existence and stability of propagating fronts for an autocatalytic reaction diffusion system we study a one dimensional reaction diffusion system which describes an isothermal autocatalytic chemical reaction involving both a quadratic a b 2b and a cubic a 2b 3b autocatalysis the parameters of this system are in the ratio d d b d a of the diffusion constants of the reactant a and the autocatalyst b and the relative activity ii of the cubic reaction first for all values of d 0 and k greater than or equal to 0 we prove the existence of a family of propagating fronts or travelling waves describing the advance of the reaction in particular in the quadratic case k 0 we recover the results of billingham and needham phil trans r sec london a 334 1991 1 24 then if d is close to 1 and k is sufficiently small we prove using energy functionals that these propagating fronts are stable against small perturbations in exponentially weighted sobolev spaces this extends part of the results that are known for the scalar equation to which our system reduces when d 1 c 1998 elsevier science b v

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

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

error propagation when approximating multi solitons the kdv equation as a case study abstract the paper studies the influence of the time discretizations when simulating some phenomena involving more than one solitary wave taking the kdv equation as a case study we obtain some conditions on the numerical method in order to get a more correct simulation of multi soliton solutions they are related to the evolution of the conserved quantities of the problem through the numerical integration it is shown that when approximating n solitons a method that preserves n invariants of the problem shows a better time propagation of the error than that of a general scheme as a consequence of this the simulation of some physical parameters that characterize the waves is more suitable when using conservative integrators we also show how these results can be extended to the approximation to multi solitons of any equation of the kdv hierarchy and more generally other integrable equations

on asymptotics in case of linear index 2 differential algebraic equations asymptotic properties of solutions of general linear differential algebraic equations daes and those of their numerical counterparts are discussed new results on the asymptotic stability in the sense of lyapunov as well as on contractive index 2 daes are given the behavior of the backward differentiation formula bdf implicit runge kutta irk and projected implicit runge kutta pirk methods applied to such systems is investigated in particular we clarify the significance of certain subspaces closely related to the geometry of the dae asymptotic properties like a stability and l stability are shown to be preserved if these subspaces are constant moreover algebraically stable irk dae are b stable under this condition the general results are specialized to the case of index 2 hessenberg systems

first order soliton perturbation theory for a generalized kdv model with stochastic forcing and damping we discuss an application of the expansion method to the soliton perturbation theory for a generalized kdv model with stochastic forcing and damping we find that the expansion method is a perturbation technique which allows one to retain a more representative linearization for a nonlinear differential equation than comparable methods of perturbation such as those relying on small parameters indeed one primary benefit of the method is that it tends to provide much more reasonable solutions with relatively few terms compared to standard perturbation methods due to the more accurate linearization we show that the method allows one to obtain first order perturbations in which the stochastic and nonlinear contributions separate allowing one to study each influence independently at least to first order we then apply the expansion method to the study of a stochastic generalized kdv model and we show that this procedure gives a natural extension to the perturbation theory for the traditional kdv model the actual method of computation is outlined and may be applied to a variety of generalized kdv models

description of two soliton collision for the quartic gkdv equation in this paper we give the rst description of the collision of two solitons for a nonintegrable equation in a special regime we consider solutions of the quartic gkdv equation tu x 2 xu u 4 0 which behave as t 1 like u t x qc1 x c1t qc2 x c2t t x where qc x ct is a soliton andk t kh1k qc2kh1k qc1kh1 the global behavior of u t is given by the following stability result for all t 2 r u t x qc1 t x y1 t qc2 t x y2 t t x where k t kh1k qc2kh1 and limt 1c1 t c limt 1c2 t c in the case where u t is a pure 2 soliton solution as t 1 i e limt 1 k t kh1 0 we obtain c c1 c 0 therefore in contrast with the integrable kdv equation or mkdv equation no global pure 2 soliton solution exists and the collision is inelastic a dierent notion of global 2 soliton is then proposed

graphical krein signature theory and evans krein functions two concepts evidently very different in nature have proved to be useful in analytical and numerical studies of spectral stability in nonlinear wave theory i the krein signature of an eigenvalue a quantity usually defined in terms of the relative orientation of certain subspaces that is capable of detecting the structural instability of imaginary eigenvalues and hence their potential for moving into the right half plane leading to dynamical instability under perturbation of the system and ii the evans function an analytic function detecting the location of eigenvalues one might expect these two concepts to be related but unfortunately examples demonstrate that there is no way in general to deduce the krein signature of an eigenvalue from the evans function for example by studying derivatives of the latter the purpose of this paper is to recall and popularize a simple graphical interpretation of the krein signature well known in the spectral theory of polynomial operator pencils once esta

new types of soliton solutions we announce a detailed investigation of limits of n soliton solutions of the korteweg devries kdv equation as n tends to infinity our main results provide new classes of kdv solutions including in particular new types of soliton like reflectionless solutions as a byproduct we solve an inverse spectral problem for one dimensional schrodinger operators and explicitly construct smooth and real valued potentials that yield a purely absolutely continuous spectrum on the nonnegative real axis and give rise to an eigenvalue spectrum that includes any prescribed countable and bounded subset of the negative real axis

