Linear stability analysis for traveling waves of second order in time PDE's

TL;DR

This paper develops a general linear stability theory for traveling waves in second order in time PDEs, providing explicit stability criteria and applying them to classical models like Boussinesq, KGZ, and beam equations, including new stability results.

Contribution

It introduces an explicit stability index for second order PDE traveling waves and applies it to classical models, deriving new stability criteria and explicit formulas.

Findings

Explicit stability index $ ext{ω}^*$ for second order PDEs.

New stability criteria for Boussinesq and KGZ models.

Explicit formulas for stability in the beam equation.

Abstract

We develop a general theory for linear stability of traveling waves of second order in time PDE's. More precisely, we introduce an explicitly computable index $\om^*\in (0, \infty]$ (depending on the self-adjoint part of the linearized operator) so that the wave is stable if and only if $|c|\geq \om^*$. The results are applicable both in the periodic case and in the whole line case. As an application, we consider three classical models - the Boussinesq equation, the Klein-Gordon-Zakharov (KGZ) system and the fourth order beam equation. For the Boussinesq model and the KGZ system (and as a direct application of the main results), we compute explicitly the set of speeds which give rise to linearly stable traveling waves (and for all powers of $p$ in the case of Boussinesq). This result is new for the KGZ system, while it generalizes the results of Alexander-Sachs, which apply to the case $p=2$. For the beam equation, we provide an explicit formula (depending of the function $\|\vp_c'\|_{L^2}$), which works for all $p$ and for both the periodic and the whole line cases. Our results complement (and exactly match, whenever they exist) the results of a long line of investigation regarding the related notion of orbital stability of the same waves.