The one-dimensional Landau-Lifshitz equation describes the dynamics of the magnetization in a ferromagnetic material. It has travelling-wave solutions called solitons.

In this talk, I first review previous result of the global existence and orbital stability of solitons. Then, I will focus on the asymptotic stability in the energy space of non-zero speed solitons. More precisely, I will 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 geometric invariances of the equation. The proof relies on the ideas developed by Martel and Merle for the generalized Korteweg-de Vries equations. We will finish by the case of multi-solitons. The solitons have non-zero speed, are ordered according to their speeds and have sufficiently separated initial positions. We will show that they are asymptotically stable.

The talk is based on my PhD work at Ecole polytechnique in France under the supervision of Philippe Gravejat and Raphael Cote.