Orbital Stability of Smooth Solitary Waves for the Novikov Equation

Abstract

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 endstate, and are encoded as critical points of a particular action functional. As an importantstep in our analysis we must study the spectrum the Hessian of this action functional, which turns out to be a nonlocal integro-differential operatoracting on $L^2(\mathbb{R})$. We provide a combination of analytical and numerical evidence that the necessary spectral hypotheses alwaysholds 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.