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

TL;DR

This paper investigates the orbital stability of smooth solitary waves in the modified Camassa-Holm equation with cubic nonlinearity, using conserved quantities and the Vakhitov-Kolokolov condition to establish stability.

Contribution

It provides a rigorous stability analysis of solitary waves for the modified Camassa-Holm equation, a topic not extensively covered before.

Findings

Orbital stability established for solitary waves on a nonzero background.

Utilized Hamiltonian structure and conserved functionals for analysis.

Stability applies to perturbations in the $H^1(\mathbb{R})$ space.

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 $H^1(\mathbb{R})$.