Instability of the solitary waves for the Generalized Benjamin-Bona-Mahony Equation

TL;DR

This paper investigates the stability of solitary waves for the generalized Benjamin-Bona-Mahony equation, establishing orbital instability at a critical wave speed where stability was previously known for other speeds.

Contribution

It proves the orbital instability of solitary waves precisely at the critical speed c=c0(p), filling a gap in the stability analysis of these solutions.

Findings

Orbital instability at the critical wave speed c=c0(p).

Extension of instability results to the critical case.

Clarification of stability behavior for all wave speeds.

Abstract

In this work, we consider the generalized Benjamin-Bona-Mahony equation $$\partial_t u+\partial_x u+\partial_x( |u|^pu)-\partial_t \partial_x^{2}u=0, \quad(t,x) \in \mathbb{R} \times \mathbb{R}, $$ with $p>4$. This equation has the traveling wave solutions $\phi_{c}(x-ct), $ for any frequency $c>1.$ It has been proved by Souganidis and Strauss \cite{Strauss-1990} that, there exists a number $c_{0}(p)>1$, such that solitary waves $\phi_{c}(x-ct)$ with $1<c<c_{0}(p) $ is orbitally unstable, while for $c>c_{0}(p), $ $\phi_{c}(x-ct)$ is orbitally stable. The linear exponential instability in the former case was further proved by Pego and Weinstein \cite{Pego-1991-eigenvalue}. In this paper, we prove the orbital instability in the critical case $c=c_{0}(p)$.