Abstract
In this paper, we study the different types of new soliton solutions to the Landau–Lifshitz equation with the aid of the auxiliary equation method. Then, we get some special soliton solutions for this equation. Without the Gilbert damping term, we present a travelling wave solution with a finite energy in the initial time. The parameters of the soliton envelope are obtained as a function of the dependent model coefficients.
1. Introduction
Nonlinear partial differential equations have different types of equations; one of them is the Landau–Lifshitz equation that is relevant to the classical and quantum mechanics. Nonlinear evolution equations (NEEs) which describe many physical phenomena are often illustrated by nonlinear partial differential equations. So, the exact solutions of NLPDE are explored in detail in order to understand the physical structure of natural phenomena that are described by such equations. A variety of powerful methods have been used to study the nonlinear evolution equations, for the analytic and numerical solutions. Some of these methods, the Riccati Equation method [1], Hirota’s bilinear operators [2], exponential rational function method [3], the Jacobi elliptic function expansion [4], the homogeneous balance method [5], the tanh-function expansion [6], first integral method [7, 8], the subequation method [9], the exp-function method [10], the Backlund transformation, and similarity reduction [11–29], are used to obtain the exact solutions of NLPDE.
In physics, the Landau–Lifshitz–Gilbert equation, named for Lev Landau, Evgeny Lifshitz, and T. L. Gilbert, is a name used for a differential equation describing the processional motion of magnetization in a solid. It is a modification by Gilbert of the original equation of LLG equation (see [30]) can be written down as here, , denotes the cross product. The term multiplying with represents the exchange interaction, while the term denotes the Gilbert damping term. Especially, two extreme cases of (1) ( and , respectively) include as special cases the well-known Schrödinger map equation and harmonic map heat flow, respectively. The well-posedness problem of the LL(G) equation are intensively studied in mathematics, to list a few, in 1986, of the weak solution of the LL(G) equation. Under the small initial value, the global existence of the solution in different spaces [31–33] was proved. The first progress on the existence of partially regular solutions to the LLG equation was found [30, 33–36]. Even for the small initial data, the exact form of the solution is still unknown. On the other hand, whether the LLG equation admits a global solution will develop a finite time singularity from the large initial data is an open equation.
2. Auxiliary Equation Method
Let us consider a typical nonlinear PDE for , given by
Under the wave transformations of and , Equation (2) becomes an ordinary differential equation given by We assume that the solution of the nonlinear Equation (18) can be presented as in which are all real constants to be determined, the balancing number is a positive integer which can be determined by balancing the highest order derivative terms with the highest power nonlinear terms in Equation (2) and expresses the solutions of the following auxiliary ordinary differential equation: where are real parameters. Equations (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), and (19) with give the following solutions:
Case 1. For
Case 2. For
Case 3. For ,
Case 4. For ,
Case 5. For ,
Case 6. For ,
Case 7. For
Case 8. For ,
Case 9. For
Case 10. For ,
Case 11. For ,
Case 12. For ,
Case 13. For
Case 14. For ,
Stage 2: substituting Equations (4) and (5) into Equation (3) and collecting all terms with the same order of together, we convert the left-hand side of Equation (3) into a polynomial in . Setting each coefficient of each polynomial to zero, we derive a set of algebraic equations for . By solving these algebraic equations, we obtain several cases of variables solutions [15, 37]
Stage 3: by substituting the obtained solutions in stage 2 into Equation (4) along with general solutions of Equations (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), and (19), it finally generates new exact solutions for the nonlinear PDE (1)
3. Travelling Wave Solutions of the LL Equation via Aem
In this section, we construct a travelling wave solution without the Gilbert term [30]. Under the condition of the auxiliary equation method, we consider the following wave transformations: where and are constants undetermined.
Here, we assume Substitute (5) into (1) [30], the separate real and the virtual parts, respectively, as where . (21) and (22) are the nonlinear constant coefficients of ordinary differential equation system with the variable . According to (22), we can obtain a relationship between and : where is the arbitrary constant. If we set =0, we have
Substituting (24) into (21), we get
4. Results
By the auxiliary equation method, the solution of (25) is assumed as
From (5), we have where and are constants. Substituting (26) into (25) along with (27) and comparing the coefficients of alike powers of provides an algebraic system of equations, and solving this set of algebraic equations by used of the Maple, we obtain several case solutions. For example is as follows.
Set 1:
From set 1 and Eq. (26), we have
So, solutions of Equation (5) are obtained in (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), and (19), we have final solutions of Equation (1) and Equation (25) as follows:
In Figure 1, the graphical behavior of solutions for in different values of and is illustrated.
In Figure 2, the graphical behavior of solutions for in different values of and is illustrated.
In Figure 3, the graphical behavior of solutions for in different values of and is illustrated.
Set 2:
From set 2 and Eq. (26), we have
So, solutions of Equation (5) are obtained in (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), and (19), we have final solutions of Equation (1) and Equation (25) as follows:
In Figure 4, the graphical behavior of solutions for in different values of and is illustrated.
In Figure 5, the graphical behavior of solutions for in different values of and is illustrated.
For more convenience and understanding, the graphical behavior of the answers is considered (see Figures 1–5).
5. Conclusions
This paper derived new optical soliton solutions of the Landau–Lifshitz equation, which describe the propagation of ultrashort pulses in nonlinear optical fibers by using the auxiliary equation method. We boldly say that the work here is valuable and may be beneficial for studying in other nonlinear science. The exact solutions obtained from the model equations provide important insight into the dynamics of solitary waves. The solutions obtained in this paper have not been reported in the old research.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.