Stability of 2-soliton solutions for the modified Camassa-Holm equation with cubic nonlinearity

TL;DR

This paper proves the nonlinear stability of 2-soliton solutions on a nonzero background for the modified Camassa-Holm equation with cubic nonlinearity, using conserved quantities related to the momentum variable in Sobolev space.

Contribution

It establishes the nonlinear stability of 2-soliton solutions for the modified Camassa-Holm equation with cubic nonlinearity, a novel result in the analysis of this equation.

Findings

2-soliton solutions are nonlinearly stable under perturbations in H^2.

Conserved quantities in terms of the momentum variable are key to the stability proof.

Stability is shown on a nonzero constant background.

Abstract

In this paper, we are concerned with the stability of 2-soliton solutions on a nonzero constant background for the modified Camassa-Holm equation with cubic nonlinearity. By employing conserved quantities in terms of the momentum variable $m$, we show that the 2-soliton, when regarded as a solution to the initial-value problem for the modified Camassa-Holm equation, is nonlinearly stable to perturbations with respect to the momentum variable in the Sobolev space $H^2$.