Energy decay rate of multidimensional inhomogeneous Landau\u2013Lifshitz\u2013Gilbert equation and Schr\xf6dinger map equation on the sphere

Penghong Zhong,Fengong Wu,Chao Zhang

doi:10.1186/s13662-018-1795-4

Abstract

We consider the multidimensional dimensional inhomogeneous Landau–Lifshitz–Gilbert (ILLG) equation and its degenerate case, the Schrödinger map equation. We investigate the special solutions (under large initial values) and their energy property of the ILLG and Schrödinger map equations. Until now, we had not seen a paper presenting an explicit dynamic solution of the multidimensional ILLG. Using the stereographic method, an equivalent equation of ILLG is obtained. Based on this equivalent system, we obtain some exact solutions of the ILLG equation and present some implicit solutions of the Schrödinger map equation. Based on these solutions, by a careful estimation we give the decay rate of energy density.

Highlights

1 Introduction Long-wavelength spin motions in diverse ferromagnetic structures are commonly described by the Landau–Lifshitz–Gilbert equation, which was first derived by Landau and Lifshitz [14]
We present some special solutions of the inhomogeneous Landau–Lifshitz–Gilbert (ILLG) equation and Schrödinger map equation and discuss their properties to enrich the solutions of these equations
Under the different ansatzs of the solutions, we obtain some explicit solutions of the modification system. The solution of this system can be changed into the solution of the ILLG equation

Summary

Introduction

Long-wavelength spin motions in diverse ferromagnetic structures are commonly described by the Landau–Lifshitz–Gilbert equation (or the LLG equation), which was first derived by Landau and Lifshitz [14]. Li and Wang [16] proved that blowup occurs in some specific format of the inhomogeneous term Their did not provide an exact form of the solution of the inhomogeneous Schrödinger map equation. 3. At the same time, we employ this form to construct an implicit solution of the Schrödinger map equation in Sect. Based on the smallness initial condition, we can prove the global existence of weak (or even smooth) solutions of the Schrödinger map equation (or even ILLG). From these tables we can see the exact form of variable separation solution, inhomogeneity terms, and decay rates.

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Conclusions