# The fascinating world of the Landau–Lifshitz–Gilbert equation: an overview

@article{Lakshmanan2011TheFW, title={The fascinating world of the Landau–Lifshitz–Gilbert equation: an overview}, author={Muthusamy Lakshmanan}, journal={Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences}, year={2011}, volume={369}, pages={1280 - 1300} }

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 Schrödinger and sine-Gordon equations. It can admit very many dynamical structures including spin waves, elliptic…

## 168 Citations

### The Landau-Lifshitz equation and related models

- Mathematics
- 2020

In this manuscript, I present my research on nonlinear dispersive PDEs, done after my Ph.D. Most of the results concern the Landau-Lifshitz equation, which is a quasilinear equation that describes…

### Recent results for the Landau–Lifshitz equation

- Mathematics
- 2021

We give a survey on some recent results concerning the Landau–Lifshitz equation, a fundamental nonlinear PDE with a strong geometric content, describing the dynamics of the magnetization in…

### Self-similar solutions of the one-dimensional Landau–Lifshitz–Gilbert equation

- Mathematics
- 2014

We consider the one-dimensional Landau–Lifshitz–Gilbert (LLG) equation, a model describing the dynamics for the spin in ferromagnetic materials. Our main aim is the analytical study of the…

### Pattern Formation in Axially Symmetric Landau–Lifshitz–Gilbert–Slonczewski Equations

- PhysicsJ. Nonlinear Sci.
- 2017

A major part of this study concerns existence and stability of wavetrains and of front- and domain wall-type coherent structures whose profiles asymptote to wavetr trains or the constant up-/down-magnetizations.

### Self-similar shrinkers of the one-dimensional Landau–Lifshitz–Gilbert equation

- MathematicsJournal of Evolution Equations
- 2020

The main purpose of this paper is the analytical study of self-shrinker solutions of the one-dimensional Landau–Lifshitz–Gilbert equation (LLG), a model describing the dynamics for the spin in…

### Nonlinear dynamics of spin transfer nano-oscillators

- Physics
- 2015

The evolution equation of a ferromagnetic spin system described by Heisenberg nearest-neighbour interaction is given by Landau–Lifshitz–Gilbert (LLG) equation, which is a fascinating nonlinear…

### From the open generalized Heisenberg model to the Landau–Lifshitz equation

- PhysicsNew Journal of Physics
- 2020

Magnetic systems can be described by the classical Landau–Lifshitz (LL) equation or the fully quantum open generalized Heisenberg model. Using the Lindblad master equation and the mean-field…

### Magnetization dynamics: path-integral formalism for the stochastic Landau–Lifshitz–Gilbert equation

- Physics
- 2014

We construct a path-integral representation of the generating functional for the dissipative dynamics of a classical magnetic moment as described by the stochastic generalization of the…

### Nonlinear Dynamics of an Array of Nano Spin Transfer Oscillators

- Physics
- 2014

The dynamics of a macrospin variable representing homogeneous magnetization of the free layer of a nanospin transfer oscillator (STNO) can be represented by the Landau–Lifshitz–Gilbert–Slonczewski…

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