On the Stability of Solitary Waves of a Generalized Ostrovsky Equation

We consider the stability of ground state solitary waves of the generalized Ostrovsky equation (ut − βuxxx + f(u)x)x = γu, with homogeneous nonlinearities of the form f(u) = ae|u| + ao|u|u. We obtain bounds on the function d whose convexity determines the stability of the solitary waves. These bounds imply that, when 2 ≤ p < 5 and ao < 0, solitary waves are stable for c near c∗ = 2 √ βγ. These bounds also imply that, for γ > 0 small, solitary waves are stable when 2 ≤ 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.