Some Blow-Up and Smooth Solutions about Landau-Lifshitz Equation and Their Spatial Curvature Behavior

Abstract

We construct the exact solution of (2+1)-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.

1. Introduction

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 𝑉(𝑥,𝑡), which is a function of position 𝑥 and time 𝑡. For instance, a subcase of the LLE is as follows: 𝑉_𝑡=𝛼𝑉×𝐻_eff−𝛽𝑉×𝑉×𝐻_eff,(1.1) where 𝑉=(𝑣₁,𝑣₂,𝑣₃). 𝛼, 𝛽 are constants. × denotes a vector cross-product. The effective magnetic field 𝐻_eff is given by the derivative of the magnetic crystal energy 𝐸 with respect to the vector 𝑉: 𝐻_eff=−𝜕𝐸𝜕𝑉,(1.2) where 𝐸 is given by the sum of the exchange energy, the anisotropy energy, and the magnetic field energy; that is, 𝐸(𝑉)=𝐸_𝑒𝑥(𝑉)+𝐸_𝑎𝑛(𝑉)+𝐸_𝑚𝑓(𝑉).(1.3)

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., 𝛽=0), anisotropy energy and the magnetic field energy (here, we set 𝛼=1): 𝑉_𝑡=𝑉×Δ𝑉_𝑅²,Δ𝑉_𝑅²=𝜕²𝑉𝜕𝑥²¹+𝜕²𝑉𝜕𝑥²².(1.4)

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 [3] of (1.4). Nakamura and Sasada constructed a solution [4] of the nonvanishing external magnetic field case in 1974. Tjon and Wright also found some solitons [5] for such case in 1977.

LLE is nonintegrable [6] in high dimensions (𝑛≥2). Furthermore, as far as we know, many famous direct methods such as Hirota bilinear method and auxiliary function method [7] 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 𝑛-dimensions LLE, some exact blow-up solutions were constructed in [12, 13]. In this situation, a solution 𝑉 blows up at time 𝑇>0 if lim_{𝑡→𝑇−}sup‖∇𝑉(⋅,𝑡)‖_∞=∞.(1.5)

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: 𝑉_𝑡=𝑉×𝑉_𝑟𝑟+1𝑟𝑉×𝑉^𝑟,(2.1) where 𝑟=𝑥²¹+𝑥²², 𝑉=(𝑣₁(𝑡,𝑟),𝑣₂(𝑡,𝑟),𝑣₃(𝑟)).

We will find the explicit solution of (2.1) in the form of 𝑣𝑣₁(𝑡,𝑟)=cos(𝑚(𝑡,𝑟))𝑓(𝑟),₂𝑣(𝑡,𝑟)=sin(𝑚(𝑡,𝑟))𝑓(𝑟),₃(𝑡,𝑟)=𝛾𝑓(𝑟),(2.2) where 𝑚(𝑡,𝑟) and 𝑓(𝑟) are functions to be determined, 𝛾 is constant.

Example 2.1. If 𝑚(𝑡,𝑟)=−143227𝐶³¹𝑟⁴+108𝐶²¹𝑟^8/3𝐶₂+144𝐶₁𝑟^4/3𝐶²²+64𝐶³²𝑡−𝑇+𝐶³𝑓(𝑟)=−81𝐶₃₁,(2.3)𝑟⁴−324𝐶²¹𝑟^8/3𝐶₂−432𝐶₁𝑟^4/3𝐶²²−192𝐶³²𝛾𝐶²¹16𝑟^1/3𝐶²²+24𝐶₁𝑟^5/3𝐶₂+9𝐶²¹𝑟³²,(2.4) where 𝛾,𝐶₁,𝐶₂,𝐶₃ are any constants, 𝑇>0, then (2.2) is an exact blow-up solution of (2.1).

