Explicit piecewise smooth solutions of Landau-Lifshitz equation with discontinuous external field

Abstract

In this paper, we shall construct some explicit piecewise smooth (global continuous) solutions as well as blow up solutions to the multidimensional Landau-Lifshitz equation, subject to the external magnetic fields being both discontinuous and unbounded. When the external magnetic field is continuous, some explicit exact smooth solutions and blow up solution are also constructed. We also establish some necessary and sufficient conditions to ensure that the solution of multidimensional Landau-Lifshitz equation with external magnetic field converges to the solution of equation without external magnetic field when the external magnetic field tends to zero.

Some New Solutions of the Landau-Lifshitz Equation in Cylindrical Symmetric System

“…According to above Lemma 4.2, some explicit exact solutions (depend only on the radius r and time t) have been obtained by Guo and Yang [15]. But we are interested in explicit dynamic solution with respect to angle θ and time t for the HLZ.…”

Limiting behavior of solutions of multidimensional Landau-Lifshitz equations with second approximation of effective field (I)

“…In 2003, Yang and Chang obtained an exact transform between with and without an external field(see [32]). By the transform, some sufficient and necessary conditions that the solution with an external field converges to a certain solution without an external field are given(see [32,35]).…”

Lie Group Classifications and Stability of Exact Solutions for Multidimensional Landau-Lifshitz Equations

“…In [9] and [10], Guo and Han as well as Yang have also obtained an exact blow up solution for the n-dimensions form. In [11] and [12], Yang considered the relations between (1) and (2).…”