the sensitivity of the stable lyapunov equation we present an analysis of the sensitivity of the solution of the lyapunov equation a x xa w where a is stable this analysis leads to a spectral norm bound on the relative perturbation of the solution which is optimal for a certain class of estimates and which is essentially equivalent to the frobenius norm bound obtained from the associated kronecker product system the latter bound can be expressed in terms of operatorname sep a a and is known to accurately reflect the sensitivity of the lyapunov problem but it is hard to interpret in terms of the original matrix a in contrast the spectral norm bound which we derive is directly related to the minimal l 2 damping of the dynamical system dot z az moreover this dynamical link with the sensitivity problem leads to a new method of systematically investigating the norm behavior of e at as well as providing a wealth of information about control theoretic aspects of dot z az when a is the closed loop state matrix

adiabatic midpoint rule for the dispersion managed nonlinear schrodinger equation the dispersion managed nonlinear schrodinger equation contains a small parameter varepsilon a rapidly changing piecewise constant coefficient function and a cubic nonlinearity typical solutions are highly oscillatory and have a discontinuous time derivative and hence solving this equation numerically is a challenging task we present and analyze a tailor made time integrator which attains the desired accuracy with a significantly larger step size than traditional methods the construction of this method is based on a favorable transformation to an equivalent problem and the explicit computation of certain integrals over highly oscillatory phases the error analysis requires the thorough investigation of various cancellation effects which result in improved accuracy for special step sizes

backward perturbation analysis of certain characteristic subspaces this paper gives optimal backward perturbation bounds and the accuracy of approximate solutions for subspaces associated with certain eigenvalue problems such as the eigenvalue problemax x the generalized eigenvalue problem ax bx and the singular value decomposition of a matrixa this paper also gives residual bounds for certain eigenvalues generalized eigenvalues and singular values

optical pulse propagation in fibers with random dispersion abstract the propagation of optical pulses in two types of fibers with randomly varying dispersion is investigated the first type refers to a uniform fiber dispersion superimposed by random modulations with a zero mean the second type is the dispersion managed fiber line with fluctuating parameters of the dispersion map application of the mean field method leads to the nonlinear schrodinger equation nlse with a dissipation term expressed by a fourth order derivative of the wave envelope the prediction of the mean field approach regarding the decay rate of a soliton is compared with that of the perturbation theory based on the inverse scattering transform ist a good agreement between these two approaches is found possible ways of compensation of the radiative decay of solitons using the linear and nonlinear amplification are explored the corresponding mean field equation coincides with the complex swift hohenberg equation the condition for the autosolitonic regime in propagation of optical pulses along a fiber line with fluctuating dispersion is derived and the existence of autosoliton dissipative soliton is confirmed by direct numerical simulation of the stochastic nlse the dynamics of solitons in optical communication systems with random dispersion management is further studied applying the variational principle to the mean field nlse which results in a system of odes for soliton parameters extensive numerical simulations of the stochastic nlse mean field equation and corresponding set of odes are performed to verify the predictions of the developed theory

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

on conformable fractional calculus recently the authors khalil et al 2014 introduced a new simple well behaved definition of the fractional derivative called conformable fractional derivative in this article we proceed on to develop the definitions there and set the basic concepts in this new simple interesting fractional calculus the fractional versions of chain rule exponential functions gronwall s inequality integration by parts taylor power series expansions laplace transforms and linear differential systems are proposed and discussed

dispersion managed solitons via an averaged variational principle constrained minimization is used as a computational strategy to approximate and study dispersion managed solitons through their characterization as minima of an averaged variational principle a basis of hermite gaussian functions is used and the constrained minimization procedure is carried out to find such pulses this method produces more accurate pulse shapes than the usual gaussian approximations propagating with less noise and providing a far better representation of the tail of the pulse the success of this procedure provides confirmation that the dispersion managed soliton is truly a minimum of the constrained variational principle when the residual dispersion is negative the dispersion managed soliton can no longer be a minimum a fact that is confirmed by our numerical simulations even in this case however approximate pulse shapes can be obtained using this finite dimensional approximation additionally we find critical points other than the ground state the most interesting is an excited bound state corresponding to a symmetric bisoliton 2003 elsevier b v all rights reserved

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

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

on estimating the separation between a b and c d associated with the generalized sylvester equation axd bxc e abstract in this paper we estimate the following quantity sep a b c d min x f 1 a x d b x c f where a b c m m c d c n n and f is the frobenius norm in terms of the generalized schur decomposition and weiestrass canonical form of the regular matrix pair lower and upper bounds of sep a b c d are derived respectively the results are illustrated by numerical examples

quartic normal forms for the periodic nonlinear schrodinger equation with dispersion management we investigate birkhoff normal forms for the periodic nonlinear schrodinger equation with dispersion management the normalization we describe is related to averaging arguments considered in the literature and has the advantage of producing fewer resonant couplings between high spatial frequency modes one consequence is that the normal form equations have invariant subspaces of large but finite dimension where we can find several classes of periodic orbits the formal arguments apply to other related dispersive systems and to normal forms of high order we also present a rigorous version of the normal form calculation and show that solutions of the quartic normal form equations remain close to solutions of the full system over a time that is inversely proportional to a small nonlinearity parameter