Proof. Substituting (2.2) into (2.1), we get 𝛾𝛾𝑟𝑟scions((𝑚𝑚))𝑚𝑚^2𝑟_2𝑟𝑓−𝛾𝑟cos(𝑚)𝑚_𝑟𝑟𝑓−2𝛾𝑟cos(𝑚)𝑚_𝑟𝑓_𝑟−𝑓𝛾cos(𝑚)𝑚_𝑟+sin(𝑚)𝑚_𝑡𝑟=0,(2.5)𝑓+𝛾𝑟sin(𝑚)𝑚𝑟_𝑟𝑟𝑚𝑓+_𝑟𝑟2𝛾𝑟sin(𝑚)𝑚_𝑟𝑓_𝑟+𝑓𝛾sin(𝑚)𝑚_𝑟+cos(𝑚)𝑚_𝑡𝑟=0,(2.6)𝑓+2𝑟𝑚_𝑟𝑓_𝑟+𝑚_𝑟𝑓=0.(2.7) Furthermore, if we insert (2.7) into (2.5) and (2.6), we have 𝛾𝑟𝑚𝑚_{𝑟𝑟2𝑟}𝑓+2𝑟𝑚_𝑟𝑓_𝑟+𝑚_𝑟𝑓=0,𝑓+𝑚_𝑡=0.(2.8)
If we settle down on the following ansatzs: 𝑚=𝑔(𝑟)𝑓=𝑔𝑡−(𝑟)𝑇𝑗(+𝑟)𝐶,,(2.9) where 𝑔(𝑟) and 𝑗(𝑟) are the functions about 𝑟 to be determined, 𝑇 and 𝐶 are constants.
From (2.8) and (2.9), we have 𝛾𝑟𝑔𝑔^2𝑟_2𝑟𝑔𝑗−𝑔=0,𝑔𝑗+2𝑟𝑔_𝑟𝑔_𝑟𝑗+𝑔𝑗_𝑟+𝑔_𝑟𝑔𝑗=0.(2.10)
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: 𝑔=164𝐶³¹𝑟⁴+116𝐶²¹𝑟^8/3𝐶₂+112𝐶¹𝑟^4/3𝐶²²+127𝐶³²𝑗=20736,𝛾𝐶²¹9𝐶²¹𝑟³+24𝐶₁𝑟^5/3𝐶₂+16𝑟^1/3𝐶²²²,(2.11) where 𝛾,𝐶₁,𝐶₂ 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 𝑟 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. (i) It holds that ||𝑉(𝑡,𝑟)||²=9144𝐶¹𝑟^4/3𝐶²²+108𝐶²¹𝑟^8/3𝐶₂+27𝐶³¹𝑟⁴+64𝐶³²²1+𝛾²𝛾²𝐶⁴¹16𝑟^1/3𝐶²²+24𝐶₁𝑟^5/3𝐶₂+9𝐶²¹𝑟³⁴(2.12) is independent of 𝑡, so ∫^𝑟1_𝑟0𝑟|𝑉|²𝑑𝑟(0<𝑟₀≤𝑟₁<∞) is a conservation quantity.
(ii) When 𝑟₀≤𝑟≤𝑟₁, 𝑇>0, it is not difficult to verify lim_𝑡→𝑇|𝑉_𝑟|²=+∞. For example, in the case of 𝑇=1 and 𝛾=𝐶₁=𝐶₃=1, 𝐶₂=0, we can verify that ||𝑉_𝑟||²=116𝑟⁸−256𝑡+128+128𝑡²𝑟⁶(𝑡−1)²(2.13) is not bounded as 𝑡→1.
(iii) For any first orthogonal matrix 𝐵 (the elements of 𝐵 are constants.), 𝑉𝐵 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 𝑆² (here, we mean that |𝑉|²=1). 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 𝑣𝑣₁(𝑡,𝑟)=𝑓(𝑡)sin(𝑞(𝑟)),₂𝑣(𝑡,𝑟)=1−(𝑓(𝑡))²sin(𝑞(𝑟)),₃(𝑡,𝑟)=cos(𝑞(𝑟)),(3.1)

Submitting (3.1) into (2.1), (2.1) can be transformed into 1−(𝑓(𝑡))²𝑟²𝑑²𝑑𝑟²𝑞(𝑟)+1−(𝑓(𝑡))²𝑑𝑑𝑟𝑞(𝑟)𝑟+𝑑𝑑𝑡𝑓(𝑡)sin(𝑞(𝑟))𝑟²=0.(3.2)

So we have 1−(𝑓(𝑡))²+1𝑟𝐶𝑑𝑑𝑡𝑓(𝑡)=0,₂𝑑²𝑑𝑟²𝑞(𝑟)+𝑑𝑑𝑟𝑞(𝑟)𝑟−𝐶sin(𝑞(𝑟))𝑟²=0.(3.3) Solving (3.3), we have 𝑓(𝑡)=sin𝐶𝑡+𝐶₁or𝑓(𝑡)=cos𝐶𝑡+𝐶₁,𝑞(𝑟1)−=𝑓±²4(,𝑡)ar=ctacnos𝑟𝐶𝑡+_±√𝐶𝐶₁or1−𝑓²(𝑡)=sin𝐶𝑡+𝐶₁,𝜆,(3.4) where 𝐶>0, 𝐶₁, 𝜆 are constants.

