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

TL;DR

This paper proves the asymptotic stability of precessing domain walls in ferromagnetic nanowires with Dzyaloshinskii-Moriya interaction, using a dimension reduction and variational approach to analyze their dynamics under small magnetic fields.

Contribution

It introduces a dimension reduction via $Gamma$-convergence and classifies critical points, establishing stability results for precessing domain walls in a new ferromagnetic model.

Findings

Asymptotic stability of precessing domain walls under small magnetic fields.

Classification of finite energy transition layers as static domain walls.

Dimension reduction results for the energy functional in nanowire models.

Abstract

We consider a ferromagnetic nanowire and we focus on an asymptotic regime where the Dzyaloshinskii-Moriya interaction is taken into account. First we prove a dimension reduction result via $\Gamma$-convergence that determines a limit functional $E$ defined for maps $m:\mathbb{R}\to \mathbb{S}^2$ in the direction $e_1$ of the nanowire. The energy functional $E$ is invariant under translations in $e_1$ and rotations about the axis $e_1$. We fully classify the critical points of finite energy $E$ when a transition between $-e_1$ and $e_1$ is imposed; these transition layers are called (static) domain walls. The evolution of a domain wall by the Landau-Lifshitz-Gilbert equation associated to $E$ under the effect of an applied magnetic field $h(t)e_1$ depending on the time variable $t$ gives rise to the so-called precessing domain wall. Our main result proves the asymptotic stability of precessing domain walls for small $h$ in $L^\infty([0, +\infty))$ and small $H^1(\mathbb{R})$ perturbations of the static domain wall, up to a gauge which is intrinsic to invariances of the functional $E$.