Orbital stability of smooth solitary waves for the modified Camassa-Holm equation

Abstract

In this paper, we explore the orbital stability of smooth solitary wave solutions to the modified Camassa-Holm equation with cubic nonlinearity. These solutions, which exist on a nonzero constant background $k$, are unique up to translation for each permissible value of $k$ and wave speed. By leveraging the Hamiltonian nature of the modified Camassa-Holm equation and employing three conserved functionals-comprising an energy and two Casimirs, we establish orbital stability through an analysis of the Vakhitov-Kolokolov condition. This stability pertains to perturbations of the momentum variable in $H^1(\mathbb{R})$.