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

benchmarking optimization software with performance profiles we propose performance profiles distribution functions for a performance metric as a tool for benchmarking and comparing optimization software we show that performance profiles combine the best features of other tools for performance evaluation

a conjugate gradient method with descent direction for unconstrained optimization a modified conjugate gradient method is presented for solving unconstrained optimization problems which possesses the following properties i the sufficient descent property is satisfied without any line search ii the search direction will be in a trust region automatically iii the zoutendijk condition holds for the wolfe powell line search technique iv this method inherits an important property of the well known polak ribiere polyak prp method the tendency to turn towards the steepest descent direction if a small step is generated away from the solution preventing a sequence of tiny steps from happening the global convergence and the linearly convergent rate of the given method are established numerical results show that this method is interesting

stability of semiclassical bound states of nonlinear schrodinger equations with potentials in this paper we study the lyapunov stabilities of some semiclassical bound states of the nonhomogeneous nonlinear schrodinger equation we prove that among those bound states those which are concentrated near local minima respectively maxima of the potentialv are stable respectively unstable we also prove that those bound states are positive if is sufficiently small

an effective co evolutionary differential evolution for constrained optimization many practical problems can be formulated as constrained optimization problems due to the simple concept and easy implementation the penalty function method has been one of the most common techniques to handle constraints however the performance of this technique greatly relies on the setting of penalty factors which are usually determined by manual trial and error and the suitable penalty factors are often problem dependent and difficult to set in this paper a differential evolution approach based on a co evolution mechanism named cde is proposed to solve the constrained problems first a special penalty function is designed to handle the constraints second a co evolution model is presented and differential evolution de is employed to perform evolutionary search in spaces of both solutions and penalty factors thus the solutions and penalty factors evolve interactively and self adaptively and both the satisfactory solutions and suitable penalty factors can be obtained simultaneously simulation results based on several benchmark functions and three well known constrained design problems as well as comparisons with some existed methods demonstrate the effectiveness efficiency and robustness of the proposed method

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

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

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

an efficient constraint handling method for genetic algorithms many real world search and optimization problems involve inequality and or equality constraints and are thus posed as constrained optimization problems in trying to solve constrained optimization problems using genetic algorithms gas or classical optimization methods penalty function methods have been the most popular approach because of their simplicity and ease of implementation however since the penalty function approach is generic and applicable to any type of constraint linear or nonlinear their performance is not always satisfactory thus researchers have developed sophisticated penalty functions specific to the problem at hand and the search algorithm used for optimization however the most difficult aspect of the penalty function approach is to find appropriate penalty parameters needed to guide the search towards the constrained optimum in this paper ga s population based approach and ability to make pair wise comparison in tournament selection operator are exploited to devise a penalty function approach that does not require any penalty parameter careful comparisons among feasible and infeasible solutions are made so as to provide a search direction towards the feasible region once sufficient feasible solutions are found a niching method along with a controlled mutation operator is used to maintain diversity among feasible solutions this allows a real parameter ga s crossover operator to continuously find better feasible solutions gradually leading the search near the true optimum solution gas with this constraint handling approach have been tested on nine problems commonly used in the literature including an engineering design problem in all cases the proposed approach has been able to repeatedly find solutions closer to the true optimum solution than that reported earlier

a hybrid particle swarm optimization with a feasibility based rule for constrained optimization during the past decade hybrid algorithms combining evolutionary computation and constraint handling techniques have shown to be effective to solve constrained optimization problems for constrained optimization the penalty function method has been regarded as one of the most popular constraint handling technique so far whereas its drawback lies in the determination of suitable penalty factors which greatly weakens the efficiency of the method as a novel population based algorithm particle swarm optimization pso has gained wide applications in a variety of fields especially for unconstrained optimization problems in this paper a hybrid pso hpso with a feasibility based rule is proposed to solve constrained optimization problems in contrast to the penalty function method the rule requires no additional parameters and can guide the swarm to the feasible region quickly in addition to avoid the premature convergence simulated annealing sa is applied to the best solution of the swarm to help the algorithm escape from local optima simulation and comparisons based on several well studied benchmarks demonstrate the effectiveness efficiency and robustness of the proposed hpso moreover the effects of several crucial parameters on the performance of the hpso are studied as well

