On the Instabilities of the Walker Propagating Domain Wall Solution

TL;DR

This paper applies a new mathematical method to analyze the stability of the Walker domain wall solution in the Landau-Lifshitz-Gilbert equation, revealing conditions under which it becomes unstable due to spin wave emission.

Contribution

It introduces a novel application of spectral analysis to the LLG equation, showing the Walker solution's potential instability under certain field and anisotropy conditions.

Findings

Walker domain wall is unstable against spin wave emission.

Different wave types are emitted under low and high fields.

Walker solution may be unstable at high fields with large transverse anisotropy.

Abstract

A powerful mathematical method for front instability analysis that was recently developed in the field of nonlinear dynamics is applied to the 1+1 (spatial and time) dimensional Landau-Lifshitz-Gilbert (LLG) equation. From the essential spectrum of the LLG equation, it is shown that the famous Walker rigid body propagating domain wall (DW) is not stable against the spin wave emission. In the low field region only stern spin waves are emitted while both stern and bow waves are generated under high fields. By using the properties of the absolute spectrum of the LLG equation, it is concluded that in a high enough field, but below the Walker breakdown field, the Walker solution could be convective/absolute unstable if the transverse magnetic anisotropy is larger than a critical value, corresponding to a significant modification of the DW profile and DW propagating speed. Since the Walker solution of 1+1 dimensional LLG equation can be realized in experiments, our results could be also used to test the mathematical method in a controlled manner.