stability analysis for pitchfork bifurcations of solitary waves in generalized nonlinear schrodinger equations abstract linear stability of both sign definite positive and sign indefinite solitary waves near pitchfork bifurcations is analyzed for the generalized nonlinear schrodinger equations with arbitrary forms of nonlinearity and external potentials in arbitrary spatial dimensions bifurcations of linear stability eigenvalues associated with pitchfork bifurcations are analytically calculated based on these eigenvalue bifurcation formulae linear stability of solitary waves near pitchfork bifurcations is then determined it is shown that the base solution branch switches stability at the bifurcation point in addition the two bifurcated solution branches and the base branch have the opposite same stability when their power slopes have the same opposite sign furthermore the stability of these solution branches can be determined almost exclusively from their power diagram especially for positive solitary waves these stability results are also compared with the hamiltonian krein index theory and they are shown to be consistent with each other lastly various numerical examples are presented and the numerical results confirm the analytical predictions both qualitatively and quantitatively

analysis of the linearization around a critical point of an infinite dimensional hamiltonian system partial differential equations that conserve energy can often be written as infinite dimensional hamiltonian systems of the following general form du dt je u where j x x is a symplectic matrix i e jj 1 and e x r is a c 2 functional defined on some hilbert space x

count of eigenvalues in the generalized eigenvalue problem we study isolated and embedded eigenvalues in the generalized eigenvalue problem defined by two self adjoint operators with a positive essential spectrum and a finite number of isolated eigenvalues the generalized eigenvalue problem determines the spectral stability of nonlinear waves in infinite dimensional hamiltonian systems the theory is based on pontryagin s invariant subspace theorem and extends beyond the scope of earlier papers of pontryagin krein grillakis and others our main results are i the number of unstable and potentially unstable eigenvalues equals the number of negative eigenvalues of the self adjoint operators ii the total number of isolated eigenvalues of the generalized eigenvalue problem is bounded from above by the total number of isolated eigenvalues of the self adjoint operators and iii the quadratic forms defined by the two self adjoint operators are strictly positive on the subspace related to the continuous spectrum of the generalized eigenvalue problem applicatio

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

uniqueness of positive radial solutions of δu f u 0 in ℝ n ii we prove a uniqueness result for the positive solution of u f u 0 in r n which goes to 0 at the result applies to a wide class of nonlinear functions f including the important model case f u u u p 1 p n 2 n 2 the result is proved by reducing to an initial boundary problem for the ode u n 1 r f u 0 and using a shooting method

linear stability of perturbed hamiltonian systems theory and a case example in this paper we present a general theory based on a lyapunov schmidt reduction for the linearized stability of a perturbed hamiltonian system with a number of symmetries the reduction leads to an eigenvalue problem in a small subspace of the original system whose eigenvalues and eigenvectors can be used to derive the ones of the perturbed original system the krein signature of the eigenvalues can also be obtained in this framework the method is applied to the case example of the coupled nonlinear schr dinger equations giving excellent agreement with the full numerical linear stability results

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

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

high dimensional random apollonian networks abstract we propose a simple algorithm which produces a new category of networks high dimensional random apollonian networks with small world and scale free characteristics we derive analytical expressions for their degree distributions and clustering coefficients which are determined by the dimension of the network the values obtained for these parameters are in good agreement with simulation results and comparable to those coming from real networks we estimate also analytically that the average path length of the networks increases at most logarithmically with the number of vertices

method of moments analysis of resonant circular arrays of cylindrical dipoles we perform a method of moments mom analysis of a circular array of cylindrical dipoles the array is known from earlier theoretical and experimental studies to possess very narrow resonances the earlier theoretical studies were carried out using the two term theory the present paper is a direct continuation of a recent work showing that the problem possesses unique and particular difficulties the main difficulties are overcome herein using a set of improved kernels in the usual hallen type integral equations these kernels had been developed in previous works and were successfully incorporated into the aforementioned two term theory analyses we make a detailed comparison of our mom results to two term theory results and also to the earlier experimental results

counting eigenvalues via the krein signature in infinite dimensional hamiltonian systems spectra of nonlinear waves in infinite dimensional hamiltonian systems are investigated we establish a connection via the krein signature between the number of negative directions of the second variation of the energy and the number of potentially unstable eigenvalues of the linearization about a nonlinear wave we apply our results to determine the effect of symmetry breaking on the spectral stability of nonlinear waves in weakly coupled nonlinear schrodinger equations

purely nonlinear instability of standing waves with minimal energy we consider hamiltonian systems with u 1 symmetry we prove that in the generic situation the standing wave that has the minimal energy among all other standing waves is unstable in spite of the absence of linear instability essentially the instability is caused by higher algebraic degeneracy of the zero eigenvalue in the spectrum of the linearized system we apply our theory to the non

persistence and stability of discrete vortices in nonlinear schrodinger lattices abstract we study discrete vortices in the two dimensional nonlinear schrodinger lattice with small coupling between lattice nodes the discrete vortices in the anti continuum limit of zero coupling represent a finite set of excited nodes on a closed discrete contour with a non zero charge using the lyapunov schmidt reductions we analyze continuation and termination of the discrete vortices for small coupling between lattice nodes an example of a square discrete contour is considered that includes the vortex cell also known as the off site vortex we classify families of symmetric and asymmetric discrete vortices that bifurcate from the anti continuum limit we predict analytically and confirm numerically the number of unstable eigenvalues associated with each family of such discrete vortices