We point out there that (3.4) is vortex solution of (2.1). Similarly, if we set (𝐶 is a constant) 𝑣𝑣₁(𝑡,𝑥,𝑦)=cos(𝑡+𝐶)sin(𝑄(𝑥,𝑦)),₂𝑣(𝑡,𝑥,𝑦)=sin(𝑡+𝐶)sin(𝑄(𝑥,𝑦)),₃(𝑡,𝑥,𝑦)=cos(𝑄(𝑥,𝑦)),(3.5)

Equation (1.4) can be transformed into 𝜕²𝑄𝜕𝑥²+𝜕²𝑄𝜕𝑦²=sin𝑄.(3.6)

Here, we propose the following two subcase solutions of (3.6).

(I) Functional separable solutions of (3.6)
let 𝑄(𝑥,𝑦)=±4arctan𝑓𝜉₁𝑔𝜉₂,(3.7) where 𝜉₁=𝑎𝑥+𝑏, 𝜉₂=𝑐𝑦+𝑑.
Substituting (3.7) into (3.6), we can deduce the following equations: 4𝑎²𝑓^𝑔+4𝑎²𝑓^𝑓²𝑔³−8𝑎²(𝑓′)²𝑓𝑔³+4𝑐²𝑔^𝑓+4𝑐²𝑔^𝑓³𝑔²−8𝑐²(𝑔′)²𝑓³𝑔−4𝑓𝑔+4𝑓³g³=0.(3.8)
Introducing two auxiliary function about 𝑓′, 𝑔^, ((𝑓𝑔′′))²₂=𝐴₁𝑓⁴+𝐵₁𝑓³+𝐶₁𝑓²+𝐷₁𝑓+𝐸₁,(3.9)=𝐴₂𝑔⁴+𝐵₂𝑔³+𝐶₂𝑔²+𝐷₂𝑔+𝐸₂.(3.10)
According to (3.9) and (3.10), we have 𝑓^=12𝐷¹+2𝐴₁𝑓³+32𝐵¹𝑔𝑓²+𝐶₁𝑓,(3.11)_=12𝐷²+2𝐴₂𝑔³+32𝐵²𝑔²+𝐶₂𝑔.(3.12)
Substituting (3.9)–(3.12) into (3.8), we have −2𝐵₁𝑎²𝑓⁴𝑔⁴−2𝐵₂𝑐²𝑓³𝑔⁴−4𝐶₁𝑎²+𝐶₂𝑐²−1𝑓³𝑔³−6𝐷₂𝑐²𝑓³𝑔²++88𝐴𝐴₁₂𝑎²−𝐸₂𝑐²𝑓³𝑔−6𝐷₁𝑎²𝑓²𝑔³+6𝐵₁𝑎²𝑓²𝑔𝑐²−𝐸₁𝑎²𝑓𝑔³+6𝐵₂𝑐²𝑓𝑔²+4𝐶₁𝑎²+𝐶₂𝑐²−1𝑓𝑔+2𝐷₂𝑐²𝑓+2𝐷₁𝑎²𝑔=0.(3.13)
Equating the coefficients of different powers of 𝑓𝑔 in (3.13), we obtain a set of algebraic equations: 𝐵𝐶₁=𝐵₂=𝐷₁=𝐷₂=0,₁𝐴𝑎²+𝐶₂𝑐²−1=0,₁𝐴𝑎²−𝐸₂𝑐²=0,₂𝑐²−𝐸₁𝑎²=0.(3.14)
If we set 𝐴=𝐴₁𝑎²,𝐵=𝐶₁𝑎²,𝐶=𝐸₁𝑎²,(3.15) according to (3.14), functions 𝑓=𝑓(𝜉₁) and 𝑔=𝑔(𝜉₂) are determined by the first-order autonomous ordinary differential equations as follows: 𝑓_𝜉1²=𝐴𝑓⁴+𝐵𝑓²+𝐶,𝑔_𝜉2²=C𝑔⁴+(1−𝐵)𝑔²+𝐴,(3.16)
Here, we omit some details about the deducing and present some exact solutions of (3.16). (A) If 𝐴=𝐶=0, 0<𝐵<1, then 𝑓=exp±√𝐵𝜉₁,𝑔=exp±√1−𝐵𝜉₂.(3.17)(B) If 𝐴=0, 𝐵=1, 𝐶>0, solving (3.16), we have 𝑓=√𝐶sinh𝜉₁,𝑔=±1√𝐶𝜉₂.(3.18)(C) If 𝐴=0, 𝐵>1, 𝐶>0, then 𝑓=𝐶𝐵sinh√𝐵𝜉₁,𝑔=𝐵−1𝐶sec√𝐵−1𝜉₂.(3.19)(D) According to Table 3.1 which is proposed in [7], we can deduce a solution easily. Defining 𝑠𝑐(𝜉,𝑅)=𝑠𝑛(𝜉,𝑅)𝑐𝑛(𝜉,R),𝑑𝑠(𝜉,𝑅)=𝑑𝑠𝑛𝑛((𝜉𝜉,,𝑅𝑅)),(3.20) where 𝑅 is the modulus of Jacobi elliptic function.
Suppose 𝑅=√3/2, 𝐴=−3/16, 𝐵=1/2, 𝐶=1, we have 𝑓=𝑠𝑑𝜉₁,√32,𝑔=𝑑𝑠𝜉²,√32.(3.21)

