On the stationary solution of the Landau-Lifshitz-Gilbert equation on a nanowire with constant external magnetic field

TL;DR

This paper analyzes stationary solutions of the Landau-Lifshitz-Gilbert equation in a ferromagnetic nanowire with a constant external magnetic field, proving existence, uniqueness, and instability of these solutions, supported by numerical simulations.

Contribution

It establishes the existence, uniqueness, and instability of stationary solutions for the LLG equation in a nanowire with a constant magnetic field, providing new insights into domain wall behaviors.

Findings

Existence of stationary solutions under certain conditions on the external field.

Uniqueness of solutions up to invariances of the equation.

Instability of the stationary solutions and their orbits.

Abstract

We consider an infinite ferromagnetic nanowire, with an energy functional $E$ with easy-axis in the direction $e_1$ and a constant external magnetic field $H_{ext} = h_0 e_1$ along the same direction. The evolution of its magnetization is governed by the Landau-Lifshitz-Gilbert equation (LLG) associated to $E$. Under some assumptions on $h_0$, we prove the existence of stationary solutions with the same limits at infinity, their uniqueness up to the invariances of the equation and the instability of their orbits with respect to the flow. This property gives interesting new insights of the behavior of the solutions of (LLG), which are completed by some numerical simulations and discussed afterwards, in particular regarding the stability of 2-domain wall structures proven in \cite{Cote_Ferriere__2DW} and more generally the interactions between domain walls.