Nonlinear stability of Gardner breathers

Abstract

We show that breather solutions of the Gardner equation, a natural generalization of the KdV and mKdV equations, are $H^2(\mathbb{R})$ 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 (\emph{spectral stability}), and $(ii)$ controlling degenerated directions by using low regularity conservation laws.

arXiv commentThis is v2. Improved with a complete stability result on the existence interval of the parameter $\mu \in (0,{\mu }_{max})$. 29 pages. arXiv admin note: text overlap with arXiv:1206.3157