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

decay estimates and smoothness for solutions of the dispersion managed non linear schrodinger equation we study the decay and smoothness of solutions of the dispersion managed non linear schrodinger equation in the case of zero residual dispersion using new x space versions of bilinear strichartz estimates we show that the solutions are not only smooth but also fast decaying

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

ground states for schrodinger type equations with nonlocal nonlinearity abstract the problem of the existence of stable solitary wave solutions for nonlinear schrodinger type equations with a generalized cubic nonlinearity is considered these types of equations have recently arisen in the context of optical communications as averaging approximations to nonlinear dispersive equations with widely separated time scales in this paper it is shown that under general conditions on the kernel of the nonlocal term stable standing wave solutions exist for these equations

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

regularity of ground state solutions of dispersion managed nonlinear schrodinger equations abstract we consider the dispersion managed nonlinear schrodinger equation dmnls in the case of zero residual dispersion using dispersive properties of the equation and estimates in bourgain spaces we show that the ground state solutions of dmnls are smooth the existence of smooth solutions in this case matches the well known smoothness of the solutions in the case of nonzero residual dispersion in the case x r 2 we prove that the corresponding minimization problem with zero residual dispersion has no solution

sensitivity analysis of the algebraic riccati equations in this paper some sharp perturbation bounds for the hermitian positive semi definite solution to an algebraic riccati equation are developed a further analysis for these bounds is done this analysis shows that there is presumably some intrinsic relation between the sensitivity of the solution to the algebraic riccati equation and the distance of the spectrum of the closed loop matrix from the imaginary axis

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

the singular perturbation limit of a variational problem from nonlinear fiber optics abstract among the fiber optical models the dispersion managed dm systems have attracted interest due to their favorable properties compared to the more traditional approaches which rely on classical solitons numerical and physical experiments have shown that in the limit of strong dispersion management there exist stable nearly periodic pulses it was later found that these breathing pulses correspond to the ground states solutions of a certain averaged dm problem with positive and even zero residual dispersion it was not clear however how dm solitons in the zero residual dispersion regime are related to those ones in the positive residual dispersion regime in this paper dm solitons in the mean zero case are obtained as dm solitons in the limit of vanishing positive residual dispersion this corresponds to studying the singular perturbation limit of energy functionals of the form e u e 2 r u 2 d x f u with a highly nonlocal operator f

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

lower bound estimation for the separation of two matrices abstract we give some lower bounds on the separations sep 1 a b sep a b and sep f a b the bounds show that the separation of two matrices a and b is closely related to the separation between the spectral sets of a and b the structure of the jordan canonical forms of a and b and the departures from normality of a and b

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 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

breather solutions in the diffraction managed nls equation we show the existence of localized breather solutions in an averaged version of the discrete nonlinear schrodinger equation nls with diffraction management a system that models coupled waveguide arrays with periodic diffraction management geometries the breather solutions are constrained extrema of the hamiltonian of the averaged system and their existence is shown by a discrete version of the concentration compactness principle the main assumptions are that the averaged diffraction is sufficiently small compared to strength of the nonlinearity and that the sign of the nonlinearity corresponds to the focusing case an interesting feature of the problem is that the nonlinear interaction between neighboring lattice sites can be large and is of infinite range on the other hand the interaction decays rapidly at sufficiently large distances and this plays an important role in the proof the results also apply to higher dimensional lattices and to the discrete nls equation

generalized schur methods with condition estimators for solving the generalized sylvester equation stable algorithms are presented for solving the generalized sylvester equation they are based on orthogonal equivalence transformations of the original problem perturbation theory and rounding error analysis are included condition estimators dif sup 1 estimators are developed which when substituted into derived error bounds give accuracy estimates of a computed solution results from numerical experiments on well conditioned and ill conditioned problems are reported

averaging of dispersion managed solitons existence and stability we consider existence and stability of dispersion managed solitons in the two ap proximations of the periodic nonlinear schrodinger nls equation i a dynamicalsystem for a gaussian pulse and ii an average integral nls equation we apply normal form transformations for finite dimensionaland infinite dimensionalhamil tonian systems with periodic coefficients first order corrections to the leading order averaged hamiltonian are derived explicitly for both approxi mations bifurcations of soliton solutions and their stability are studied by analysis of critical points of the first order averaged hamiltonians the validity of the averaging procedure is verified and the presence of ground states corresponding to dispersion managed solitons in the averaged hamiltonian is established

