  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.
