Stability of periodic waves in Hamiltonian PDEs of either long wavelength or small amplitude

Abstract

Stability criteria have been derived and investigated in the last decades for many kinds of periodic traveling wave solutions to Hamiltonian PDEs. They turned out to depend in a crucial way on the negative signature -or Morse index -of the Hessian matrix of action integrals associated with those waves. In a previous paper (published in Nonlinearity in 2016), the authors addressed the characterization of stability of periodic waves for a rather large class of Hamiltonian partial differential equations that includes quasilinear generalizations of the Korteweg-de Vries equation and dispersive perturbations of the Euler equations for compressible fluids, either in Lagrangian or in Eulerian coordinates. They derived a sufficient condition for orbital stability with respect to co-periodic perturbations, and a necessary condition for spectral stability, both in terms of the negative signature of the Hessian matrix of the action integral. Here the asymptotic behavior of this matrix is investigated in two asymptotic regimes, namely for small amplitude waves and for waves approaching a solitary wave as their wavelength goes to infinity. The special structure of the matrices involved in the expansions makes possible to actually compute the negative signature of the action Hessian both in the harmonic limit and in the soliton limit. As a consequence, it is found that nondegenerate small amplitude waves are orbitally stable with respect to co-periodic perturbations in this framework. For waves of long wavelength, the negative signature of the action Hessian is found to be exactly governed by M (c), the second derivative with respect to the wave speed of the Boussinesq momentum associated with the limiting solitary wave. This gives an alternate proof of Gardner's result [J. Reine Angew. Math. 1997] according to which the instability of the limiting solitary wave, when M (c) is negative, implies the instability of nearby periodic waves. Interestingly enough, it is found here that in the inverse situation, when M (c) is positive, nearby periodic waves are orbitally stable with respect to co-periodic perturbations.

Modulated equations of Hamiltonian PDEs and dispersive shocks

“…The main upshots are that • Near the harmonic limit, we derive explicit conditions determining modulational instability (see theorem 7 and appendix A), those being known in some cases as the Benjamin-Feir criteria. • Near the soliton limit, we prove that modulational instability is determined by exactly the same condition ruling stability of solitary waves and, as proved in [3], co-periodic stability of nearby periodic waves, that is, it is decided by the sign of the second derivative-with respect to speed, fixing the endstate-of the Boussinesq moment of instability.…”

“…For recent accounts of this variational derivation the reader is referred to [Kam00,Bri17]. As for the class of systems considered here, the corresponding form in terms of an action integral along periodic wave pro les was explicited in [BGNR13] and subsequently crucially used in [BGMR16,BGMRar].…”

“…• linear part essentially 3 given by the linearisation about U of the original system, in the frame moving with speed c, acting on functions of ξ with the same period as U; • source terms depending only on U 0 , φ 0 and φ 1 .…”

Modulated equations of Hamiltonian PDEs and dispersive shocks

“…For recent accounts of this variational derivation the reader is referred to [Kam00,Bri17]. As for the class of systems considered here, the corresponding form in terms of an action integral along periodic wave profiles was explicited in [BGNR13] and subsequently crucially used in [BGMR16,BGMRar].…”

About plane periodic waves of the nonlinear Schrödinger equations

“…The main upshot here is that instead of the rather long list of assumptions that would be required by directly applying the abstract general theory [GSS90,DBRN19], assumptions are both simple and expressed in terms of the finite-dimensional Θ. Then, as in [BGMR20], we elucidate these criteria in two limits of interest, the solitary-wave limit when the spatial period tends to infinity and the harmonic limit when the amplitude of the wave tends to zero. To describe the solitary-wave regime, let us point out that solitary wave profiles under consideration are naturally parametrized by pc x , ρ, k φ q where ρ ą 0 is the limiting value at spatial infinities of its mass and that families of solitary waves also come with an action integral Θ psq pc x , ρ, k φ q, known as the Boussinesq momentum of stability [Bou72,Ben72,Ben84] and associated for Schrödinger-like equations with the famous Vakhitov-Kolokolov slope condition [VK73].…”