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

Abstract

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 ferromagnetic materials. We show that there is a unique smooth family of backward self-similar solutions to the LLG equation, up to symmetries, and we establish their asymptotics. Moreover, we obtain that in the presence of damping, the trajectories of the self-similar profiles converge to great circles on the sphere S 2 , at an exponential rate.In particular, the results presented in this paper provide examples of blow-up in finite time, where the singularity develops due to rapid oscillations forming limit circles.

The Use of Hypergeometric Functions in Hysteresis Modeling

“…It is worth mentioning that the well-known Landau-Lifshitz-Gilbert (LLG) equation can be used to describe the hysteresis phenomenon with a simplified micromagnetic model that can be combined with a magnetic circuit simulation [24], yet it requires a relatively long computation time. Reference [25] points out that some solutions to the LLG equation can be explicitly expressed with confluent hypergeometric functions, which are also included in the present model.…”