Instability of two-pulse periodic waves with long wavelength in some Hamiltonian PDEs

TL;DR

This paper investigates the spectral stability of long-wavelength two-pulse periodic waves in certain Hamiltonian PDEs, revealing their instability for large periods through asymptotic analysis and spectral convergence methods.

Contribution

It introduces a combined analytical framework to determine the spectral stability of two-pulse waves in Hamiltonian PDEs as the wavelength becomes large.

Findings

Long-wavelength two-pulse waves are spectrally unstable for large periods.

The asymptotic expansion of the Hessian matrix of the action integral is used to analyze stability.

Convergence of the spectrum is established via renormalized Evans function analysis.

Abstract

We consider 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. In particular, our framework includes hydrodynamic formulation of the nonlinear Schr\"odinger equations. The periodic waves we study exhibit on each period two pulses, one converging to a bright soliton and one converging to a dark soliton, when wavelength goes to infinity. We show that such waves, for sufficiently large periods, are spectrally unstable. To do so, we combine two approaches. The first one is to calculate the asymptotic expansion of the Hessian matrix of the action integral and concludes using arXiv:1505.01382 as in arXiv:1710.03936 . This shows instability when both limiting solitary waves are stable. The second approach studies the convergence of the spectrum when the period goes to infinity and is applied in remaining cases, when one of the solitary waves is unstable. To carry out the latter, we prove the convergence of an appropriate renormalization of the periodic Evans function as in arXiv:1802.02830 .