(II)Another functional separable solution of (3.6)
𝑄(𝑥,𝑦)=4arctancot𝐶₁cosh𝑓cosh𝑔,𝑓=cos𝐶¹1+𝐶²²𝑔=sin𝐶₁𝑥−𝐶₂𝑦,1+𝐶²²𝑦−𝐶₂𝑥,(3.22) where 𝐶₁ and 𝐶₂ 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. (i) For the solution (3.1) and (3.4), ||𝑉_𝑟(𝑡,𝑟)||²=16𝑟^2√𝐶−2𝐶𝜆²𝜆²+𝑟^2√𝐶²(3.23) is independent of 𝑡. Furthermore, ∫^∞0|𝑉_𝑟|²𝑟𝑑𝑟 is a conservation quantity. For example, if we set 𝐶=4 and 𝜆=1 in (3.23), we have ∫^∞0|𝑉_𝑟|²𝑟𝑑𝑟=16≫0.
(ii) For the solutions proposed in (ii)-(iii) in Example 3.1, it is a tedious expression for |∇𝑉(𝑡,𝑥,𝑦)|² (∇ is about the spacial directions) to each of them. Therefore, if we omit the details of 𝑄(𝑡,𝑥,𝑦), obviously |∇𝑉(𝑡,𝑥,𝑦)|² of (ii) and (iii) will adopt the same form as follows: ||∇𝑉(𝑡,𝑥,𝑦)||²=𝜕𝜕𝑥𝑄(𝑥,𝑦)²+𝜕𝜕𝑦𝑄(𝑥,𝑦)².(3.24) The graphic demonstration will be provided in Section 6.
(iii) For any first orthogonal matrix 𝐵 (the elements of 𝐵 are constants.), 𝑉𝐵 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 [13], 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).

Similar to Section 2, we can find out an exact solution for (2.1) as follows: 𝑣₁(𝑡,𝑟)=−cos𝐶³𝑡+(3/4)𝑟𝐶₁√𝑟^2/3+𝐶₂𝐶³𝐶¹√𝑟^4/3𝑣₂,(𝑡,𝑟)=−sin𝐶³𝑡+(3/4)𝑟𝐶₁√𝑟^2/3+𝐶₂𝐶³𝐶¹√𝑟^4/3𝑣₃,(𝑟)=−𝐶³𝐶¹√𝑟^4/3,(4.1) where 𝐶₁,𝐶₂,𝐶₃ are any constants.

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: 𝑈_𝑡=𝑈×(Δ𝑈+𝑈𝐽),(4.2) where 𝐽=diag{𝑎,𝑎,𝑏}, 𝑎,𝑏 are constants.

