On the instability of solitary waves solutions of the generalized Benjamin equation

Abstract

This work is concerned with instability properties of solutions $u(x,t)=\phi(x-ct)$ of the equation $u_t+(u^p)_x + l H u_{xx}+u_{xxx}=0$ in $\mathbb R$, where $p\in \mathbb N$, $p\geqq 2$, and $H$ is the Hilbert transform. Here, $\phi$ will be a solution of the pseudo-differential equation $\phi''+l H \phi'-c\phi=-\phi^{p}$ solving a certain variational problem. We prove that the set $$ \Omega_{\phi}=\{\phi(\cdot+y) : y\in \mathbb R\;\} $$ is unstable by the flow of the evolution equation above provided $l$ is small, $c>\frac14 l^2$ and $p>5$. Moreover, the trajectories used to exhibit instability are global and uniformly bounded. Finally, we extend these results for a natural generalization of the evolution equation above with general forms of competing dispersion, in particular, we obtain instability results for some Korteweg-de Vries type equations without requiring spectral conditions.