Some Blow-Up and Smooth Solutions about Landau-Lifshitz Equation and Their Spatial Curvature Behavior
We construct the exact solution of ()-dimensional space-time Landau-Lifshitz equation (LLE) without the Gilbert term. Under suitable transformations, some exact solutions are obtained in the radially symmetric coordinates and nonsymmetric coordinates. The type of solutions cover the finite-time blow-up solution, smooth solution in time and vortex solution. At the end, some properties about these solutions and their spatial curvature are illustrated by the graphs.
It is well known that Landau-Lifshitz equation (LLE) [1, 2] is one of the most important nonlinear equations in physics which bears a fundamental role in the understanding of nonequilibrium magnetism, just as the Navier-Stokes equation does in that of fluid dynamics. Under the principal assumption of the macroscopic theory of ferromagnetism, the state of a magnetic crystal is described by the magnetization vector , which is a function of position and time . For instance, a subcase of the LLE is as follows: where . , are constants. denotes a vector cross-product. The effective magnetic field is given by the derivative of the magnetic crystal energy with respect to the vector : where is given by the sum of the exchange energy, the anisotropy energy, and the magnetic field energy; that is,
Equation (1.1) exhibits a rich variety of dynamical properties of a spin vector in different backgrounds. In this letter, we discuss the following LLE which vanishes the Gilbert term (i.e., ), anisotropy energy and the magnetic field energy (here, we set ):
In the 1-dimensional motion LLE, the solition solutions have been studied by many physicists and mathematicians, see, for example, in 1976, Lakshmanan et al. constructed a class of solutions  of (1.4). Nakamura and Sasada constructed a solution  of the nonvanishing external magnetic field case in 1974. Tjon and Wright also found some solitons  for such case in 1977.
LLE is nonintegrable  in high dimensions (). Furthermore, as far as we know, many famous direct methods such as Hirota bilinear method and auxiliary function method  are difficult for constructing the exact solution of (1.4). So, we can only find out some particular exact solutions [8, 9] by various direct methods which base on some special ansatz about the solution. In 2000 and 2001, Guo et al. constructed some exact blow-up solutions [10, 11] for 2-dimension radially symmetric LLE. Similarly, for -dimensions LLE, some exact blow-up solutions were constructed in [12, 13]. In this situation, a solution blows up at time if
The present work is organized as follows. We firstly deduce a blow-up solution of (1.4) in Section 2. Then, we construct a vortex solution and some other periodic solutions about LLE in Section 3. In Section 4, some other exact solutions which different from blow-up solution and vortex solution are presented. In Section 6, some solutions and their spatial curvature are illustrated in a graphic way.
2. Blow-Up Solution
We deduce an exact blow-up solution in this section. Under radially symmetric coordinates, (1.4) takes the form: where , .
We will find the explicit solution of (2.1) in the form of where and are functions to be determined, is constant.
Proof. Substituting (2.2) into (2.1), we get
Furthermore, if we insert (2.7) into (2.5) and (2.6), we have
If we settle down on the following ansatzs: where and are the functions about to be determined, and are constants.
From (2.8) and (2.9), we have
To construct the exact solution of the ordinary differential equations (ODEs for short), we can resort to the mathematic software. Concretely, we find out the exact solution of (2.10) as follows: where are any constants.
This completes the proof of Example 2.1.
In the case of ODE, we mention here that deducing process of software tries to solve it using either classification methods or symmetry methods. In these process, the coefficient about in (2.11) will be extend to a general form. It is why the forms of (2.3) and (2.4) are so specific.
For this finite-time blow-up solution (2.2), it is easy to verify the following.
Remark 2.2. It holds that
is independent of , so is a conservation quantity.
When , , it is not difficult to verify . For example, in the case of and , , we can verify that is not bounded as .
For any first orthogonal matrix (the elements of are constants.), is also a blow up solution of (2.1).
3. Vortex Solution and Periodic Solutions
In this section, we concentrate on the solution on the (here, we mean that ). We firstly consider the vortex solution in the radially symmetric coordinates. Here, these time periodic solutions, which under the radially symmetric coordinates are called magnetic vortices or vortex solutions are of particular importance in manifesting interesting physical phenomena and topological structures of LLE. After that, some exact solutions which under the nonradially symmetric coordinates are proposed.