Under radially symmetric coordinates, (4.2) takes the form: 𝑈_𝑡=𝑈×𝑈_𝑟𝑟+1𝑟𝑈×𝑈^𝑟+𝑈×𝑈𝐽,(4.3) where 𝑟=𝑥²¹+𝑥²², 𝑈=(𝑢₁(𝑡,𝑟),𝑢₂(𝑡,𝑟),𝑢₃(𝑡,𝑟)).

We deduce a solution of (4.3) in the following example.

Example 4.1. If 𝑍(𝑟) is the root of 𝑢𝑍(𝑟)⁴−𝐶₁√𝑟𝑍(𝑟)−𝑎+𝑏=0,(4.4)₁(𝑡,𝑟)=cos𝐶³𝑡+∫𝑍(𝑟)²𝑑𝑟+𝐶₂𝐶₃𝑢−𝑍(𝑟)⁴−𝑏+𝑎₂,(𝑡,𝑟)=sin𝐶³𝑡+∫𝑍(𝑟)²𝑑𝑟+𝐶₂𝐶₃𝑢−𝑍(𝑟)⁴−𝑏+𝑎₃,(𝑟)=𝐶³−𝑍(𝑟)⁴−𝑏+𝑎,(4.5) where 𝐶₁,𝐶₂,𝐶₃ are any constants, then (4.4)-(4.5) is a solution of (4.3).

Proof. We settle down on the following setting: 𝑢𝑢₁(𝑡,𝑟)=cos(𝑚(𝑡,𝑟))𝑓(𝑟),₂𝑢(𝑡,𝑟)=sin(𝑚(𝑡,𝑟))𝑓(𝑟),₃(𝑟)=𝑓(𝑟),(4.6) where 𝑚(𝑡,𝑟) and 𝑓(𝑟) are functions to be determined.
Similar to the deducing of Example 2.1, if we insert (4.6) into (4.3), we have 𝛾𝑟𝑚𝑚_{𝑟𝑟2𝑟}𝑓+2𝑟𝑚_𝑟𝑓_𝑟+𝑚_𝑟𝑓=0,𝑓+(𝑏−𝑎)𝑓+𝑚_𝑡=0.(4.7) Solving (4.7), we get 𝑚𝑓(𝑡,(𝑟𝑟))==𝐶₃𝑡+𝐶𝑍₃(𝑟)²𝑑𝑟+𝐶₂,−𝑍(𝑟)⁴−𝑏+𝑎,(4.8) where 𝑍(𝑟) satisfies 𝑍(𝑟)⁴−𝐶₁√𝑟𝑍(𝑟)−𝑎+𝑏=0.
Therefore, we have completed the proof of Example 4.1.

Remark 4.2. (i) For (4.4), we can find out the exact solution of it. Let 𝐺=108𝐶²¹𝑟+127𝑄68=𝑎³−2304𝑎²𝑏+2304𝑎𝑏²−768𝑏³+81𝐶⁴¹𝑟²,𝐺^2/3−48𝑎+48𝑏𝐺^1/3,(4.9)
Then 𝑍(𝑟)=±√612𝑄±112⎷−6𝑄𝐺^2/3−288𝑄𝑎+288𝑄𝑏−72√6𝐶₁√𝑟𝐺^1/3𝐺^1/3𝑄.(4.10)
(ii) it holds that ||𝑈(𝑡,𝑟)||²=2𝐶²³𝐶²¹𝑟𝑍(𝑟)²(4.11) is independent of 𝑡, so ∫^𝑟1_𝑟0𝑟|𝑈|²𝑑𝑟(0<𝑟₀≤𝑟₁<∞) is a conservation quantity.
(iii) When 0<𝑟≤𝑟₁, it will not a difficult work to verify |𝑉_𝑟|²≤𝐶 (𝐶 is a positive constant). For instant, in the case of 𝑏=0 and 𝑎=𝐶₁=𝐶₃=1, we can verify that ||𝑈_𝑟||²=𝑟^1/2𝑍(𝑟)+18+16𝑟²+9𝑟³𝑍(𝑟)²+8𝑟^1/2𝑍(𝑟)+24𝑟^5/2𝑍(𝑟)𝑟³24𝑟^1/2𝑍(𝑟)³+16𝑍(𝑟)²+9𝑟^3/2𝑍(𝑟)+9𝑟(4.12) is bounded as 0<𝑟≤𝑟₁. In fact, |𝑈_𝑟|² decay quickly as 𝑟→+∞. We can see this property in Figure 6.
(iv) For any first orthogonal matrix 𝐵 (the elements of 𝐵 are constants.), 𝑈𝐵 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 [13] as follows.

