Contents Online

# Communications in Mathematical Sciences

## Volume 19 (2021)

### Number 1

### A spin-wave solution to the Landau–Lifshitz–Gilbert equation

Pages: 193 – 204

DOI: https://dx.doi.org/10.4310/CMS.2021.v19.n1.a8

#### Authors

#### Abstract

Magnetic materials possess the intrinsic spin order, whose disturbance leads to spin waves. From the mathematical perspective, a spin wave is known as a traveling wave, which is often seen in wave and transport equations. The dynamics of intrinsic spin order is modeled by the Landau–Lifshitz–Gilbert equation, a nonlinear parabolic system of equations with a pointwise length constraint. In this paper, a spin wave for this equation is obtained based on the assumption that the spin wave maintains its periodicity in space when propagating at a varying velocity. In the absence of magnetic field, an explicit form of spin wave is provided. When a magnetic field is applied, the spin wave does not have such an explicit form but its stability is justified rigorously. Moreover, an approximate explicit solution is constructed with approximation error depending quadratically on the strength of magnetic field and being uniform in time.

#### Keywords

Landau–Lifshitz–Gilbert equation, spin wave, asympotic analysis

#### 2010 Mathematics Subject Classification

34E10, 35B40, 35C07, 35C20, 35K55

J. Chen acknowledges the financial support by National Natural Science Foundation of China via grant 11971021, National Key R&D Program of China (No. 2018YFB0204404) , and FDCT of Macao S.A.R. (0070/2019/A2).

Y. Wang acknowledges the financial support by National Natural Science Foundation of China via grant 11671289.

L. Yang acknowledges support from FDCT of Macao S.A.R. (0070/2019/A2) and National Natural Science Foundation of China via grant 11701598.

Received 11 May 2020

Accepted 12 August 2020

Published 24 March 2021