Decay Rate and Energy Gap for the Singularity Solution of the Inhomogeneous Landau-Lifshitz Equation on S2

Abstract

The singularity solution for the inhomogeneous Landau-Lifshitz (ILL) equation without damping term in n-dimensional space was investigated. The implicit singularity solution was obtained for the case where the target space is on S2. This solution can be classified into four types that cover the global and local solutions. An estimation of the energy density of one of these types indicates its exact decay rate, which allows a global solution with finite initial energy under n>3. Analysis of the four aperiodic solutions indicates that energy gaps that are first contributions to the literature of ILL will occur for particular coefficient settings, and these are shown graphically.