Instability and stability properties of traveling waves for the double dispersion equation

TL;DR

This paper analyzes the stability and instability of traveling wave solutions in a double dispersion equation with two dispersion sources, providing explicit conditions for stability and instability based on parameters.

Contribution

It introduces explicit criteria for wave stability and instability in the double dispersion equation, extending known results for the Boussinesq case.

Findings

Explicit instability condition for wave velocities in terms of parameters a, b, p.

Identification of parameter regimes for orbital stability of traveling waves.

Numerical analysis of stability regions as parameters vary.

Abstract

In this article we are concerned with the instability and stability properties of traveling wave solutions of the double dispersion equation $~u_{tt} -u_{xx}+a u_{xxxx}-bu_{xxtt} = - (|u|^{p-1}u)_{xx}~$ for $~p>1$, $~a\geq b>0$. The main characteristic of this equation is the existence of two sources of dispersion, characterized by the terms $u_{xxxx}$ and $u_{xxtt}$. We obtain an explicit condition in terms of $a$, $b$ and $p$ on wave velocities ensuring that traveling wave solutions of the double dispersion equation are strongly unstable by blow up. In the special case of the Boussinesq equation ($b=0$), our condition reduces to the one given in the literature. For the double dispersion equation, we also investigate orbital stability of traveling waves by considering the convexity of a scalar function. We provide both analytical and numerical results on the variation of the stability region of wave velocities with $a$, $b$ and $p$ and then state explicitly the conditions under which the traveling waves are orbitally stable.