the direct spectral problem via local derivative including truncated mittag leffler function abstract in this article we propose some substantial spectral data for sturm liouville problem with separated boundary conditions in frame of newly defined truncated m derivative which contains truncated mittag leffler function the proposed such spectral data like representation of solution for sturm liouville problem subjected to the both boundary and initial conditions asymptotic formulas of the eigenfunctions and eigenvalues and their normalized counterparts are implemented in terms of underlying efficient limit based local derivative we establish all these mentioned data for two different representation of solution under different conditions in order to obtain these crucial results certain tools like integration by parts formula leibniz rule variation of parameters method and so forth are offered in view of the truncated m derivative moreover uniqueness of the solution of the sturm liouville equation is leadingly showed substantially the main advantage of this potent derivative allowing arbitrary change of order is that it contains truncated mittag leffler function enabling us to deal better with the behavior of the matter in hand and to consider it as a generalized version of other extensive local derivatives in literature owing to the additional parameter inside definition thereby the objective of this paper is pivotally dedicated to observe the behavior of the spectral structure of sturm liouville problem by supporting with the simulation analysis

the nonlinear schrodinger equation with white noise dispersion abstract under certain scaling the nonlinear schrodinger equation with random dispersion converges to the nonlinear schrodinger equation with white noise dispersion the aim of this work is to prove that this latter equation is globally well posed in l 2 or h 1 the main ingredient is the generalization of the classical strichartz estimates additionally we justify rigorously the formal limit described above

on estimating the separation of two regular matrix pairs let a b and c d be two n times n n n and m times m m m regular matrix pairs respectively the main contribution of this paper is to derive lower and upper bounds of the separation between a b and c d in terms of their generalized schur decompositions and weierstrass canonical forms the results are illustrated by numerical examples

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 fractional sturmliouville problemnumerical approximation and application in fractional diffusion the numerical method of solving the fractional eigenvalue problem is derived in the case when the fractional sturmliouville equation is subjected to the mixed boundary conditions this non integer order differential equation is discretized to the scheme with the symmetric matrix representing the action of the numerically expressed composition of the left and the right caputo derivative the numerical eigenvalues are thus real and the eigenvectors associated to distinct eigenvalues are orthogonal in the respective finite dimensional hilbert space the advantage of the proposed method is the formulation which allows us to construct the approximate eigenfunctions which form an orthonormal function system in the infinite dimensional weighted lebesgue integrable function space the developed numerical method of calculation of the eigenvalues and eigenfunctions is then applied in construction of the approximate solution to the 1d space time fractional diffusion problem in a bounded domain a new numerical method for solving a fractional eigenvalue problem has been proposed the approximate eigenfunctions are orthogonal at each step of the proposed method the method is applied in the approximate solution to fractional diffusion problem

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

stabilizing effects of dispersion management a cubic nonlinear schrodinger equation nls with periodically varying dispersion coefficient as it arises in the context of fiber optics communication is considered for sufficiently strong variation corresponding to the so called strong dispersion management regime the equation possesses pulse like solutions which evolve nearly periodically this phenomenon is explained by constructing ground states for the averaged variational principle and justifying the averaging procedure furthermore it is shown that in certain critical cases e g quintic nonlinearity in one dimension and cubic nonlinearity in two dimensions the dispersion management technique stabilizes the pulses which otherwise would be unstable this observation seems to be new and is reminiscent of the well known kapitza s effect of stabilizing the inverted pendulum by rapidly moving its pivot 2001 published by elsevier science b v

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

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

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

a new definition of fractional derivative we give a new definition of fractional derivative and fractional integral the form of the definition shows that it is the most natural definition and the most fruitful one the definition for email protected email protected 1 coincides with the classical definitions on polynomials up to a constant further if a 1 the definition coincides with the classical definition of first derivative we give some applications to fractional differential equations

periodic solutions for the schrodinger equation with nonlocal smoothing nonlinearities in higher dimension abstract we consider the nonlinear schrodinger equation in higher dimension with dirichlet boundary conditions and with a nonlocal smoothing nonlinearity we prove the existence of small amplitude periodic solutions in the fully resonant case we find solutions which at leading order are wave packets in the sense that they continue linear solutions with an arbitrarily large number of resonant modes the main difficulty in the proof consists in a small divisor problem which we solve by using a renormalisation group approach

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

blow up solutions to the nonlinear schrodinger equation with oscillating nonlinearities abstract in this article we investigate the possibility of finite time blow up in h 1 r 2 for solutions to critical and supercritical nonlinear schrodinger equations with an oscillating nonlinearity we prove that despite the oscillations some solutions blow up in finite time conversely we observe that for a given initial data oscillations can extend the local existence time of the corresponding solution