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

stability of discrete solitons in nonlinear schrodinger lattices we consider the discrete solitons bifurcating from the anti continuum limit of the discrete nonlinear schrodinger nls lattice the discrete soliton in the anti continuum limit represents an arbitrary finite superposition of in phase or anti phase excited nodes separated by an arbitrary sequence of empty nodes by using stability analysis we prove that the discrete solitons are all unstable near the anti continuum limit except for the solitons which consist of alternating anti phase excited nodes we classify analytically and confirm numerically the number of unstable eigenvalues associated with each family of the discrete solitons

letter to the editor spurious localized highest frequency modes in schrodinger type equations solved by finite difference methods high frequency solutions of one or several schrodinger type equations are well known to differ very little from the plane wave solutions exp ikx that is the potential terms impact the envelope of a high frequency plane wave to only a small extent however when such equations are solved by a finite difference method the highest frequency solutions may under certain conditions turn out to be localized this may puzzle the researcher and suggest that the code may have an error however this is not an error but a numerical artifact and in this note we explain it

instability of a class of dispersive solitary waves this paper studies the stability and instability properties of solitary wave solutions x ct of a general class of evolution equations of the form mu t f u x 0 which support weakly nonlinear dispersive waves it turns out that depending on their speed c and the relation between the dispersion i e the order of the pseudodifferential operator and the nonlinearity travelling waves may be stable or unstable sharp conditions to that effect are given

instability indices for matrix polynomials abstract there is a well established instability index theory for linear and quadratic matrix polynomials for which the coefficient matrices are hermitian and skew hermitian this theory relates the number of negative directions for the matrix coefficients which are hermitian to the total number of unstable eigenvalues for the polynomial herein we extend the theory to even matrix polynomials of any finite degree in particular unlike previously known cases we show that the instability index depends upon the size of the matrices when the degree of the polynomial is greater than two we also consider hermitian matrix polynomials and derive an index which counts the number of eigenvalues with nonpositive imaginary part the results are refined if we consider the hermitian matrix polynomial to be a perturbation of a even polynomials however this refinement requires additional assumptions on the matrix coefficients

stable manifolds for all monic supercritical focusing nonlinear schrodinger equations in one dimension keywords critical schrodinger equation stable manifolds modulation theory spectral theory klein gordon equations non linear schrodinger blow up rate standing waves nonintegrable equations invariant manifolds stability theory solitary waves ground states instability reference epfl article 155183doi 10 1090 s0894 0347 06 00524 8 record created on 2010 11 19 modified on 2017 05 12

existence of perturbed solitary wave solutions to a model equation for water waves abstract we prove the existence of travelling wave solutions to a fifth order partial differential equation which is a formal asymptotic approximation for water waves with surface tension these travelling waves are arbitrarily small perturbations of solitary waves but are not solitary waves themselves because they approach small amplitude oscillations for large values of the independent variable this result suggests that for bond numbers less than one third there are branches of travelling wave solutions to the water wave equations which are perturbations of supercritical elevation solitary waves and which bifurcate from froude number one and bond number one third

dispersive analysis of charge transfer models we prove dispersive estimates for the time dependent schrodinger equation with a charge transfer hamiltonian as a by product we also obtain another proof of asymptotic completeness of the wave operators for a charge transfer model established earlier by k yajima and j m graf we also consider a more general matrix non self adjoint charge transfer problem this model appears naturally in the study of nonlinear multisoliton systems and is specifically motivated by the problem of asymptotic stability of multisoliton states of a nonlinear schrodinger equation 2004 wiley periodicals inc

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

slow damping of internal waves in a stably stratified fluid we study the damping of internal gravity waves in a stably stratified fluid with constant viscosity in two and three dimensional bounded domains for the linearized navier stokes equations for incompressible flow with no slip boundary conditions that model this fluid we prove there are non oscillatory normal modes with arbitrarily small exponential decay rates the proof is very different from that for a horizontally periodic layer and depends on a structure theorem for compact operators which are self adjoint with respect to an indefinite scalar product in a hilbert space we give a complete proof of this theorem which is closely related to results of pontrjagin

inertia law for spectral stability of solitary waves in coupled nonlinear schrodinger equations spectral stability analysis for solitary waves is developed in the context of the hamiltonian system of coupled nonlinear schrodinger equations the linear eigen value problem for a non self adjoint operator is studied with two self adjoint matrix schrodinger operators sharp bounds on the number and type of unstable eigen values in the spectral problem are found from the inertia law for quadratic forms associated with the two self adjoint operators symmetry breaking stability analysis is also developed with the same method

linearized instability for nonlinear schrodinger and klein gordon equations in this paper i am introducing a new technique for proving instability of bound states for hamiltonian systems there are already two disparate types of instability results in the literature the approach developed by strauss shatah 20 gave an instability criterion coming from the variational structure of the problem on the other hand jones approach 11 produced a complementary criterion related to the difference between the number of negative eigenvalues of two selfadjoint operators using quite different techniques it turns out that with the methods developed in this paper these two criteria can be derived within a single framework that also leads to a generalization of the previous results finally in order to demonstrate how this method works i apply the instability criterion in some specific examples