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

TL;DR

This paper investigates the stability of periodic waves in Hamiltonian PDEs, analyzing the negative signature of the action Hessian in small amplitude and long wavelength regimes, and establishing stability criteria.

Contribution

It provides explicit computations of the action Hessian's negative signature in asymptotic regimes, linking stability to wave parameters and extending previous stability characterizations.

Findings

Small amplitude waves are orbitally stable in this framework.

The negative signature for long wavelength waves is governed by the second derivative of Boussinesq momentum.

Explicit formulas for the action Hessian's signature are derived in asymptotic limits.

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 of the Hessian matrix of action integrals associated with those waves. In a previous paper (Nonlinearity 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 - or Morse index - 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 the second derivative with respect to the wave speed of the Boussinesq momentum associated with the limiting solitary wave.