spectral stability and time evolution of n solitons in the kdv hierarchy this paper concerns spectral stability and time evolution of n solitons in the korteweg de vries kdv hierarchy with mixed commuting time flows spectral stability problem is analysed by using a pair of self adjoint operators with finite numbers of negative eigenvalues we show that the absence of unstable eigenvalues in the stability problem is related to the absence of negative eigenvalues of these operators in the constrained function spaces time evolution of n solitons is uniquely characterized from the inverse scattering transform technique

synchronization and relaxation for a class of globally coupled hamiltonian systems abstract a class of coupled hamiltonian systems is examined in which identical nonlinear oscillators are coupled through a mean field the system is shown to have a steady desynchronized solution which becomes linearly unstable as the coupling strength is increased we observe in the stable case that the order parameter of the system decays to zero for a wide class of initial conditions the decay is exponential onan intermediate time scale and then as t 3 2 as t this system shares many similarities to the vlasov poisson equation and as well as kuramoto s model

dynamics of neuronal waves first of all by studying the existence and stability of traveling wave fronts of the following nonlinear nonlocal model equation u t au x u alpha mathop int mathbb r k x y h u y t theta rm d y beta mathop int mathbb r k x y h u y t theta rm d y we derive relation between speed index function and stability index function for each of the waves this model was derived when studying working memory in synaptically coupled neuronal networks which incorporates low persistent activity rate and high saturating rate we will investigate dynamics of neuronal waves for this purpose we will be concerned with the equation for several different kinds of symmetric and asymmetric kernels and will compare speeds of the waves our analysis and results on the speed index functions facilitate our investigation on stability of the waves and the estimates of speeds secondly we are concerned with standing waves of the nonlinear nonhomogeneous system of integral differential equations begin array lll u t u w alpha mathop int limits mathbb r k x y h u y t theta rm d y beta mathop int limits mathbb r k x y h u y t theta rm d y mathcal i x t w t varepsilon u gamma w end array and the scalar equation begin array lll u t u alpha mathop int limits mathbb r k x y h u y t theta rm d y beta mathop int limits mathbb r k x y h u y t theta rm d y mathcal i x t end array

asymptotic stability of high dimensional zakharov kuznetsov solitons we prove that solitons or solitary waves of the zakharov kuznetsov zk equation a physically relevant high dimensional generalization of the korteweg de vries kdv equation appearing in plasma physics and having mixed kdv and nonlinear schrodinger nls dynamics are strongly asymptotically stable in the energy space we also prove that the sum of well arranged solitons is stable in the same space orbital stability of zk solitons is well known since the work of de bouard proc r soc edinburgh 126 89 112 1996 our proofs follow the ideas of martel siam j math anal 157 759 781 2006 and martel and merle math ann 341 391 427 2008 applied for generalized kdv equations in one dimension in particular we extend to the high dimensional case several monotonicity properties for suitable half portions of mass and energy we also prove a new liouville type property that characterizes zk solitons and a key virial identity for the linear and nonlinear part of the zk dynamics obtained independently of the mixed kdv nls dynamics this last virial identity relies on a simple sign condition which is numerically tested for the two and three dimensional cases with no additional spectral assumptions required possible extensions to higher dimensions and different nonlinearities could be obtained after a suitable local well posedness theory in the energy space and the verification of a corresponding sign condition

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

linear stability of solitary waves of the green naghdi equations we investigate the eigenvalue problem obtained from linearizing the green naghdi equations about solitary wave solutions unlike weakly nonlinear water wave models the physical system considered here has nonlinearity in its highest derivative term this results in more detailed asymptotic analysis of the eigenvalue problem in the presence of a large parameter combining the technique of singular perturbation with the evans function we show that for solitary waves of small amplitude the problem has no eigenvalues of positive real part and the evans function is nonvanishing everywhere except the origin this fact then leads to the linear stability of these solitary waves 2001 john wiley sons inc

finite wavelength stability of capillary gravity solitary waves we consider the euler equations describing nonlinear waves on the free surface of a two dimensional inviscid irrotational fluid layer of finite depth for large surface tension bond number larger than 1 3 and froude number close to 1 the system possesses a one parameter family of small amplitude traveling solitary wave solutions we show that these solitary waves are spectrally stable with respect to perturbations of finite wave number in particular we exclude possible unstable eigenvalues of the linearization at the soliton in the long wavelength regime corresponding to small frequency and unstable eigenvalues with finite but bounded frequency arising from non adiabatic interaction of the infinite wavelength soliton with finite wavelength perturbations

long time decay for 2d klein gordon equation abstract we obtain a dispersive long time decay in weighted energy norms for solutions of the 2d klein gordon equations the decay extends the results obtained by jensen kato and murata for the equations of schrodinger s type by the spectral approach for the proof we modify the approach to make it applicable to relativistic equations

error propagation in numerical approximations near relative equilibria we study the propagation of errors in the numerical integration of perturbations of relative equilibrium solutions of hamiltonian differential equations with symmetries first it is shown that taking an initial perturbation of a relative equilibrium the corresponding solution is related in a first approximation to another relative equilibrium with the parameters perturbed from the original then this is used to prove that for stable relative equilibria error growth with respect to the perturbed solution is in general quadratic but only linear for schemes that preserve the invariant quantities of the problem in this sense the conclusion is similar to the one obtained when integrating unperturbed relative equilibria numerical experiments illustrate the results

