Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers

TL;DR

This paper investigates the long-term behavior of complex-valued mKdV solitons and breathers, establishing their stability properties using Bäcklund transformations, without relying on inverse scattering, applicable even under rough perturbations.

Contribution

The paper proves H^1 stability of complex-valued mKdV breathers and introduces a new approach using Bäcklund transformations, extending stability results beyond previous H^2 analyses.

Findings

H^1 stability of mKdV breathers established

Negative energy breathers shown to be asymptotically stable

Method applicable under rough perturbations without inverse scattering

Abstract

We study the long-time dynamics of complex-valued modified Korteweg-de Vries (mKdV) solitons, which are recognized because they blow-up in finite time. We establish stability properties at the H^1 level of regularity, uniformly away from each blow-up point. These new properties are used to prove that mKdV breathers are H^1 stable, improving our previous result, where we only proved H^2 stability. The main new ingredient of the proof is the use of a B\"acklund transformation which links the behavior of breathers, complex-valued solitons and small real-valued solutions of the mKdV equation. We also prove that negative energy breathers are asymptotically stable. Since we do not use any method relying on the Inverse Scattering Transformation, our proof works even under rough perturbations, provided a corresponding local well-posedness theory is available.