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

Abstract

We prove the asymptotic stability in energy space of nonzero speed solitons for the one-dimensional Landau–Lifshitz equation with an easy-plane anisotropy

$${\partial }_{t}m+m\times \left({\partial }_{xx}m-m_3{e}_3\right)=0$$

for a map $m=\left(m_1,m_2,m_3\right):ℝ\times ℝ\to {\mathbb{S}}^2$, where $e_3=\left(0,0,1\right)$. More precisely, we show that any solution corresponding to an initial datum close to a soliton with nonzero speed is weakly convergent in energy space as time goes to infinity to a soliton with a possible different nonzero 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 Schrödinger equations. Finally, we prove a Liouville type theorem, which shows that only the solitons enjoy these properties in their neighbourhoods.