Nonlinear stability of mKdV breathers

TL;DR

This paper proves the global stability of mKdV breather solutions in H^2 space using a novel Lyapunov functional, effectively managing perturbations and instability modes.

Contribution

It introduces a new Lyapunov functional at the H^2 level to analyze the nonlinear stability of mKdV breathers, including control of degenerate directions.

Findings

Breather solutions are globally stable in H^2 topology.

A new Lyapunov functional effectively describes perturbation dynamics.

Degenerate directions are controlled via low-regularity conservation laws.

Abstract

Breather solutions of the modified Korteweg-de Vries equation are shown to be globally stable in a natural H^2 topology. Our proof introduces a new Lyapunov functional, at the H^2 level, which allows to describe the dynamics of small perturbations, including oscillations induced by the periodicity of the solution, as well as a direct control of the corresponding instability modes. In particular, degenerate directions are controlled using low-regularity conservation laws.