Lemma 4.3. Let 𝐵=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑏²²+𝑏²³𝑏²¹+𝑏²²+𝑏²³−𝑏¹𝑏₂𝑏²²+𝑏²³𝑏²¹+𝑏²²+𝑏²³−𝑏¹𝑏₃𝑏²²+𝑏²³𝑏²¹+𝑏²²+𝑏²³0𝑏³𝑏²²+𝑏²³−𝑏²𝑏²²+𝑏²³𝑏₁𝑏²¹+𝑏²²+𝑏²³𝑏₂𝑏²¹+𝑏²²+𝑏²³𝑏₃𝑏²¹+𝑏²²+𝑏²³𝐵=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝑏²²+𝑏²³−𝑏¹𝑏₂𝑏²²+𝑏²³−𝑏¹𝑏₃𝑏²²+𝑏²³0𝑏³𝑏²²+𝑏²³−𝑏²𝑏²²+𝑏²³𝑏₁𝑏₂𝑏₃𝐴=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,ℎ₂₂+ℎ₃₂cos(𝜌̃)||𝐻||−ℎ¹ℎ₂cos(𝜌̃)||𝐻||ℎ₂₂+ℎ₃₂−ℎ³sin(𝜌̃)ℎ₂₂+ℎ₃₂−ℎ¹ℎ₃cos(𝜌̃)||𝐻||ℎ₂₂+ℎ₃₂+ℎ²sin(𝜌̃)ℎ₂₂+ℎ₃₂ℎ₂₂+ℎ₃₂sin(𝜌̃)||𝐻||−ℎ¹ℎ₂sin(𝜌̃)||𝐻||ℎ₂₂+ℎ₃₂+ℎ³cos(𝜌̃)ℎ₂₂+ℎ₃₂−ℎ¹ℎ₃sin(𝜌̃)||𝐻||ℎ₂₂+ℎ₃₂−ℎ²cos(𝜌̃)ℎ₂₂+ℎ₃₂ℎ₁||𝐻||||𝐻ℎ²||||𝐻ℎ³||⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(4.13) where 𝐻=(ℎ₁,ℎ₂,ℎ₃)=(ℎ₁,ℎ₂,ℎ₃)𝐵^𝑇, |𝐻|=ℎ²¹+ℎ²²+ℎ²³, and 𝜌̃=|𝐻|∫^𝑡0𝜇(𝑠)𝑑𝑠.
Considering the following two LLEs (ℙ₁) and (ℙ₂): ℙ₁𝜕𝜕𝑉𝑡=𝑉×(Δ𝑉+𝜇(𝑡)𝐻+𝐻(𝑉)),in𝑅^𝑛𝑉∣_𝑡=0×(0,∞),=𝑉₀,in𝑅^𝑛,(4.14) where ∫^𝑡0𝜇(𝑠)𝑑𝑠∈𝐶[0,∞) and 𝜇(𝑡)∈𝐿²(0,∞), 𝐻(𝑉)=∑^3𝑖,𝑗=1𝑏_𝑖𝑗𝑢_𝑖(𝑏₁,𝑏₂,𝑏₃), 𝑏₃≠0, 𝑏_𝑖𝑗=𝛼+(𝛽−𝛼)𝑏^𝑖(𝛽−𝛼)𝑏_𝑖𝑏_𝑗,𝑖=𝑗,ℙ𝑏_𝑗,𝑖≠𝑗.(4.15)₂𝜕𝜕𝑈𝑡=𝑈×Δ𝑈+𝐻¹𝑈∣_𝑡=0(𝑈),in𝑅^𝑛×(0,∞),=𝑈₀,in𝑅^𝑛,(4.16) where 𝐻₁(𝑈)=(0,0,(𝛽−𝛼)(𝑏²¹+𝑏²²+𝑏²³)𝑢₃).
By the transform 𝑉=𝑈𝐵𝐴𝐵, (4.14) is equivalent to (4.16).

Combining Example 4.1 with Lemma 4.3, one has the following Proposition, in which one also adopts the notations of Lemma 4.3.

