# Nonlinear stability of Gardner breathers

## TL;DR

This paper proves the nonlinear stability of Gardner breathers, special wave solutions, by using variational methods, spectral analysis, and conservation laws, extending stability results to a broader class of equations.

## Contribution

It introduces a variational characterization of Gardner breathers as minimizers of a new Lyapunov functional and analyzes their spectral stability.

## Key findings

- Gardner breathers are proven to be $H^2$ stable.
- Spectral analysis confirms stability through explicit linear systems.
- Low regularity conservation laws control degenerated directions.

## 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.