Orbital Stability of Smooth Solitary Waves for the Novikov Equation

We study the orbital stability of smooth solitary wave solutions of the Novikov equation, which is a Camassa-Holm-type equation with cubic nonlinearities. These solitary waves are shown to exist as a one-parameter family (up to spatial translations) parameterized by their asymptotic end state, and are encoded as critical points of a particular action functional. As an important step in our analysis, we must study the spectrum the Hessian of this action functional, which turns out to be a nonlocal integro-differential operator acting on L2(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>2({\mathbb {R}})$$\end{document}. We provide a combination of analytical and numerical evidence that the necessary spectral hypotheses always hold for the Novikov equation. Together with a detailed study of the associated Vakhitov-Kolokolov condition, our analysis indicates that all smooth solitary wave solutions of the Novikov equation are nonlinearly orbitally stable.