The Difference Format of Landau-Lifshitz Equation in Two-dimensional Case

In this paper, the author considers a difference scheme of Laudau-Lifshitz equation (LL for short) and modulus of u n j which are constantly remaining equal to 1. Using this iteration format error which is ordered to t / 2h 2 , the author comes to a conclusion based on several initial simulations. According to some conditions, the author gives the numerical solution, the examples of exact solution and the error comparisons of the solutions.


INTRODUCTION
The continuous Heisenberg spin chain has aroused considerable interest among physicists in the classical study of one-dimensional motion of ferromagnetic chain. In 1907, Weiss explained the property of the ferromagnetic bodies through a study about elements of the hypothesis. In 1928, Heisenberg explained the molecular field theory which is proffered by Weiss. In 1935, L. D. Landau When the external magnetic field is vanish, the L-L equation is shown as follows: ) ( . It performs a fundamental role in the research of non-equilibrium magnetism.
A lot of work on the Landau-Lifshitz system of 1-dimensional motion spin chain has been made by some physicists and mathematicians.
The energy critical Schrodinger map is related to the Landau-Lifshitz equation for ferromagnetism, and it belongs to a class of geometric evolution equations including wave maps and the harmonic heat flow. It has attracted a considerable attention in the past ten years. 2 2 So will the solutions of minimization problem on the energy function of Landau-Lifshitz equation be smooth? In 2000, Nai-Heng Chang, J. Shatah and K. Uhlenbeck considered an initial value problem for the two-dimensional cylindrical symmetric Landau-Lifshitz equations [2]. There are some papers to consider the partial regularity and the existence of weak solutions [3], [4], analyze the concentration sets [5] study the relation between weak solution and harmonic map [5] [6] and discuss the stability.
Whether there exists the global smooth solution for Landau-Lifshitz equation is an open problem. Indeed, the multidimensional Landau-Lifshitz equation is hard to solve exactly, even two-dimensional Heisenberg's equations. Only when there is a lack of the external field or the anisotropy field, many authors had attempted Hirota's famous method, Lie algebraic structure, and Backlund transform, Double transform and so on, but exact solutions were still not established.
In 2000-2001, Guo, Han and Yang established several exact solutions of the cylindrical symmetric n-dimension Landau-Lifshitz equation in [7] and [8]. In 2001, Guo and Yang constructed a family of exact disc solutions of the two-dimensional Landau-Lifshitz equation in [9].
In 2003, Yang and Chang obtained an exact transform between with an external field and without an external field [10]. Based on the transform, some sufficient and necessary conditions from the solution with an external field converges to a certain solution without an external field were given [10].
When both external field and anisotropy field are ti-dimensional Landau-Lifshitz equation with an uprush external field and anisotropy field. We gave a method to find explicit solutions and provided some explicit solutions for two-dimensional case [11]. Mu and Liu et. gave a method to find that an external field and anisotropy field such as the solution remained unvarying [12].
In 2011, we presented a method to solve the Landau-Lifshitz equations with the external magnetic field and anisotropy field. Using the method, we obtained a new family of symmetric solutions, that is, the spherical cone symmetric solution. We also showed how those solutions can evolve with time and gave a series of explicit dynamic spherical cone symmetric solutions [13]. The continuous Heisenberg spin chain has aroused considerable interest among physicists. In the classical study of one-dimensional motion of ferromagnetic chain, the so-called Landau-Lifshitz equation [1] for the isotropic Heisenberg chain is a special case in the following system: And the following system with the Gilbert damping term is shown as follows: Where " " u denotes the vector cross product in 3 R , 0 D ! is the Gilbert damping constant, 1 2 3 ( , , ) z z z z : is the unknown vector-valued function. There have been many contributions to study the initial value problem with periodic boundary condition for a class of general systems in the ferromagnetic chain [14]: In this paper, we want to establish sooth solutions of the initial value problems for one class of general systems of the ferromagnetic chain, which takes the form as follows: . It is with the periodic boundary condition and the initial condition as follows: Here the given function 0 ( ) z x satisfies as follows: (11) We also study the initial value problem with the Gilbert damping term as follows: Then we obtain it as follows:

NUMERICAL TESTS
In order to verify the theoretical results, we come to a conclusion by several initial simulations.   graph of solution of (16)  Figure 1 shows the irregular change of initial value.

Web of Conferences MATEC
Observed from Figure 1 and Figure 2, for the 40 space-and-time steps, the numerical solution is a part of the round. And this conclusion may help with the later researches.

Graph of solution of Landau-Lifshitz equation
In 1974, Nakamura and Sasada research the following equation in the case of one-dimensional motion in [15]: They found the following solitary wave. Tjon and Wright also found some solutions based on non-vanishing external magnetic field in 1977 [16]. .Therefore, we can conclude that the method is stable. Figure 3 We can conclude that the graphs will be blowup when time 1 2 t , thus we will improve and change designs of the numerical differential scheme. When the differential scheme is unstable, it will not be computed in the long term. Figure 4 is the numerical solution and the exact solution error of (16) without damping when ; Figure 5 shows the graph of solution of (15) with damping. Figure 6 is the graph of solution of (13) with damping; Figure 7 displays the numerical solution and exact solution. The graph of solution of (7) is without damping when 0.0001 t '  Web of Conferences MATEC