Asymptotic stability in the energy space for dark solitons of the Landau-Lifshitz equation

TL;DR

This paper proves the asymptotic stability of non-zero speed dark solitons in the one-dimensional Landau-Lifshitz equation with easy-plane anisotropy, showing solutions close to a soliton converge to a (possibly different) soliton over time.

Contribution

It introduces a novel stability analysis for Landau-Lifshitz solitons using hydrodynamical transformations and monotonicity formulas, extending techniques from KdV equations.

Findings

Solutions near a soliton converge weakly to a (possibly different) soliton.

The limit profile is localized with exponential decay.

A Liouville theorem confirms solitons are the only stable localized solutions.

Abstract

We prove the asymptotic stability in the energy space of non-zero speed solitons for the one-dimensional Landau-Lifshitz equation with an easy-plane anisotropy. More precisely, we show that any solution corresponding to an initial datum close to a soliton with non-zero speed, is weakly convergent in the energy space as time goes to infinity, to a soliton with a possible different non-zero speed, up to the invariances of the equation. Our analysis relies on the ideas developed by Martel and Merle for the generalized Korteweg-de Vries equations. We use the Madelung transform to study the problem in the hydrodynamical framework. In this framework, we rely on the orbital stability of the solitons and the weak continuity of the flow in order to construct a limit profile. We next derive a monotonicity formula for the momentum, which gives the localization of the limit profile. Its smoothness and exponential decay then follow from a smoothing result for the localized solutions of the Schr\"odinger equations. Finally, we prove a Liouville type theorem, which shows that only the solitons enjoy these properties in their neighbourhoods.