Existence and Uniqueness of Domain Walls for Notched Ferromagnetic Nanowires

TL;DR

This paper proves the existence and uniqueness of a magnetic domain wall in notched ferromagnetic nanowires using calculus of variations, critical point theory, and symmetry analysis, providing a rigorous mathematical foundation for magnetic domain behavior.

Contribution

It introduces a novel mathematical approach to establish existence and uniqueness of domain walls in notched nanowires, combining lifting arguments and Mountain-Pass techniques.

Findings

Existence of a unique critical point corresponding to a domain wall

Asymptotic decay of the solution at infinity

Symmetry properties of the domain wall solution

Abstract

In this article, we investigate a simple model of notched ferromagnetic nanowires using tools from calculus of variations and critical point theory. Specifically, we focus on the case of a single unimodal notch and establish the existence and uniqueness of the critical point of the energy. This is achieved through a lifting argument, which reduces the problem to a generalized Sturm-Liouville equation. Uniqueness is demonstrated via a Mountain-Pass argument, where the assumption of two distinct critical points leads to a contradiction. Additionally, we show that the solution corresponds to a system of magnetic spins characterized by a single domain wall localized in the vicinity of the notch. We further analyze the asymptotic decay of the solution at infinity and explore the symmetric case using rearrangement techniques.