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

asymptotic stability of the toda m soliton we prove that multi soliton solutions of the toda lattice are both linearly and nonlinearly stable our proof uses neither the inverse spectral method nor the lax pair of the model but

asymptotic stability of solitons for subcritical generalized kdv equations we prove in this paper the asymptotic completeness of the family of solitons in the energy space for generalized korteweg de vries equations in the subcritical case this includes in particular the kdv equation and the modified kdv equation this result is obtained as a consequence of a rigidity theorem on the flow close to a soliton up to a scaling and a translation which has its own interest the proofs use some tools introduced in a previous paper to prove similar results in the case of critical generalized kdv equation

asymptotic stability of solitary waves we show that the family of solitary waves 1 solitons of the korteweg de vries equation partial t u u partial x u partial x 3 u 0 is asymptotically stable our methods also apply for the solitary waves of a class of generalized korteweg de vries equations partial t u partial x f u partial x 3 u 0 in particular we study the case wheref u up 1 p 1 p 1 2 3 and 3 0 withf c4 the same asymptotic stability result for kdv is also proved for the casep 2 the modified korteweg de vries equation we also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4 the solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep 4 the solution is decomposed into a modulating solitary wave with time varying speedc t and phase t bound state part and an infinite dimensional perturbation radiating part the perturbation is shown to decay exponentially in time in a local sense relative to a frame moving with the solitary wave asp 4 the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations

convergence of difference approximations of quasilinear evolution equations abstract the very successful theory of quasilinear evolution equations which applies to many problems of mathematical physics has been developed by t kato the theory obtains solutions of quasilinear problems via contraction mappings which are defined by means of a theory of linear evolution equations also developed by kato in the current work we show how the existence and continuous dependence theorems obtained by kato can be proved by discretization in time as opposed to earlier work in this direction the current results are much sharper concerning the continuity properties of the solutions of the discretized problem and the strength of the norms in which they converge

on the spectra of pulses in a nearly integrable system an eigenvalue problem for the pulse solutions to perturbed generalized korteweg de vries equations is studied evans function techniques with the aid of the geometric approach to singular perturbation problems are employed this approach leads to a clear understanding of the spectrum of the pulses to the perturbed equations

application of spectral deformation to the vlasov poisson system ii mathematical results this paper presents a mathematical description of the linearized vlasov poisson operator lk acting on a family of banach spaces xp related to l p r and the application of the method of spectral deformation to this model it is shown that a type a analytic family of operators lk c lk 0 lk can be associated with lk by means of this family the landau damped modes of the plasma are identified as the spectral resonances of lk existence and uniqueness of solutions to the initial value problem for the evolution equation g lk g is proven an expansion of any solution to the initial value problem with sufficiently smooth initial data is obtained in terms of the eigenfunctions of lk and a spectral integral over the essential spectrum this is applied to derive an expansion for solutions to the vlasov equation in which the landau damped portions of the distribution function are manifestly exhibited a self contained discussion of the spectral deformation method and an extension of it to ce

pulse dynamics in low reynolds number interfacial hydrodynamics experiments and theory abstract we analyze interaction of nonlinear pulses in active dispersive dissipative nonlinear media a particular example of such media is a viscous thin film coating a vertical fibre experiments for this system reveal that the interface evolves into a train of droplike solitary pulses in which numerous inelastic coalescence events take place in such events larger pulses catch up with smaller ones and annihilate them however for certain flow conditions and after a certain distance from the inlet no more coalescence is observed and the flow is described by quasi equilibrium solitary pulses interacting continuously with each other through attractions and repulsions and eventually they form bound states of groups of pulses in which the pulses travel with the same velocities as a whole this experimental study represents the first evidence of formation of bound states in low reynolds number interfacial hydrodynamics to gain theoretical insight into the interaction of the pulses and formation of bound states we derive a weakly nonlinear model for the flow the generalized kuramoto sivashinsky gks equation that retains the fundamental mechanisms of the wave evolution namely dominant nonlinearity instability stability and dispersion much like in the experiments the spatio temporal evolution of the gks equation is dominated by quasi stationary solitary pulses which continuously interact with each other through coalescence events or attractions repulsions to understand the latter case we utilize a weak interaction theory for the solitary pulses of the gks equation the theory is based on representing the solution of the equation as a superposition of the pulses and an overlap function and leads to a coupled system of ordinary differential equations describing the evolution of the locations of the pulses or alternatively the evolution of the separation distances by analyzing the fixed points of this system we obtain bound states of interacting pulses for two pulses we provide a criterion for the existence of a countable infinite or finite number of bound states depending on the strength of the dispersive term in the equation the interaction theory and resulting bound states are corroborated by computations of the full equation we also find qualitative agreement between the theory and the experiments

