Nonlinear stability of Gardner breathers

We show that breather solutions of the Gardner equation, a natural generalization of the KdV and mKdV equations, are H2(R) stable. Through a variational approach, we characterize Gardner breathers as minimizers of a new Lyapunov functional and we study the associated spectral problem, through (i) the analysis of the spectrum of explicit linear systems (spectral stability), and (ii) controlling degenerated directions by using low regularity conservation laws.