Abstract
We used the complex method and the exp(-)-expansion method to find exact solutions of the (2+1)-dimensional mKdV equation. Through the maple software, we acquire some exact solutions. We have faith in that this method exhibited in this paper can be used to solve some nonlinear evolution equations in mathematical physics. Finally, we show some simulated pictures plotted by the maple software to illustrate our results.
1. Introduction
Nonlinear science is basic science to study the generality of nonlinear phenomena. It is a comprehensive discipline which has been gradually developed by various branch disciplines characterized by nonlinearity since the 1960s [1]. It was known as the “Third Revolution” of Natural Science in the 20th century. The scientific community believes that the research of nonlinear science has not only great scientific significance but also broad application prospects. It involves almost all fields of natural science and social science, including engineering application, basic physical research, biological research, control theory, and management [2, 3]. And the nonlinear science is changing people’s traditional view of the real world. It is more and more important to find the exact solution of the nonlinear evolution equation. Therefore, a variety of solutions have emerged.
In 1967, Gardner and others [4] first proposed scattering inversion method for KdV equation. Since then, many methods and techniques for constructing nonlinear partial differential equations have been gradually proposed for seeking soliton solutions [5–7], such as Bäcklund transform [8], Hirota method [9–11], and Darboux transform [12]. But these methods are complex and difficult to use in solving processes. With the deepening of research and the continuous development of mathematical computing software, in recent years, some analytical tools and direct algebraic methods have gradually emerged, such as fixed point theorems [13–16], variational methods [17–20], topological degree method [21–24], homogeneous balance method [25], tanh function method [26, 27], Jacobi elliptic function method [28, 29], F-expansion method [30, 31], and -expansion method [32–34].
In 2012, Alejo [35] got some numerical results which showed a new family of solutions of the geometric mKdV equation. In 2014, Huang Y, Wu Y, and Meng F, et al. [36] used the complex method to get the meromorphic solutions of complex combined KdV-mKdV equation. Singh [37] used the Jacobian elliptic function expansion method to get the exact solutions of Wick-type stochastic Kersten-Krasil’ shchik coupled KdV-mKdV equations.
Consider the following:
In (1)(see [38]), and are constants.
Submitting , , into (1) and after integrating, we get
In order to get the exact solution of mKdV equation, we use the complex method which was suggested by Yuan et al. [39–42] to get solutions of (2), and then we also get some exact solutions by the exp-expansion method.
Theorem 1. By using the complex method, we suppose that and then all meromorphic solutions belong to the class W. And we found that there will be three forms of solutions of (2):
(1) The Elliptic Function Solutionsin (3), , , and and are arbitrary constants.
(2) The Simply Periodic Solutionsin (4), , , , and .
And the other solution is in (5), , .
(3) The Rational Function Solutionsandin (6), , or in (7), , .
Theorem 2. By using -expansion method, there will be three forms of solutions of (2).
If , , If , , If , , If , , If , ,
2. Preliminary Lemmas, Complex Method and -Expansion Method
2.1. Introduction of Complex Method
For the introduction of complex method, we have to know some concepts and symbols.
Lemma 3 (see [34]). First we set , , , . Then we can get a differential monomial by and are regarded as the weight and degree of , separately.
The differential polynomial can be defined as follows: In (23), are constants, and is a finite index set.
The total weight and degree of are marked as and , separately.
Considering the complex, ordinary differential equations In (24), , are constants, and .
We take , and we regard the meromorphic solutions of (24) to have one or more poles. We can say that (24) is satisfied the condition, where means that the equation has distinct meromorphic solutions and means that their multiplicity of the pole at is .
It is difficult for us to find the condition of (24), so we need a method to find the weak condition showed as follows.
To find out the weak condition of (24), we need to substitute Laurent series into (24); then we can find out the distinct Laurent singular parts as below:
Given two complex numbers , , and , are discrete subset , which is isomorphic to . Let the discriminant and
A meromorphic function means that is holomorphic in the complex plane except for poles. is the Weierstrass elliptic function [43, 44] with invariants and .
If is an elliptic function, or a rational function of , or a rational function of , then we say that the meromorphic function belongs to the class .