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

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

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

perturbation expansion and nth order fermi golden rule of the nonlinear schrodinger equations in this paper we consider generalized nonlinear schrodinger equations with external potentials we find the expressions for the fourth and the sixth order fermi golden rules fgrs conjectured in gang and sigal rev math phys 17 1143 1207 2005 geom funct anal 16 no 7 1377 1390 2006 the fgr is a key condition in a study of the asymptotic dynamics of trapped solitons

multichannel nonlinear scattering for nonintegrable equations ii the case of anisotropic potentials and data abstract the nonlinear scattering and stability results of soffer and weinstein comm math phys 1990 are extended to the case of anisotropic potentials and data the range of nonlinearities for which the theory is shown to be valid is also extended considerably

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

on scattering of small energy solutions of non autonomous hamiltonian nonlinear schrodinger equations abstract we revisit a result by cuccagna kirr and pelinovsky about the cubic nonlinear schrodinger equation nls with an attractive localized potential and a time dependent factor in the nonlinearity we show that under generic hypotheses on the linearization at 0 of the equation small energy solutions are asymptotically free this is yet a new application of the hamiltonian structure continuing a program initiated in a paper by bambusi and cuccagna

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 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

on dispersion of small energy solutions of the nonlinear klein gordon equation with a potential in this paper we study small amplitude solutions of nonlinear klein gordon equations with a potential under suitable smoothness and decay assumptions on the potential and a genericity assumption on the nonlinearity we prove that all small energy solutions are asymptotically free in cases where the linear system has at most one bound state the result was already proved by soffer and weinstein we obtain here a result valid in the case of an arbitrary number of possibly degenerate bound states the proof is based on a combination of birkhoff normal form techniques and dispersive estimates

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

speeding up continuous grasp abstract continuous grasp c grasp is a stochastic local search metaheuristic for finding cost efficient solutions to continuous global optimization problems subject to box constraints hirsch et al 2007 like a greedy randomized adaptive search procedure grasp a c grasp is a multi start procedure where a starting solution for local improvement is constructed in a greedy randomized fashion in this paper we describe several improvements that speed up the original c grasp and make it more robust we compare the new c grasp with the original version as well as with other algorithms from the recent literature on a set of benchmark multimodal test functions whose global minima are known hart s sequential stopping rule 1998 is implemented and c grasp is shown to converge on all test problems

continuous global optimization through the generation of parametric curves in this paper we develop a new approach for solving a large class of global optimization problems the objective function is only continuous non smooth and non lipschitzian defined on a rectangle of r n this approach is based on the generation in the feasible set of a family of parametrized curves satisfying certain properties combined with the one dimensional evtushenko algorithm to accelerate the corresponding mixed algorithm we have incorporated in a variant a pattern search type deterministic local optimization method and in another variant a new stochastic local optimization method both variants have been applied to several typical test problems a comparison with some well known methods is highlighted through numerical experiments

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

a class of gradient unconstrained minimization algorithms with adaptive stepsize in this paper the development convergence theory and numerical testing of a class of gradient unconstrained minimization algorithms with adaptive stepsize are presented the proposed class comprises four algorithms the first two incorporate techniques for the adaptation of a common stepsize for all coordinate directions and the other two allow an individual adaptive stepsize along each coordinate direction all the algorithms are computationally efficient and possess interesting convergence properties utilizing estimates of the lipschitz constant that are obtained without additional function or gradient evaluations the algorithms have been implemented and tested on some well known test cases as well as on real life artificial neural network applications and the results have been very satisfactory

a three terms polak ribiere polyak conjugate gradient algorithm for large scale nonlinear equations in this paper a conjugate gradient algorithm for systems of large scale nonlinear equations is designed by the following steps i a three terms conjugate gradient direction d k is presented where the direction possesses the sufficient descent property and the trust region property independent of line search technique ii a backtracking line search technique along the direction is proposed to get the step length k and construct a point iii if the point satisfies the given condition then it is the next point otherwise the projection proximal technique is used and get the next point both the direction and the line search technique are the derivative free approaches then the large scale nonlinear equations are successfully solved 100 000 variables the global convergence of the given algorithm is established under suitable conditions numerical results show that the proposed method is efficient for large scale problems

