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

compositions and averages of two resolvents relative geometry of fixed points sets and a partial answer to a question by c byrne abstract we show that the set of fixed points of the average of two resolvents can be found from the set of fixed points for compositions of two resolvents associated with scaled monotone operators recently the proximal average has attracted considerable attention in convex analysis our results imply that the minimizers of proximal average functions can be found from the set of fixed points for compositions of two proximal mappings associated with scaled convex functions when both convex functions in the proximal average are indicator functions of convex sets least squares solutions can be completely recovered from the limiting cycles given by compositions of two projection mappings this provides a partial answer to a question posed by c byrne a novelty of our approach is to use the notion of resolvent average and proximal average

hamiltonian dynamics of vortex and magnetic lines in hydrodynamic type systems vortex line and magnetic line representations are introduced for a description of flows in ideal hydrodynamics and magnetohydrodynamics mhd respectively for incompressible fluids it is shown with the help of this transformation that the equations of motion for vorticity omega and magnetic field follow from a variational principle by means of this representation it is possible to integrate the hydrodynamic type system with the hamiltonian h integral omega dr and some other systems it is also demonstrated that these representations allow one to remove from the noncanonical poisson brackets defined in the space of divergence free vector fields the degeneracy connected with the vorticity frozenness for the euler equation and with magnetic field frozenness for ideal mhd for mhd a new weber type transformation is found it is shown how this transformation can be obtained from the two fluid model when electrons and ions can be considered as two independent fluids the weber type transformation for ideal mhd gives the whole lagrangian vector invariant when this invariant is absent this transformation coincides with the clebsch representation analog introduced by v e zakharov and e a kuznetsov dokl ajad nauk 194 1288 1970 sov phys dokl 15 913 1971

the asymptotic behavior of the composition of two resolvents abstract the asymptotic behavior of the composition of two resolvents in a hilbert space is investigated connections are made between the solutions of associated monotone inclusion problems and their dual versions the applications provided include a study of an alternating minimization procedure and a new proof of von neumann s classical result on the method of alternating projections

linear and strong convergence of algorithms involving averaged nonexpansive operators abstract we introduce regularity notions for averaged nonexpansive operators combined with regularity notions of their fixed point sets we obtain linear and strong convergence results for quasicyclic cyclic and random iterations new convergence results on the borwein tam method btm and on the cyclically anchored douglas rachford algorithm cadra are also presented finally we provide a numerical comparison of btm cadra and the classical method of cyclic projections for solving convex feasibility problems

prox penalization and splitting methods for constrained variational problems this paper is concerned with the study of a class of prox penalization methods for solving variational inequalities of the form ax n c x ni0 where mathcal h is a real hilbert space a mathcal h rightrightarrows mathcal h is a maximal monotone operator and n c is the outward normal cone to a closed convex set c subset mathcal h given psi mathcal h to mathbb r cup infty which acts as a penalization function with respect to the constraint x in c and a penalization parameter beta n we consider a diagonal proximal algorithm of the form x n i lambda n a beta n partial psi 1 x n 1 and an algorithm which alternates proximal steps with respect to a and penalization steps with respect to c and reads as x n i lambda n beta n partial psi 1 i lambda na 1 x n 1 we obtain weak ergodic convergence for a general maximal monotone operator a and weak convergence of the whole sequence x n when a is the subdifferential of a proper lower semicontinuous convex function mixing with passty s idea we can extend the ergodic convergence theorem thus obtaining the convergence of a prox penalization splitting algorithm for constrained variational inequalities governed by the sum of several maximal monotone operators our results are applied to an optimal control problem where the state variable and the control are coupled by an elliptic equation we also establish robustness and stability results that account for numerical approximation errors

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

stability of periodic traveling waves for complex modified korteweg de vries equation abstract we study the existence and stability of periodic traveling wave solutions for complex modified korteweg de vries equation we also discuss the problem of uniform continuity of the data solution mapping

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 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

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

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

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

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

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

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

nonlinear stability of periodic traveling wave solutions to the schrodinger and the modified korteweg de vries equations abstract this work is concerned with stability properties of periodic traveling waves solutions of the focusing schrodinger equation i u t u x x u 2 u 0 posed in r and the modified korteweg de vries equation u t 3 u 2 u x u x x x 0 posed in r our principal goal in this paper is the study of positive periodic wave solutions of the equation 3 0 called dnoidal waves a proof of the existence of a smooth curve of solutions with a fixed fundamental period l 2 2 l 2 h per 0 l is given it is also shown that these solutions are nonlinearly stable in the energy space h per 1 0 l and unstable by perturbations with period 2l in the case of the schrodinger equation

