Asymptotic stability of 2-domain walls for the Landau-Lifshitz-Gilbert equation in a nanowire with Dzyaloshinskii-Moriya interaction

We consider a ferromagnetic nanowire, with an energy functional E with easy-axis in the direction e 1 , and which takes into account the Dzyaloshinskii-Moriya interaction. We consider configurations of the magnetization which are perturbations of two well separated domain wall, and study their evolution under the Landau-Lifshitz-Gilbert flow associated to E . Our main result is that, if the two walls have opposite speed, these configurations are asymptotically stable, up to gauges intrinsic to the invariances of the energy E . Our analysis builds on the framework developed in [4], taking advantage that it is amenable to space localisation.