Proposition 4.4. Let 𝑉=𝑈𝐵𝐴^≈𝐵(4.17) be a transform between the initial problem of the isotropic Landau-Lifshitz equation: ℚ₁𝜕𝜕𝑉𝑡=𝑉×(Δ𝑉),in𝑅^𝑛𝑉∣_𝑡=0×(0,∞),=𝑉₀,in𝑅^𝑛,(4.18) and the initial problem of the LLE with an anisotropic magnetic field: ℚ₂𝜕𝜕𝑈𝑡=𝑈×Δ𝑈+𝐻¹𝑈|_𝑡=0(𝑈),in𝑅^𝑛×(0,∞),=𝑈₀,in𝑅^𝑛,(4.19) where 𝐻₁(𝑈)=(0,0,(𝛽−𝛼)(𝑏²¹+𝑏²²+𝑏²³)𝑢₃), 𝐴=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ℎ₂₂+ℎ₃₂||𝐻||−ℎ¹ℎ₂||𝐻||ℎ₂₂+ℎ₃₂−ℎ¹ℎ₃||𝐻||ℎ₂₂+ℎ₃₂0ℎ³ℎ₂₂+ℎ₃₂−ℎ²ℎ₂₂+ℎ₃₂ℎ₁||𝐻||||𝐻ℎ²||||𝐻ℎ³||⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(4.20)
If 𝛽=𝛼−1+𝑏²³𝑏²³,𝑏₁=𝑏₂=𝑏₃or𝛼=0,𝑏¹=−𝑏₂−b₃,(4.21) then 𝑉 is a solution of (ℚ₁) if and only if 𝑈 is a solution of (ℚ₂).

Proof. According to Lemma 4.3, to prove Proposition 4.4, we just need to set 𝜇(𝑡)𝐻=0 and 𝐻(𝑉)=∑^3𝑖,𝑗=1𝑏_𝑖𝑗𝑢_𝑖(𝑏₁,𝑏₂,𝑏₃)=0 in (4.19).
𝐻(𝑉)=0 if and only if 33𝛼𝛼++((𝛽𝛽−−𝛼𝛼))𝑏𝑏²¹₁+𝑏₁𝑏₂+𝑏₁𝑏₃=0,3𝛼+(𝛽−𝛼)𝑏₁𝑏₂+𝑏²²+𝑏₂𝑏₃=0,𝑏₃+𝑏₂𝑏₃+𝑏²³=0.(4.22) Solving (4.22), we have {𝛽=𝛼(−1+𝑏²³)/𝑏²³,𝑏₁=𝑏₂=𝑏₃} or {𝛼=0,𝑏₁=−𝑏₂−𝑏₃}.
On the other hand, if 𝜇(𝑡)=0, 𝐴 will degenerate as 𝐴^≈.

Remark 4.5. According to Proposition 4.4, we can construct a smooth solution about (2.1) on the disc from (4.17), (4.21), and (4.4)-(4.5).

5. Discussions

In Sections 2, and 4, we construct some solutions in which |𝑉| is not a constant. In the original physical model [1, 2] about LLE, LLE is under the constrain of 𝑉∈𝑆² (here, we means that |𝑉|=𝐶>0.). However, many papers [10–13] do some investigations about the case that |𝑉| is not a constant. If |𝑉| 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, (2+1)-dimensional inhomogeneous LLE [14] takes the following form: 𝑈_𝑡=q(𝑟)𝑈×𝑈_𝑟𝑟+1𝑟𝑈^𝑟+𝑞(𝑟)^𝑟𝑈×𝑈_𝑟,(5.1) where 𝑞(𝑟) is the inhomogeneous term.

Suppose 𝑉=(𝑣₁(𝑡,𝑟),𝑣₂(𝑡,𝑟),𝑣₃(𝑡,𝑟)) is a solution of (2.1), 𝑍=(𝑧₁(𝑡,𝑟),𝑧₂(𝑡,𝑟),𝑧₃(𝑡,𝑟)),𝑧²¹(𝑡,𝑟)+𝑧²²(𝑡,𝑟)+𝑧²³(𝑡,𝑟)=𝐶>0. Furthermore, the relationship between 𝑉 and 𝑍 is in the form of 𝑣𝑣₁(𝑡,𝑟)=𝑧₁(𝑡,𝑟)𝐺(𝛼𝑟),₂𝑣(𝑡,𝑟)=𝑧₂(𝑡,𝑟)𝐺(𝛼𝑟),₃(𝑡,𝑟)=𝑧₃(𝑡,𝑟)𝐺(𝛼𝑟),(5.2) where 𝐺(𝛼𝑟) is a functions, 𝛼 is constant.

