Instability of the solitary waves for the generalized Boussinesq equations

Abstract

In this work, we consider the following generalized Boussinesq equation \begin{align*} \partial_{t}^2u-\partial_{x}^2u+\partial_{x}^2(\partial_{x}^2u+|u|^{p}u)=0,\qquad (t,x)\in\mathbb R\times \mathbb R, \end{align*} with $0<p<\infty$. This equation has the traveling wave solutions ${\varphi }_{\omega }(x-\omega t)$, with the frequency $\omega \in (-1,1)$ and ${\varphi }_{\omega }$ satisfying \begin{align*} -\partial_{xx}ϕ_ω+(1-{ω^2})ϕ_ω-ϕ_ω^{p+1}=0. \end{align*} Bona and Sachs (1988) proved that the traveling wave ${\varphi }_{\omega }(x-\omega t)$ is orbitally stable when $0<p<4,$ $\frac{p}{4}<{\omega }^2<1$. Liu (1993) proved the orbital instability under the conditions $0<p<4,$ ${\omega }^2<\frac{p}{4}$ or $p\ge 4,$ ${\omega }^2<1$. In this paper, we prove the orbital instability in the degenerate case $0<p<4,{\omega }^2=\frac{p}{4}$ .

30 pages