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

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

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 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

a new use of douglas rachford splitting for identifying infeasible unbounded and pathological conic programs in this paper we present a method for identifying infeasible unbounded and pathological conic programs based on douglas rachford splitting when an optimization program is infeasible unbounded or pathological the iterates of douglas rachford splitting diverge somewhat surprisingly such divergent iterates still provide useful information which our method uses for identification in addition for strongly infeasible problems the method produces a separating hyperplane and informs the user on how to minimally modify the given problem to achieve strong feasibility as a first order method the proposed algorithm relies on simple subroutines and therefore is simple to implement and has low per iteration cost

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

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

orbital stability of one parameter periodic traveling waves for dispersive equations and applications abstract this paper establishes sufficient conditions for the orbital stability of one parameter spatially periodic traveling wave solutions for one dimensional dispersive equations our method of proof combines known techniques with some new ideas as a consequence of our result we give several applications for well known dispersive equations extension of the theory to regularized equations is also established

soliton solution and gauge equivalence for an integrable nonlocal complex modified korteweg de vries equation in this paper we prove that an integrable nonlocal complex modified korteweg de vries mkdv equation introduced by ablowitz and musslimani nonlinearity 29 915 946 2016 is gauge equivalent to a spin like model from the gauge equivalence one can see that there exists significant difference between the nonlocal complex mkdv equation and the classical complex mkdv equation through constructing the darboux transformation for nonlocal complex mkdv equation a variety of exact solutions including dark soliton w type soliton m type soliton and periodic solutions are derived

boundary invariants of pseudoconvex domains the purpose of this paper is to describe an invariant ax z which is associated with each point z in the boundary of a smoothly bounded pseudoconvex domain in cn the motivation for defining the invariant ax z referred to as the multitype of z comes from the study of the boundary regularity properties of solutions of the d neumann problem on domains of finite type by a domain of finite type we mean one that satisfies the definition given by d angelo 5 namely a domain such that at any boundary point the maximum order of contact of one dimensional complex analytic varieties with the boundary is bounded in 4 the author showed how one can give a proof of global regularity of the a neumann problem for pseudoconvex domains of finite type if there is an invariant such as the one described in this paper in a forthcoming article this invariant will be used to obtain a proof of subelliptic estimates for the same class of domains before giving the definition of the multitype ax z we must introduce some

sufficient conditions for stability of solitary wave solutions of model equations for long waves abstract we consider solitary wave solutions of model equations for long waves that feature a general form of linear dispersion sufficient conditions for the non linear stability of such solutions are derived these conditions are shown to obtain for the korteweg de vries equation and certain of its generalizations such as the benjamin ono equation and the intermediate long wave equation

on the range of the douglas rachford operator the problem of finding a minimizer of the sum of two convex functions or more generally that of finding a zero of the sum of two maximally monotone operators is of central importance in variational analysis perhaps the most popular method of solving this problem is the douglas rachford splitting method surprisingly little is known about the range of the douglas rachford operator in this paper we set out to study this range systematically we prove that for 3 monotone operators a very pleasing formula can be found that reveals the range to be nearly equal to a simple set involving the domains and ranges of the underlying operators a similar formula holds for the range of the corresponding displacement mapping we discuss applications to subdifferential operators to the infimal displacement vector and to firmly nonexpansive mappings various examples and counterexamples are presented including some concerning the celebrated brezis haraux theorem

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

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

generalized solutions for the sum of two maximally monotone operators a common theme in mathematics is to define generalized solutions to deal with problems that potentially do not have solutions a classical example is the introduction of least squares solutions via the normal equations associated with a possibly infeasible system of linear equations in this paper we introduce a normal problem associated with finding a zero of the sum of two maximally monotone operators if the original problem admits solutions then the normal problem returns this same set of solutions the normal problem may yield solutions when the original problem does not admit any furthermore it has attractive variational and duality properties several examples illustrate our theory

on the douglas rachford algorithm the douglas rachford algorithm is a very popular splitting technique for finding a zero of the sum of two maximally monotone operators the behaviour of the algorithm remains mysterious in the general inconsistent case i e when the sum problem has no zeros however more than a decade ago it was shown that in the possibly inconsistent convex feasibility setting the shadow sequence remains bounded and its weak cluster points solve a best approximation problem in this paper we advance the understanding of the inconsistent case significantly by providing a complete proof of the full weak convergence in the convex feasibility setting in fact a more general sufficient condition for the weak convergence in the general case is presented our proof relies on a new convergence principle for fejer monotone sequences numerous examples illustrate our 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

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

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

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