conditional stability of unstable viscous shocks abstract continuing a line of investigation initiated by texier and zumbrun on dynamics of viscous shock and detonation waves we show that a linearly unstable lax type viscous shock solution of a semilinear strictly parabolic system of conservation laws possesses a translation invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small l 1 h 2 perturbations converging time asymptotically to a translate of the unperturbed wave that is for a shock with p unstable eigenvalues we establish conditional stability on a codimension p manifold of initial data with sharp rates of decay in all l p for p 0 we recover the result of unconditional stability obtained by howard mascia and zumbrun

local stability theory of solitary pulses in an active medium abstract if the spatio temporal chaos of an extended active medium is driven by repulsive or attractive interaction among localized coherent structures such as kinks and pulses its dynamics can often be described by a low dimensional dynamical system this welcomed reduction in the degrees of freedom only occurs when the localized structures remain stable during their mutual interaction it is shown that the structures can be destroyed by two types of local disturbance an expanding radiation packet and a localized disturbance which travels with the structure these two instabilities and the bifurcations they induce select the proper coherent structure and specify the dynamics of the generalized kuramoto sivashinsky gks equation in an extended domain the possibility of describing pulse radiation interaction by resonance poles also suggests that a more general coherent structure theory can be developed

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

asymptotic stability of solitons for mkdv abstract we prove a full asymptotic stability result for solitary wave solutions of the mkdv equation we consider small perturbations of solitary waves with polynomial decay at infinity and prove that solutions of the cauchy problem evolving from such data tend uniformly on the real line to another solitary wave as time goes to infinity we describe precisely the asymptotics of the perturbation behind the solitary wave showing that it satisfies a nonlinearly modified scattering behavior this latter part of our result relies on a precise study of the asymptotic behavior of small solutions of the mkdv equation

on asymptotic stability of moving kink for relativistic ginzburg landau equation we prove the asymptotic stability of kink for the nonlinear relativistic wave equations of the ginzburg landau type in one space dimension for any odd initial condition in a small neighborhood of the kink the solution asymptotically in time is the sum of the kink and dispersive part described by the free klein gordon equation the remainder converges to zero in a global norm

asymptotic stability of solitons of the gkdv equations with general nonlinearity we consider the generalized korteweg de vries equation gkdv partial t u partial x partial x 2 u f u 0 quad t x in 0 t times mathbb r with general c3 nonlinearity f under an explicit condition on f and c 0 there exists a solution in the energy space h1 of the type u t x qc x x0 ct called soliton in this paper under general assumptions on f and qc we prove that the family of solitons around qc is asymptotically stable in some local sense in h1 i e if u t is close to qc for all t 0 then u t locally converges in the energy space to some qc as t note in particular that we do not assume the stability of qc this result is based on a rigidity property of the gkdv equation around qc in the energy space whose proof relies on the introduction of a dual problem these results extend the main results in martel siam j math anal 38 759 781 2006 martel and merle j math pures appl 79 339 425 2000 arch ration mech anal 157 219 254 2001 nonlinearity 1 55 80 devoted to the pure power case

stability of solitary waves in nonlinear composite media the system of equations for planar waves in elastic composite media in the presence of anisotropy is considered in anisotropic case two two parametric families of solitary waves are found in an explicit form in case of the absence of anisotropy these two families coalesce into the unique three parametric family the solitary wave solutions are found to be orbitally stable in a certain range of their phase speeds range of stability both in an anisotropic as well as in an isotropic materials it is also shown that the initial value problem for the governing equations is locally well posed which is needed to prove the stability result the local well posedness of the initial value problem along with stability of solitary waves implies global existence result provided the initial data lie in a neighbourhood of a stable solitary wave this complements the previous results of blow up for this type of equations

existence uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks abstract the author is concerned with the existence and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks previous methods do not apply in solving these problems because there is no maximum principle or conservation laws available to the integral differential equations he applies fixed point theorems to prove the existence of the traveling waves then he makes use of linearization technique as well as eigenvalue functions to study the exponential stability of the waves

