The Fascinating World of Landau-Lifshitz-Gilbert Equation: An Overview

TL;DR

This paper reviews the Landau-Lifshitz-Gilbert (LLG) equation, highlighting its mathematical and physical significance, diverse dynamical structures, and recent applications in spintronics, emphasizing its broad relevance across physics and mathematics.

Contribution

It provides a comprehensive overview of the LLG equation's mathematical properties, physical applications, and recent developments in spintronics, connecting classical and modern research.

Findings

Multiple dynamical structures including solitons and chaos

Connections with integrable soliton equations

Recent application in spintronics and nanoferromagnets

Abstract

The Landau-Lifshitz-Gilbert (LLG) equation is a fascinating nonlinear evolution equation both from mathematical and physical points of view. It is related to the dynamics of several important physical systems such as ferromagnets, vortex filaments, moving space curves, etc. and has intimate connections with many of the well known integrable soliton equations, including nonlinear Schr\"odinger and sine-Gordon equations. It can admit very many dynamical structures including spin waves, elliptic function waves, solitons, dromions, vortices, spatio-temporal patterns, chaos, etc. depending on the physical and spin dimensions and the nature of interactions. An exciting recent development is that the spin torque effect in nanoferromagnets is described by a generalization of the LLG equation which forms a basic dynamical equation in the field of spintronics. This article will briefly review these developments as a tribute to Robin Bullough who was a great admirer of the LLG equation.