Asymptotic stability of solitons for mKdV

TL;DR

This paper proves that small perturbations of solitary waves in the mKdV equation asymptotically evolve into a shifted solitary wave, with detailed analysis of the nonlinear scattering behavior of the perturbations.

Contribution

It establishes a full asymptotic stability result for mKdV solitary waves, including precise asymptotics of the perturbations and their nonlinear scattering behavior.

Findings

Solutions tend to a solitary wave as time goes to infinity.

Perturbations exhibit nonlinearly modified scattering.

The asymptotic behavior of small solutions is characterized precisely.

Abstract

We prove a full asymptotic stability result for solitary wave solutions of the mKdV equation. We consider small perturbations of solitary waves with polynomial decay at infinity and prove that solutions of the Cauchy problem evolving from such data tend uniformly, on the real line, to another solitary wave as time goes to infinity. We describe precisely the asymptotics of the perturbation behind the solitary wave showing that it satisfies a nonlinearly modified scattering behavior. This latter part of our result relies on a precise study of the asymptotic behavior of small solutions of the mKdV equation.