a modified polak ribiere polyak conjugate gradient algorithm for nonsmooth convex programs the conjugate gradient cg method is one of the most popular methods for solving smooth unconstrained optimization problems due to its simplicity and low memory requirement however the usage of cg methods is mainly restricted to solving smooth optimization problems so far the purpose of this paper is to present efficient conjugate gradient type methods to solve nonsmooth optimization problems by using the moreau yosida regulation smoothing approach and a nonmonotone line search technique we propose a modified polak ribiere polyak prp cg algorithm for solving a nonsmooth unconstrained convex minimization problem our algorithm possesses the following three desired properties i the search direction satisfies the sufficient descent property and belongs to a trust region automatically ii the search direction makes use of not only the gradient information but also the function value information and iii the algorithm inherits an important property of the well known prp method the tendency to turn towards the steepest descent direction if a small step is generated away from the solution preventing a sequence of tiny steps from happening under standard conditions we show that the algorithm converges globally to an optimal solution numerical experiment shows that our algorithm is effective and suitable for solving large scale nonsmooth unconstrained convex optimization problems

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

dispersive estimates for schrodinger operators in dimensions one and three we consider l 1 l estimates for the time evolution of hamiltonians h v in dimensions d 1 and d 3 with bound t 2 we require decay of the potentials but no regularity in d 1 the decay assumption is 1 x v x dx whereas in d 3 it is v x c 1 x 3

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

global optimization through a stochastic perturbation of the polak ribiere conjugate gradient method abstract we develop a new modified polak ribiere conjugate gradient method by considering a random perturbation our approach is suitable for solving a large class of optimization problems on a rectangle of r n or unconstrained problems theoretical results ensure that the proposed method converges to a global minimizer numerical experiments are achieved on some typical test problems particularly the engineering problem of lennard jones clusters a comparison with well known methods is carried out to show the performance of our algorithm

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

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

on asymptotic stability in energy space of ground states for nonlinear schrodinger equations we consider nonlinear schrodinger equations iu t delta u beta u 2 u 0 text for t x in mathbb r times mathbb r d where d 3 and is smooth we prove that symmetric finite energy solutions close to orbitally stable ground states converge to a sum of a ground state and a dispersive wave as t assuming the so called the fermi golden rule fgr hypothesis we improve the sign condition required in a recent paper by gang zhou and i m sigal

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

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

asymptotic dynamics of nonlinear schrodinger equations resonance dominated and dispersion dominated solutions we consider a linear schrodinger equation with a nonlinear perturbation in r 3 assume that the linear hamiltonian has exactly two bound states and its eigenvalues satisfy some resonance condition we prove that if the initial data is sufficiently small and is near a nonlinear ground state then the solution approaches to certain nonlinear ground state as the time tends to infinity furthermore the difference between the wave function solving the nonlinear schrodinger equation and its asymptotic profile can have two different types of decay the resonancedominated solutions decay as t 1 2 or the dispersion dominated solutions decay at least like t 3 2 c 2002 john wiley sons inc

non linear wave and schrodinger equations i instability of periodic and quasiperiodic solutions we investigate stability of periodic and quasiperiodic solutions of linear wave and schrodinger equations under non linear perturbations we show in the case of the wave equations that such solutions are unstable for generic perturbations for the schrodinger equations periodic solutions are stable while the quasiperiodic ones are not we extend these results to periodic solutions of non linear equations

convergence of descent method without line search line search method is sometimes a useful and efficient technique for solving unconstrained optimization problems especially for solving small and middle scale problems however a line search procedure is necessary at each iteration leading to a significant amount of computation in order to reduce the evaluations of objective functions and gradients the line search procedure should be avoided in algorithm design in this paper we propose a new descent method without line search for unconstrained optimization problems the algorithm is required to estimate some parameters at each iteration we analyze theoretically the global convergence of the new algorithm under mild conditions these theoretical conclusions can help us in designing new efficient methods for optimization problems

accelerating adaptive trade off model using shrinking space technique for constrained evolutionary optimization adaptive trade off model atm is a constraint handling mechanism proposed recently the main advantages of this model are its simplicity and adaptation moreover it can be easily embedded into evolutionary algorithms for solving constrained optimization problems this paper proposes a novel method for constrained optimization which aims at accelerating the atm using shrinking space technique eighteen benchmark test functions and five engineering design problems are used to test the performance of the method proposed experimental results suggest that combining the atm with the shrinking space technique is very beneficial the method proposed can promptly converge to competitive results without loss of the quality and the precision of the final results performance comparisons with some other state of the art approaches from the literature are also presented

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