Periodic solutions for the Landau-Lifshitz-Gilbert equation

TL;DR

This paper constructs time-periodic solutions for the Landau-Lifshitz-Gilbert equation modeling complex magnetization patterns in small ferromagnetic particles, using perturbation and spectral analysis techniques.

Contribution

It introduces a method to find periodic solutions for LLG in small, shaped ferromagnetic particles, expanding understanding of their dynamic behavior.

Findings

Existence of time-periodic solutions under specific shape conditions.

Application of spectral analysis to the linearized problem.

Use of perturbation methods for solution construction.

Abstract

Ferromagnetic materials tend to develop very complex magnetization patterns whose time evolution is modeled by the so-called Landau-Lifshitz-Gilbert equation (LLG). In this paper, we construct time-periodic solutions for LLG in the regime of soft and small ferromagnetic particles which satisfy a certain shape condition. Roughly speaking, it is assumed that the length of the particle is greater than its hight and its width. The approach is based on a perturbation argument and the spectral analysis of the corresponding linearized problem as well as the theory of sectorial operators.