The Gardner equation and the stability of multi-kink solutions of the mKdV equation

TL;DR

This paper proves the global stability of multi-kink solutions for the defocusing mKdV equation using transformations to Gardner-like equations, and explores the dynamics and inelastic collisions of generalized multi-kinks in non-integrable gKdV equations.

Contribution

It establishes the global $H^1$-stability of multi-kink solutions for the defocusing mKdV and extends the analysis to generalized non-integrable gKdV equations, including collision dynamics.

Findings

Multi-kink solutions are globally $H^1$-stable.

Transformations link mKdV to Gardner-like equations for stability analysis.

Inelastic collision behavior observed in certain regimes of generalized gKdV equations.

Abstract

Multi-kink solutions of the defocusing, modified Korteweg-de Vries equation (mKdV) found by Grosse are shown to be globally $H^1$-stable. Stability in the one-kink case was previously established by Zhidkov, and Merle-Vega. The proof uses transformations linking the mKdV equation with focusing, Gardner-like equations, where stability and asymptotic stability in the energy space are known. We generalize our results by considering the existence, uniqueness and the dynamics of generalized multi-kinks of defocusing, non-integrable gKdV equations, showing the inelastic character of the 4-kink collision in some regimes.