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.