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

invariant manifolds for parabolic partial differential equations on unbounded domains in this paper finite dimensional invariant manifolds for nonlinear parabolic partial differential equations of the form

stability of post fertilization traveling waves article history this paper studies the stability of a family of traveling wave solutions to the system proposed by lane et al d c lane j d murray v s manoranjan analysis of wave phenomena in a morphogenetic mechanochemical model and an application to post fertilization waves on eggs ima j math appl med biol 4 4 1987 309 331 to model a pair of mechanochemical phenomena known as post fertilization waves on eggs the waves consist of an elastic deformation pulse on the egg s surface and a free calcium concentration front the family is indexed by a coupling parameter measuring contraction stress effects on the calcium concentration this work establishes the spectral linear and nonlinear orbital stability of these post fertilization waves for small values of the coupling parameter the usual methods for the spectral and evolution equations cannot be applied because of the presence of mixed partial derivatives in the elastic equation nonetheless exponential decay of the directly constructed semigroup on the complement of the zero eigenspace is established we show that small perturbations of the waves yield solutions to the nonlinear equations decaying exponentially to a phase modulated traveling

metastability of solitary roll wave solutions of the st venant equations with viscosity we study by a combination of numerical and analytical evans function techniques the stability of solitary wave solutions of the st venant equations for viscous shallowwater ow down an incline and related models our main result is to exhibit examples of metastable solitary waves for the st venant equations with stable point spectrum indicating coherence of the wave prole but unstable essential spectrum indicating oscillatory convective instabilities shed in its wake we propose a mechanism based on dynamic spectrum of the wave prole by which a wave train of solitary pulses can stabilize each other by de amplication of convective instabilities as they pass through successive waves we present numerical time evolution studies supporting these conclusions which bear also on the possibility of stable periodic solutions close to the homoclinic for the closely related viscous jin xin model by contrast for which the essential spectrum is stable we show using the stability index of gardner zumbrun that solitary wave pulses are always exponentially unstable possessing point spectra with positive real part

absolute and convective instabilities of waves on unbounded and large bounded domains abstract instabilities of nonlinear waves on unbounded domains manifest themselves in different ways an absolute instability occurs if the amplitude of localized wave packets grows in time at each fixed point in the domain in contrast convective instabilities are characterized by the fact that even though the overall norm of wave packets grows in time perturbations decay locally at each given point in the unbounded domain wave packets are convected towards infinity in experiments as well as in numerical simulations bounded domains are often more relevant we are interested in the effects that the truncation of the unbounded to a large but bounded domain has on the aforementioned in stability properties of a wave these effects depend upon the boundary conditions that are imposed on the bounded domain we compare the spectra of the linearized evolution operators on unbounded and bounded domains for two classes of boundary conditions it is proved that periodic boundary conditions reproduce the point and essential spectrum on the unbounded domain accurately spectra for separated boundary conditions behave in quite a different way firstly separated boundary conditions may generate additional isolated eigenvalues secondly the essential spectrum on the unbounded domain is in general not approximated by the spectrum on the bounded domain instead the so called absolute spectrum is approximated that corresponds to the essential spectrum on the unbounded domain calculated with certain optimally chosen exponential weights we interpret the difference between the absolute and the essential spectrum in terms of the convective behavior of the wave on the unbounded domain in particular it is demonstrated that the stability of the absolute spectrum implies convective instability of the wave but that convectively unstable waves can destabilize under domain truncation the theoretical predictions are compared with numerical computations

weighted energy decay for 1d wave equation we obtain a dispersive long time decay in weighted energy norms for solutions to the 1d wave equation with generic potential the decay extends the results obtained by murata for the 1d schrodinger equation

instabilities in threshold diffusion equations with delay the introduction of delays into ordinary or partial differential equation models is well known to facilitate the production of rich dynamics ranging from periodic solutions through to spatio temporal chaos in this paper we consider a class of scalar partial differential equations with a delayed threshold nonlinearity which admits exact solutions for equilibria periodic orbits and travelling waves importantly we show how the spectra of periodic and travelling wave solutions can be determined in terms of the zeros of a complex analytic function using this as a computational tool to determine stability we show that delays can have very different effects on threshold systems with negative as opposed to positive feedback direct numerical simulations are used to confirm our bifurcation analysis and to probe some of the rich behaviour possible for mixed feedback

on eigenfunction expansion of solutions to the hamilton equations we establish a spectral representation for solutions to linear hamilton equations with positive definite energy in a hilbert space our approach is a special version of m krein s spectral theory of j selfadjoint operators in the hilbert spaces with indefinite metric our main result is an application to the eigenfunction expansion for the linearized relativistic ginzburg landau equation

dynamics near the solitary waves of the supercritical gkdv equations abstract this work is devoted to study the dynamics of the supercritical gkdv equations near solitary waves in the energy space h 1 we construct smooth local center stable center unstable and center manifolds near the manifold of solitary waves and give a detailed description of the local dynamics near solitary waves in particular the instability is characterized as follows any forward flow not starting from the center stable manifold will leave a neighborhood of the manifold of solitary waves exponentially fast moreover orbital stability is proved on the center manifold which implies the uniqueness of the center manifold and the solutions on it exist globally and asymptotically approach the solitary waves