Under the radially symmetric coordinates, we set
So we have Solving (3.3), we have where , , are constants.
Equation (1.4) can be transformed into
Here, we propose the following two subcase solutions of (3.6).
Functional separable solutions of (3.6)
let where , .
Substituting (3.7) into (3.6), we can deduce the following equations:
Introducing two auxiliary function about , ,
According to (3.9) and (3.10), we have
Substituting (3.9)–(3.12) into (3.8), we have
Equating the coefficients of different powers of in (3.13), we obtain a set of algebraic equations:
If we set according to (3.14), functions and are determined by the first-order autonomous ordinary differential equations as follows:
Here, we omit some details about the deducing and present some exact solutions of (3.16). (A) If , , then (B) If , , , solving (3.16), we have (C) If , , , then (D) According to Table 3.1 which is proposed in , we can deduce a solution easily. Defining where is the modulus of Jacobi elliptic function.
Suppose , , , , we have
Another functional separable solution of (3.6)
where and are arbitrary constants.
According to the above derivation, we conclude our main result in the following example.
Example 3.1. (i) (3.1) and (3.4) are a vortex solution of (2.1);
(ii) (3.5), (3.7) (3.17)–(3.19), and (3.21) are exact periodic solutions about (1.4);
(iii) (3.5) and (3.22) are a periodic solution about (1.4).
Remark 3.2. For the solution (3.1) and (3.4),
is independent of . Furthermore, is a conservation quantity. For example, if we set and in (3.23), we have .
For the solutions proposed in (ii)-(iii) in Example 3.1, it is a tedious expression for ( is about the spacial directions) to each of them. Therefore, if we omit the details of , obviously of (ii) and (iii) will adopt the same form as follows: The graphic demonstration will be provided in Section 6.
For any first orthogonal matrix (the elements of are constants.), is also a blow up solution of (1.4) or (2.1).
4. Non-Blow-Up Solutions on the Disc
In Section 1, we propose some blow-up solutions on the disc. However, whether or not there are non-blow-up solutions (here the non-blow-up solution contains discontinuous one) on the disc is not so clear as far as we know. By constructing some solutions, we give a positive answer about it.
According to results which provided in , some exact solutions can be constructed via an explicit transform between LLE with effective magnetic field and the one without external magnetic field. So, if we meet some difficulties in searching the solutions about (1.4), we can firstly try to find out some solutions about (1.1).
Although (4.1) is an exact solution, we can find out some other exact solutions about (2.1) by adding an effective magnetic field. This fact will be demonstrated in the following illustration which concentrates on Proposition 4.4.
Imposing a special anisotropy energy on (1.4), we have the following system: where , are constants.
Under radially symmetric coordinates, (4.2) takes the form: where , .
We deduce a solution of (4.3) in the following example.
Proof. We settle down on the following setting:
where and are functions to be determined.
Similar to the deducing of Example 2.1, if we insert (4.6) into (4.3), we have Solving (4.7), we get where satisfies .
Therefore, we have completed the proof of Example 4.1.
Remark 4.2. For (4.4), we can find out the exact solution of it. Let
it holds that is independent of , so is a conservation quantity.
(iii) When , it will not a difficult work to verify ( is a positive constant). For instant, in the case of and , we can verify that is bounded as . In fact, decay quickly as . We can see this property in Figure 6.
(iv) For any first orthogonal matrix (the elements of are constants.), is also a solution of (4.3).
In order to get the exact solution of (2.1), we firstly propose an important Lemma which borrow from  as follows.
Proposition 4.4. Let
be a transform between the initial problem of the isotropic Landau-Lifshitz equation:
and the initial problem of the LLE with an anisotropic magnetic field:
If then is a solution of if and only if is a solution of .
