Asymptotic Stability of 2-Domain Walls for the Landau–Lifshitz–Gilbert Equation in a Nanowire With Dzyaloshinskii–Moriya Interaction

Abstract

Abstract 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 that 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.