(In-)Stability of Singular Equivariant Solutions to the Landau-Lifshitz-Gilbert Equation

TL;DR

This paper investigates the stability of equivariant solutions to the Landau-Lifshitz-Gilbert equation, revealing that blowup solutions are generally unstable and non-generic, with detailed asymptotic and numerical analyses supporting these findings.

Contribution

It provides a detailed asymptotic and numerical analysis of blowup solutions, establishing their instability and co-dimension one nature in the Landau-Lifshitz-Gilbert model.

Findings

Blowup solutions are co-dimension one and unstable.

Solutions deviating from radial symmetry remain global but can approach blowup.

Blowup corresponds to a saddle fixed point in a low-dimensional dynamical system.

Abstract

In this paper we use formal asymptotic arguments to understand the stability proper- ties of equivariant solutions to the Landau-Lifshitz-Gilbert model for ferromagnets. We also analyze both the harmonic map heatflow and Schrodinger map flow limit cases. All asymptotic results are verified by detailed numerical experiments, as well as a robust topological argument. The key result of this paper is that blowup solutions to these problems are co-dimension one and hence both unstable and non-generic. Solutions permitted to deviate from radial symmetry remain global for all time but may, for suitable initial data, approach arbitrarily close to blowup. A careful asymptotic analysis of solutions near blowup shows that finite-time blowup corresponds to a saddle fixed point in a low dimensional dynamical system. Radial symmetry precludes motion anywhere but on the stable manifold towards blowup. A similar scenario emerges in the equivariant setting: blowup is unstable. To be more precise, blowup is co-dimension one both within the equivariant symmetry class and in the unrestricted class of initial data. The value of the parameter in the Landau-Lifshitz-Gilbert equation plays a very subdued role in the analysis of equivariant blowup, leading to identical blowup rates and spatial scales for all parameter values. One notable exception is the angle between solution in inner scale (which bubbles off) and outer scale (which remains), which does depend on parameter values. Analyzing near-blowup solutions, we find that in the inner scale these solution quickly rotate over an angle {\pi}. As a consequence, for the blowup solution it is natural to consider a continuation scenario after blowup where one immediately re-attaches a sphere (thus restoring the energy lost in blowup), yet rotated over an angle {\pi}. This continuation is natural since it leads to continuous dependence on initial data.