In Sections 2, and 4, we construct some solutions in which is not a constant. In the original physical model [1, 2] about LLE, LLE is under the constrain of (here, we means that .). However, many papers [10–13] do some investigations about the case that is not a constant. If is not a constant, is (1.4) still a physical significance model? To answer this question, we firstly mention another LLE. Exactly speaking, under radially symmetric coordinates, ()-dimensional inhomogeneous LLE  takes the following form: where is the inhomogeneous term.
Suppose is a solution of (2.1), ,. Furthermore, the relationship between and is in the form of where is a functions, is constant.
Furthermore, suppose , (5.3) can be rearranged to give (for simplicity, we write for here. So we can regard (5.4) as a modification version of (5.1). According to the above deducing, the following theorem holds
One mentions here that if , , (5.3) will adopt the following form:
So, (5.5) will not equal to (5.1) because of the coefficient before the . And one cannot deduce a the solution of (5.1) from (2.1) according to Theorem 5.1. One has to choose another way to construct the exact solution of (5.1) which is now beyond the topic of this paper.
6. Geometric Significance
In this section, we investigate the evolution of solutions and the spacial gradient about them. Five different kinds of solutions are demonstrated in the following notations.(1)Figure 1 display part shape of solutions on solution (2.2)–(2.4) where , , , , , , and . Accordingly, from Figure 2, we can descry the blow up behavior of (2.13) near . Different from Figure 1, here we set in Figure 2.(2)For vortex solution (3.1) and (3.4), as shown in Figure 3, we can see the decaying behavior of square spatial curvature (3.23). Here, we set and , respectively. In Figure 3, begin from , we can see that the shape of (3.23) has only a declining shift of the peak when increases.(3)From Figures 4–5 we can apperceive the periodic behavior of (3.24) which is determined by the periodic function: respectively. A singularity behavior of (3.24) can be seen in Figure 4. (4) Considering the solution (4.1), and setting , and , we draw the picture of where in Figure 6. From Figure 6, we can find out that the energy cannot be bounded near . Setting and , we draw the picture of (4.1) in Figure 7. Amazedly, we point out here that (2.2)–(2.4) will perform a blow-up behavior while (4.1) will not under the infinite energy initial condition.
In the present work, we have obtained the exact solutions of the LLE by using some ansatzs about the solutions. By using the functional separable technique and general Jacobi elliptic-function method, we have obtained the blow-up solution, vortex solution, various periodic solutions, and non-blow-up solutions in the radially symmetric coordinates or not. For a better understanding about the property of the solutions, some solutions and their spatial curvature which cover five different types of solutions have been also illustrated in a graphic way. From Sections 2 and 4, we can see that the solution of Section 2 will perform a blow-up behavior while the solutions in Section 4 will not under the infinite energy initial condition. Similarly, from Section 3, we can find out the same conclusion that the vortex solution will preserve the finite energy while the corresponding periodic solutions will not.
This work is supported by NSFC (Grant no. 11071009), Funding Project for Academic Human Resources Development in Institutions of Higher Leading Under the Jurisdiction of Beijing Municipality (PHR-IHLB 200906103), Subjects and Education Program for Graduate Student-Project for Innovative Human (Ph.D.), and Innovation Fund of Beijing University of Technology-YB201211 (held by P. Zhong).
L. Landau and E. Lifshitz, “On the theory of the dispersion of magnetic permeability in ferromagnetic bodies,” Physikalische Zeitschrift der Sowjetunion, vol. 8, pp. 153–169, 1935.View at: Google Scholar
L. D. Landau and E. M. Lifshitz, “On the theory of the dispersion of magnetic permeability in ferromagnetic bodies,” in Collected Papers, L. D. Landau, Ed., pp. 101–114, Pergamon, New York, NY, USA, 1965.View at: Google Scholar
K. Nakamura and T. Sasada, “Solitons and wave trains in ferromagnets,” Physics Letters A, vol. 48, no. 5, pp. 321–322, 1974.View at: Google Scholar
B. L. Guo, Y. Q. Han, and G. S. Yang, “Exact blow-up solutions for multidimensional Landau-Lifshitz equations,” Advances in Mathematics, vol. 30, no. 1, pp. 91–93, 2001 (Chinese).View at: Google Scholar