Spectral stability of constrained solitary waves for a generalized Ostrovsky equation

Abstract

We consider the existence and stability of constrained solitary wave solutions to the generalized Ostrovsky equation \begin{align*} \partial_x\left(\partial_t u+ \alpha\partial_x u+\partial_x(f(u))+\beta \partial_x^3u\right)=\gamma u,\quad \|u\|_{L^2}^2=\lambda \gt 0, \end{align*} where the homogeneous nonlinearities $f(s)=\alpha_0|s|^p+\alpha_1|s|^{p-1}s$ , with p > 1. If $\alpha_0,\alpha_1 \gt 0$ , $\alpha\in\mathbb{R}$ , and γ < 0 satisfying $\beta\gamma=-1$ , we show that for $1 \lt p \lt 5$ , there exists a constrained ground state traveling wave solution with travelling velocity $\omega \gt \alpha-2$ . Furthermore, we obtain the exponential decay estimates and the weak non-degeneracy of the solution. Finally, we show that the solution is spectrally stable. This is a continuation of recent work [1] on existence and stability for a water wave model with non-homogeneous nonlinearities.