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

Abstract

In this work, we consider the generalized Benjamin-Bona-Mahony equation $${\partial }_{t}u+{\partial }_{x}u+{\partial }_x(|u{|}^{p}u)-{\partial }_t{\partial }_x^{2}u=0,(t,x)\in \mathbb{R}\times \mathbb{R},$$ with $p>4$. This equation has the traveling wave solutions ${\varphi }_c(x-ct),$ 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 ${\varphi }_c(x-ct)$ with $1<c<c_0(p)$ is orbitally unstable, while for $c>c_0(p),$ ${\varphi }_c(x-ct)$ 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 $c=c_0(p)$.

32 pages, 1 ficture