Nonlinear stability of Gardner breathers

Miguel A. Alejo

Published 2017-05-10Version 1

We show that breather solutions of the Gardner equation, a natural generalization of the KdV and mKdV equations, are globally stable in a \emph{natural} $H^2$ topology. Our proof presentsa systematic and simple account that put in evidence that the Gardner breather solutions satisfy a suitable variational elliptic equation, which also implies that the stability problem reduces in some sense to $(i)$ the study of the spectrum of explicit linear systems (\emph{spectral stability}), and $(ii)$ the understanding of how degenerated directions can be controlled using low regularity conservation laws.