On the asymptotic stability in the energy space for multi-solitons of the Landau-Lifshitz equation

TL;DR

This paper proves the long-term stability of multi-soliton solutions in the energy space for the one-dimensional Landau-Lifshitz equation with easy-plane anisotropy, showing solutions converge to solitons or zero depending on their position.

Contribution

It establishes the asymptotic stability of multi-solitons in the Landau-Lifshitz equation, including convergence properties and stability in the energy space.

Findings

Solutions near multi-solitons converge weakly to a soliton or zero.

Multi-solitons with separated initial positions are asymptotically stable.

The stability is proven in the energy space for the Landau-Lifshitz equation.

Abstract

We establish the asymptotic stability of multi-solitons for the one-dimensional Landau-Lifshitz equation with an easy-plane anisotropy. The solitons have non-zero speed, are ordered according to their speeds and have sufficiently separated initial positions. We provide the asymptotic stability around solitons and between solitons. More precisely, we show that for an initial datum close to a sum of $N$ dark solitons, the corresponding solution converges weakly to one of the solitons in the sum, when it is translated to the centre of this soliton, and converges weakly to zero when it is translated between solitons.