 geometrically distinct brake orbits on every c 2 compact convex symmetric hypersurface in r 2 n satisfying the reversible condition n furthermore if all brake orbits on this hypersurface are nondegenerate then there are at least n geometrically distinct brake orbits on it as a consequence we show that there exist at least n 2 1 geometrically distinct brake orbits in every bounded convex symmetric domain in r n furthermore if all brake orbits in this domain are nondegenerate then there are at least n geometrically distinct brake orbits in it in the symmetric case we give a positive answer to the seifert conjecture of 1948 under a generic condition

global existence and asymptotic behavior of solutions to the fourth order nonlinear schrodinger equation in the critical case abstract we consider the cauchy problem for the nonlinear fourth order nonlinear schrodinger equation i t u 1 4 x 4 u i x u 3 u t 0 x r u 0 x u 0 x x r with critical nonlinearity where r we assume that the initial data are such that u 0 h 1 1 with sufficiently small norm u 0 h 1 1 we prove that there exists a unique global solution e i t 4 x 4 u c 0 h 1 1 of the cauchy problem for the nonlinear fourth order nonlinear schrodinger equation such that u t l c 1 t 1 4 moreover we show that if the total mass 1 2 r u 0 x d x 0 then the large time asymptotics is determined by the self similar solution

stability of solitons described by nonlinear schrodinger type equations with higher order dispersion abstract steady solitons described by the fourth order nonlinear schrodinger type equations in one two and three dimensions with power law nonlinearities are studied conditions of existence and stability of such solitons are found numerical results demonstrating soliton existence stability and instability are described they agree with the developed theory

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

existence of multiple periodic orbits on star shaped hamiltonian surfaces consider the hamiltonian system hs i 1 n here h c2 2n in this paper we investigate the existence of periodic orbits of hs on a given energy surface z 2n h z c c o is a constant the surface is required to verify certain geometric assumptions bounds a star shaped compact region and for some ellipsoid 2n o o depending in an explicit fashion on the lengths of the main axes of and one other geometrical parameter of such that if furthermore 2 2 1 then hs has at least n distinct geometric orbits on this result is shown to extend and unify several earlier works on this subject among them works by weinstein rabinowitz ekeland lasry and ekeland in proving this result we construct index theories for an s1 action from which we derive abstract critical point theorems for s1 invariant functionals we also derive an estimate for the minimal period of solutions to differential equatious