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

a nonlinear nonlocal wave equation arising in combustion theory the initial value problem for the equation 2 t2 2 x2 t t 2 t 2 x2 et 1 is considered it is proved that under some restrictions on the initial data there is a curve denoted by t x which is positive lipschitz continuous and satisfies x 0 the case of 1 is also studied the solution for 1 blows up on t x and it is proved that under certain conditions the solutions for 1 converge to the one for 1 as 1 and lim inf 1 x x

one parameter localized traveling waves in nonlinear schrodinger lattices we address the existence of traveling single humped localized solutions in the spatially discrete nonlinear schr nls equation a mathematical technique is developed for analysis of persistence of these solutions from a certain limit in which the dispersion relation of linear waves contains a triple zero the technique is based on using the implicit function theorem for solution of an appropriate differential advance delay equation in exponentially weighted spaces the resulting melnikov calculation relies on a number of assumptions on the spectrum of the linearization around the pulse which are checked numerically we apply the technique to the so called salerno model and the translationally invariant discrete nls equation with a cubic nonlinearity we show that the traveling solutions terminate in the salerno model whereas they generally persist in the translationally invariant nls lattice as a one parameter family of solutions these results are found to be in a close correspondence with numerical approximations of traveling solutions with zero radiation tails analysis of persistence also predicts the spectral stability of the one parameter family of traveling solutions under time evolution of the discrete nls equation c 2007 elsevier b v all rights reserved

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

evans functions jost functions and fredholm determinants the principal results of this paper consist of an intrinsic definition of the evans function in terms of newly introduced generalized matrix valued jost solutions for general first order matrix valued differential equations on the real line and a proof of the fact that the evans function a finite dimensional determinant by construction coincides with a modified fredholm determinant associated with a birman schwinger type integral operator up to an explicitly computable nonvanishing factor

instability of pole singularities for the chazy equation we prove that the negative resonances of the chazy equation in the sense of painleve analysis can be related directly to its group invariance properties these resonances indicate in this case the instability of pole singularities depending on the value of a parameter in the equation an unstable isolated pole may turn into the familiar natural boundary or split into several isolated singularities in the first case a convergent series representation involving exponentially small corrections can be given this reconciles several earlier approaches to the interpretation of negative resonances on the other hand we also prove that pole singularities with the maximum number of positive resonances are stable the proofs rely on general properties of nonlinear fuchsian equations

asymptotics of step like solutions for the camassa holm equation abstract we study the long time asymptotics of solution of the cauchy problem for the camassa holm equation with a step like initial datum by using the nonlinear steepest descent method and the so called g function approach we show that the camassa holm equation exhibits a rich structure of sharply separated regions in the x t half plane with qualitatively different asymptotics which can be described in terms of a modulated finite gap elliptic function and a finite number of solitons

linear asymptotic stability and modulation behavior near periodic waves of the korteweg de vries equation abstract we provide a detailed study of the dynamics obtained by linearizing the korteweg de vries equation about one of its periodic traveling waves a cnoidal wave in a suitable sense linearly analogous to space modulated stability 15 we prove global in time bounded stability in any sobolev space and asymptotic stability of dispersive type furthermore we provide both a leading order description of the dynamics in terms of slow modulation of local parameters and asymptotic modulation systems and effective initial data for the evolution of those parameters this requires a global in time study of the dynamics generated by a non normal operator with non constant coefficients on the road we also prove estimates on oscillatory integrals particularly suitable to derive large time asymptotic systems that could be of some general interest

exponential stability of large amplitude traveling fronts for quasi linear relaxation systems with diffusion a b s t r a c t this paper is concerned with the stability of traveling front solutions for 2 2 quasi linear relaxation systems with small diffusion rate by applying geometric singular perturbation method special evans function estimates detailed spectral analysis and c0 semigroup theories we prove that all the non degenerate waves for semi linear relaxation systems are locally exponentially stable in some exponentially weighted spaces we also obtain the linear exponential stability of the non degenerate waves for quasi linear relaxation systems where the wave strengths can be large

asymptotic stability of lattice solitons in the energy space orbital and asymptotic stability for 1 soliton solutions of the toda lattice equations as well as for small solitary waves of the fpu lattice equations are established in the energy space unlike analogous hamiltonian pdes the lattice equations do not conserve the adjoint momentum in fact the toda lattice equation is a bidirectional model that does not fit in with the existing theory for the hamiltonian systems by grillakis shatah and strauss to prove stability of 1 soliton solutions we split a solution around a 1 soliton into a small solution that moves more slowly than the main solitary wave and an exponentially localized part we apply a decay estimate for solutions to a linearized toda equation which has been recently proved by mizumachi and pego to estimate the localized part we improve the asymptotic stability results for fpu lattices in a weighted space obtained by friesecke and pego

asymptotic n soliton like solutions of the subcritical and critical generalized korteweg de vries equations we consider the generalized korteweg de vries equations ut uxx upx 0t x r inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i in the subcritical and critical cases p 2 3 4 or 5 let r j t x qc j x c j t x j where j 1 n be n soliton solutions of this equation with corresponding speeds 0 c 1 c 2 c n in this paper we construct a solution u t of the generalized korteweg de vries equation such that inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i this solution behaves asymptotically as t as the sum of n solitons without loss of mass by dispersion this is an exceptional behavior indeed being given the parameters c j 1 j n x j 1 j n we prove uniqueness of such a solution in the integrable cases p 2 and 3 such solutions are explicitly known and their properties were extensively studied in the literature they are called n soliton solutions therefore the existence result is new only for the nonintegrable cases the uniqueness result is new for all cases

weighted energy decay for 3d klein gordon equation abstract we obtain a dispersive long time decay in weighted energy norms for solutions of the 3d klein gordon equation with generic potential the decay extends the results obtained by jensen and kato for the 3d schrodinger equation for the proof we modify the spectral approach of jensen and kato to make it applicable to relativistic equations

