Linear stability analysis for periodic traveling waves of the Boussinesq equation and the KGZ system

TL;DR

This paper provides a comprehensive linear stability analysis of spatially periodic traveling waves for the Boussinesq equation and the Klein-Gordon-Zakharov system, explicitly characterizing stability and instability for various solutions.

Contribution

It offers a complete and explicit characterization of the linear stability of periodic waves for these systems, including the limit to the whole line case.

Findings

Explicit stability and instability criteria for periodic waves.

Recovery of whole line stability results as period tends to infinity.

Applicability to a wide class of solutions.

Abstract

The question for linear stability of spatially periodic waves for the Boussinesq equation (the cases $p=2,3$) and the Klein-Gordon-Zakharov system is considered. For a wide class of solutions, we completely and explicitly characterize their linear stability (instability respectively), when the perturbations are taken with the same period $T$. In particular, our results allow us to completely recover the linear stability results, in the limit $T\to \infty$, for the whole line case.