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