Spherical cone symmetric families generated by Landau-Lifshitz equation and their evolution

Abstract

Until 2008, we had not seen a paper about the explicit dynamic solution of the multi-dimensional Landau-Lifshitz equation with the external magnetic field and anisotropy field. In the absence of an external magnetic field or anisotropy field, many experts have tried Hirotas famous method, Lie algebra structure and Backlund transformations, bilinear transform, but the exact solution is still not constructed. This paper presents a method to solve the Landau-Lifshitz equations with the external magnetic field and anisotropy field, and a new family of symmetric solutions, which we call it as spherical cone symmetric solution. Then, we shall show how these solutions evolve as time. Finally, we give a series of explicit dynamic spherical cone symmetric solutions.