Time decay estimates for localized perturbations around a helical state for the Landau-Lifshitz-Gilbert equation

TL;DR

This paper proves the stability and decay of localized perturbations around helical states in the Landau-Lifshitz-Gilbert equation with Dzyaloshinskii--Moriya interaction, using spectral analysis and Floquet theory.

Contribution

It provides the first rigorous analysis of the stability and decay rates of helical states under small perturbations in this model.

Findings

Global existence of solutions under small initial perturbations

Time decay estimates demonstrating stability

Spectral analysis via Bloch--Fourier-wave decomposition

Abstract

We study the dynamics of the Landau--Lifshitz--Gilbert equation with the Dzyaloshinskii--Moriya interaction. The equation admits a family of exact stationary solutions, referred to as helical states, which are periodic in one spatial variable and constant in the others. We investigate the dynamical stability of a helical state with respect to perturbations belonging to suitable Lebesgue and Sobolev spaces. Under a smallness assumption on the initial perturbation, we prove global existence and time decay estimates for solutions, demonstrating that the above helical state is stable. The analysis of the relevant linear operator is carried out via the Bloch--Fourier-wave decomposition, where the eigenvalue problem for the reduced operator is characterized by certain Mathieu equations.