On the stability of solitary waves of a generalized Ostrovsky equation

Abstract

We consider the stability of ground state solitary waves of the generalized Ostrovsky equation $( u_t - \beta u_{xxx} + f(u)_x)_x = \gamma u$, with homogeneous nonlinearities of the form $f(u)=a_e|u|^p+a_o|u|^{p-1}u$. We obtain bounds on the function $d$ whose convexity determines the stability of the solitary waves. These bounds imply that, when $2\le p<5$ and $a_o<0$, solitary waves are stable for $c$ near $c_*=2\sqrt{\beta \gamma }$. These bounds also imply that, for $\gamma >0$ small, solitary waves are stable when $2\le p<5$ and unstable when $p>5$. We also numerically compute the function $d$, and thereby determine precise regions of stability and instability, for several nonlinearities.