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

on scattering of solitons for the klein gordon equation coupled to a particle we establish the long time soliton asymptotics for the translation invariant nonlinear system consisting of the klein gordon equation coupled to a charged relativistic particle the coupled system has a six dimensional invariant manifold of the soliton solutions we show that in the large time approximation any finite energy solution with the initial state close to the solitary manifold is a sum of a soliton and a dispersive wave which is a solution of the free klein gordon equation it is assumed that the charge density satisfies the wiener condition which is a version of the fermi golden rule the proof is based on an extension of the general strategy introduced by soffer and weinstein buslaev and perelman and others symplectic projection in hilbert space onto the solitary manifold modulation equations for the parameters of the projection and decay of the transversal component

spectral properties and time decay for an airy operator with potential abstract the linearization of the korteweg de vries equation about a solitary wave is studied with respect to algebraic spatial weights the decay rate of a solution to the linearized equation is shown to depend on the degree of its initial spatial localization the operator has the entire imaginary axis as its spectrum so that special techniques are required to invert the laplace transform and thereby obtain decay estimates

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

galloping instability of viscous shock waves abstract motivated by physical and numerical observations of time oscillatory galloping spinning and cellular instabilities of detonation waves we study poincare hopf bifurcation of traveling wave solutions of viscous conservation laws the main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum preventing standard reduction to a finite dimensional center manifold we overcome this by direct lyapunov schmidt reduction using detailed pointwise bounds on the linearized solution operator to carry out a nonstandard implicit function construction in the absence of a spectral gap the key computation is a space time stability estimate on the transverse linearized solution operator reminiscent of duhamel estimates carried out on the full solution operator in the study of nonlinear stability of spectrally stable traveling waves

diffusive stability of spatial periodic solutions of the swift hohenberg equation we are interested in the nonlinear stability of the eckhaus stable equilibria of the swift hohenberg equation on the infinite line with respect to small integrable perturbations the difficulty in showing stability for these stationary solutions comes from the fact that their linearizations possess continuous spectrum up to zero the nonlinear stability problem is treated with renormalization theory which allows us to show that the nonlinear terms are irrelevant i e that the fully nonlinear problem behaves asymptotically as the linearized one which is under a diffusive regime

stability of gasless combustion fronts in one dimensional solids for gasless combustion in a one dimensional solid we show a type of nonlinear stability of the physical combustion front if a perturbation of the front is small in both a spatially uniform norm and an exponentially weighted norm then the perturbation stays small in the spatially uniform norm and decays in the exponentially weighted norm provided the linearized operator has no eigenvalues in the right half plane other than zero using the evans function we show that the zero eigenvalue must be simple factors that complicate the analysis are 1 the linearized operator is not sectorial and 2 the linearized operator has good spectral properties only when the weighted norm is used but then the nonlinear term is not lipschitz the result is nevertheless physically natural to prove it we first show that when the weighted norm is used the semigroup generated by the linearized operator decays on a subspace complementary to the operator s kernel by showing that it is a compact perturbation of the semigroup generated by a more easily analyzed triangular operator we then use this result to help establish that solutions stay small in the spatially uniform norm which in turn helps establish nonlinear convergence in the weighted norm

an evans function approach to spectral stability of internal solitary waves in stratified fluids frequently encountered in nature internal solitary waves in strat ified fluids are well observed and well studied from the experimental the the oretical and the numerical perspective from the mathematical point of view these waves are exact solutions of the 2d euler equations for incompressible inviscid fluids contrasting with a rich theory for their existence and the de velopment of methods for computing these waves their stability analysis has hardly received attention at a rigorous mathematical level this paper proposes a new approach to the investigation of stability of in ternal solitary waves in a continuously stratified fluidic medium and carries out the following four steps of this approach i to formulate the eigenvalue problem as an infinite dimensional spatial dynamical system ii to intro duce finite dimensional truncations of the spatial dynamics description iii to demonstrate that each truncation of any order permits a well defined evans function iv to prove absence of small zeros of the evans function in the small amplitude limit the latter notably implies the low frequency spectral stability of small amplitude internal solitary waves to arbitrarily high trunca tion order

