An efficient and highly accurate spectral method for modeling the propagation of solitary magnetic spin waves in thin films

Abstract

In previous works, it was shown that the propagation of magnetic spin waves in thin films can be approximated by a nonlinear Schrödinger-type equation. The formulation begins with the magnetostatic equations (Gauss and Ampere's laws of magnetism) and the Landau-Lifshitz equation. The solution of this system is a potential function whose dimensionless amplitude is the solution of a nonlinear Schrödinger. In the current work, we are demonstrating an efficient infinite series solution using the Christov functions. This is the first time the functions are used in problems involving complex arithmetics. The solutions of the time-independent and time-dependent problems are given in complex series form. Keywords Magnetic thin solitons • Spectral method • Galerkin Mathematics Subject Classification 35J10 • 65M70 • 81V60 1 Introduction Solitary waves were discovered in the mid-nineteenth century in the form of localized persistent shallow water waves, in what now has become a famous account by Russell (1845). Following this initial discovery, Boussinesq (1871a, b) proved that permanent waves exist in nonlinear systems as a result of the balance between nonlinearity and dispersion. A few decades later, Zabusky and Kruskal (1965) used numerical methods to discover wave solu-Communicated by Pierangelo Marcati.