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.