on the stability of solitary water waves with a point vortex this paper investigates the stability of traveling wave solutions to the free boundary euler equations with a submerged point vortex we prove that sufficiently small amplitude waves with small enough vortex strength are conditionally orbitally stable in the process of obtaining this result we develop a quite general stability instability theory for bound state solutions of a large class of infinite dimensional hamiltonian systems in the presence of symmetry this is in the spirit of the seminal work of grillakis shatah and strauss gss but with hypotheses that are relaxed in a number of ways necessary for the point vortex system and for other hydrodynamical applications more broadly in particular we are able to allow the poisson map to have merely dense range as opposed to being surjective and to be state dependent as a second application of the general theory we consider a family of nonlinear dispersive pdes that includes the generalized korteweg de vries kdv and benjamin ono equations the stability or instability of solitary waves for these systems has been studied extensively notably by bona souganidis and strauss who used a modification of the gss method we provide a new more direct proof of these results as a straightforward consequence of our abstract theory at the same time we allow fractional dispersion and obtain a new instability result for fractional kdv less

instability of nonlinear dispersive solitary waves we consider linear instability of solitary waves of several classes of dispersive long wave models they include generalizations of kdv bbm regularized boussinesq equations with general dispersive operators and nonlinear terms we obtain criteria for the existence of exponentially growing solutions to the linearized problem the novelty is that we dealt with models with nonlocal dispersive terms for which the spectra problem is out of reach by the evans function technique for the proof we reduce the linearized problem to study a family of nonlocal operators which are closely related to properties of solitary waves a continuation argument with a moving kernel formula is used to find the instability criteria these techniques have also been extended to study instability of periodic waves and of the full water wave problem

asymptotic stability of n solitary waves of the fpu lattices we study stability of n solitary wave solutions of the fermi pasta ulam fpu lattice equation solitary wave solutions of the fpu lattice equation cannot be characterized as critical points of conservation laws due to the lack of infinitesimal invariance in the spatial variable in place of standard variational arguments for hamiltonian systems we use an exponential stability property of the linearized fpu equation in a weighted space which is biased in the direction of motion the dispersion of the linearized fpu equation balances the potential term for low frequencies whereas the dispersion is superior for high frequencies we approximate the low frequency part of a solution of the linearized fpu equation by a solution to the linearized korteweg de vries kdv equation around an n soliton solution we prove an exponential stability property of the linearized kdv equation around n solitons by using the linearized backlund transformation and use the result to analyze the linearized fpu equation

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

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

large time behavior and asymptotic stability of the 2d euler and linearized euler equations abstract we study the asymptotic behavior and the asymptotic stability of the 2d euler equations and of the 2d linearized euler equations close to parallel flows we focus on flows with spectrally stable profiles u y and with stationary streamlines y y 0 such that u y 0 0 a case that has not been studied previously we describe a new dynamical phenomenon the depletion of the vorticity at the stationary streamlines an unexpected consequence is that the velocity decays for large times with power laws similarly to what happens in the case of the orr mechanism for base flows without stationary streamlines the asymptotic behaviors of velocity and the asymptotic profiles of vorticity are theoretically predicted and compared with direct numerical simulations we argue on the asymptotic stability of this ensemble of flow profiles even in the absence of any dissipative mechanisms

accuracy and conservation properties in numerical integration the case of the korteweg de vries equation when numerically integrating time dependent differential equations it is often recommended to employ methods that preserve some of the invariant quantities mass energy etc of the problem being considered this recommendation is usually justified on the grounds that conservation of invariant quantities may ensure that the numerical solution possesses some important qualitative features however there are cases where schemes that preserve invariants are also advantageous in that they possess favourable error propagation mechanisms that render them superior from a quantitative point of view in the present paper we consider the korteweg de vries equation as a case study we show rigorously that for soliton problems and at leading order the error of conservative schemes consists of a phase error that grows linearly with time plus a complementary term that is bounded in the h 1 norm uniformly in time for general nonconservative schemes the error involves a linearly growing amplitude error a quadratically growing phase error and a complementary term that grows linearly in the h 1 norm numerical experiments are presented

asymptotic stability of solitary wave solutions to the regularized long wave equation abstract we study the asymptotic stability of solitary wave solutions to the regularized long wave equation rlw in h 1 r rlw is an equation which describes the long waves in water to prove the result we make use of the monotonicity of the local h1 norm and apply the liouville property of rlw as in merle and martel j math pures appl 79 2000 339 arch rational mech anal 157 2001 219

an index theorem for the stability of periodic travelling waves of korteweg de vries type we consider the stability of periodic travelling wave solutions to a generalized korteweg de vries gkdv equation and prove an index theorem relating the number of unstable and potentially unstable eigenvalues to geometric information on the classical mechanics of the travelling wave ordinary differential equation we illustrate this result with several examples including the integrable kdv and modified kdv equations the l 2 critical kdv 4 equation that arises in the study of blow up and the kdv equation which is an idealized model for plasmas

inelastic interaction of nearly equal solitons for the quartic gkdv equation this paper describes the interaction of two solitons with nearly equal speeds for the quartic gkdv equation partial tu partial x partial x 2u u 4 0 quad t x in mathbb r 0 1 we call soliton a solution of 0 1 of the form u t x qc x ct y0 where c 0 y0 and q c q c 4 cq c since 0 1 is not an integrable model the general question of the collision of two given solitons q c 1 x c 1 t q c 2 x c 2 t with c1 c2 is an open problem