Substituting (5.2) into (2.1), we get 𝑍_𝑡=𝐺(𝛼𝑟)𝑍×𝑍_𝑟𝑟+1𝑟𝑍^𝑟+2𝛼𝐺(𝛼𝑟)^𝑍×𝑍_𝑟,(5.3) where “′” is a derivative about 𝛼𝑟.

Furthermore, suppose 𝛼=1/2, (5.3) can be rearranged to give (for simplicity, we write 𝐺 for 𝐺(⋅) here. 𝑍_𝑡=𝐺𝑍×𝑍_𝑟𝑟+1𝑟𝑍^𝑟+𝐺^𝑍×𝑍_𝑟.(5.4) So we can regard (5.4) as a modification version of (5.1). According to the above deducing, the following theorem holds

Theorem 5.1. Under the constrain (5.2), 𝑉=(𝑣₁(𝑡,𝑟),𝑣₂(𝑡,𝑟),𝑣₃(𝑡,𝑟)) is a solution of (2.1) if and only if 𝑍=(𝑧₁(𝑡,𝑟),𝑧₂(𝑡,𝑟),𝑧₃(𝑡,𝑟))(𝐺(⋅)=𝐺(𝑟/2)) is a solution of (5.4) where 𝑧²¹(𝑡,𝑟)+𝑧²²(𝑡,𝑟)+𝑧²³(𝑡,𝑟)=𝐶>0.

One mentions here that if 𝛼=1, 𝐺(𝛼𝑟)=𝐺(𝑟), (5.3) will adopt the following form: 𝑈_𝑡=𝐺(𝑟)𝑍×𝑍_𝑟𝑟+1𝑟𝑍^𝑟+2𝐺(𝑟)^𝑟𝑈×𝑈_𝑟.(5.5)

So, (5.5) will not equal to (5.1) because of the coefficient 2 before the 𝐺(𝑟)_𝑟[𝑍×𝑍_𝑟]. 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 𝑟∈[1,3], 𝑡∈[0,9/10], 𝑇=1, 𝛾=1, 𝐶₁=1, 𝐶₂=0, 𝐶₃=1 and 𝐶₄=−1. Accordingly, from Figure 2, we can descry the blow up behavior of (2.13) near 𝑇=1. Different from Figure 1, here we set 𝑟∈[3/5,10] 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 𝐶=4 and 𝜆=0,1,2,3,4, respectively. In Figure 3, begin from 𝜆=1, we can see that the shape of (3.23) has only a declining shift of the peak when 𝜆 increases.(3)From Figures 4–5 we can apperceive the periodic behavior of (3.24) which is determined by the periodic function: 𝑄(𝑥,𝑦)=4arctan𝑠𝑑𝑥+1,√32𝑑𝑠𝑦+1,√3𝑄(𝑥,𝑦)=4arctan⎛⎜⎜⎝cot(1)cosh√2,(6.1)2/2cos(1)(𝑥−𝑦)cosh√2/2sin(1)(𝑦−𝑥)⎞⎟⎟⎠,(6.2)     respectively. A singularity behavior of (3.24) can be seen in Figure 4. (4) Considering the solution (4.1), and setting 𝐶₁=1, 𝐶₂=1 and 𝐶₃=1, we draw the picture of |𝑈_𝑟|²=(1/9)(9𝑟^8/3+8/𝑟^10/3) where 𝑟∈[1/2,25] in Figure 6. From Figure 6, we can find out that the energy cannot be bounded near 𝑡=0. Setting 𝑡∈[0,3] and 𝑟∈[1/10,3], 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.

Figure 1 

(2.2)–(2.4) with 𝑟∈[1,3], 𝑡∈[0,9/10].

Figure 2 

Part shape of (2.13) with 𝑇=1.

Figure 3 

(3.23) as 𝜆=0,1,2,3,4, respectively.

Figure 4 

Part shape of periodic function (3.24) and (6.1).

Figure 5 

Part shape of periodic function (3.24) and (6.2).

Figure 6 

(4.1) with 𝑟∈[1/2,25] near 𝑡=0.

Figure 7 

Part shape of (4.1) as 𝑡∈[0,3] and 𝑟∈[1/10,3].

7. Conclusions

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.

Acknowledgments

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).