Weierstrass elliptic function is a meromorphic function with double periods and defined as which satisfies the following:and in (29), and .
Or alternating (29) to the form in (30),
Contrarily, given two complex numbers and and , then there will have double periods Weierstrass elliptic function which the solutions will possess.
In 2009, Eremenko [45] et al. investigated the -order Briot-Bouquet equation (BBEq) as follows:
In (31), are constant coefficients polynomials, . For the order BBEq, there are the following lemmas.
Lemma 4 (see [39–42]). Let , and . Considering that a -order Briot-Bouquet equation satisfies weak condition, then all the meromorphic solutions will belong to the class . For some values of parameters, if the solution exists, then other meromorphic solutions will form a one-parametric family . Furthermore, it can be written as the following forms of each elliptic solution with pole at : In (33), are given by equation (23), and , .
Each rational function solution can be show as the following form: with distinct poles of multiplicity .
Every simply periodic solution is a rational function of . has distinct poles of multiplicity and can be show as the following form:
Lemma 5 (see [43, 44]). Weierstrass elliptic functions have two successive degeneracies and addition formula:
(I) Degeneracy to simply periodic functions (i.e., rational functions of one exponential ) according to if one root is double ().
(II) Degeneracy to rational functions of according to if one root is triple ().
(III) Addition formula By the above lemma and results, we introduce complex method to find exact solutions of some PDEs. The detailed five steps are as follows:
(1) Put the transform into a given PDE to produce a nonlinear ODE.
(2) Put (25) into (24) or (32) to find out the weak condition.
(3) By determinant relation equation (33)–(35) we will, respectively, find the elliptic, rational and simply periodic solutions of (24) or (32) with pole at .
(4) By Lemmas 3 and 4, we obtain meromorphic solutions and the addition formula.
(5) Put the inverse transform into these meromorphic solutions , all exact solutions of the original PDE will be found.
More details of complex method can be found in [46–50].
2.2. Introduction of exp-Expansion Method
Consider that a nonlinear partial differential equation (PDE) in the following form:In (39), P is a polynomial with an unknown function and its derivatives in which nonlinear terms and highest order derivatives are involved. And it can be processed as follows.
Step 1. Insert the traveling wave transform into (39) alternating it to the following ordinary differential equation (ODE): and in (40), is a polynomial of and its derivatives.
Step 2. Regarding that (40) has the following traveling wave solution: in (41), are constants and will be determined later and and satisfies the ODE as follows:
Equation (42) has different style solutions as follows:
If , then
If , then
If , then
If , then
If , then
In above equations, , , and are constants and will be determined later and is an arbitrary constant. We consider the homogeneous balance between nonlinear terms and highest order derivatives of (40), so we can find the positive integer .
Step 3. Putting (41) into (40) and accounting the function , we get a polynomial of . Calculating all the coefficients of the same power of to zero and then we get a set of algebraic equations. By solving the algebraic equations, we get the values of , , and , and then we put these into (34) along with(43)-(49) to get the determination of the solutions of (39).
3. Proof of Theorems 1 and 2
3.1. Proof of Theorem 1
Putting (25) into (2), we have , , , , , , and and is arbitrary constant.
Because (2) satisfies weak condition and is a two-order mKdv equation, (2) satisfies the dominant condition. By Lemma 4, we know that all meromorphic solutions of (2) belong to . Now we will give the forms of all meromorphic solutions of (2).
By (2), we infer the indeterminant rational solutions of (2) with pole at that
Putting into (2), we get two classes: one is in (51), .
And the other one is in (52), .
All rational solutions of (2) are as follows:and
In order to get simply periodic solutions, we set and then put into (2). We get
By putting into (55), we get in (57), .
And the other solution is in (58),
So for , all simply periodic solutions of (2) are gotten, which are and
So all simply periodic solutions of (2) are gotten: in (61), , , , and .
And the other solution is in (62), .
By (33) of Lemma 4, we have indeterminant relations of elliptic solutions of (2) with the pole at in (63), . Applying the conclusion II of Lemma 5 to and noting that the results of rational solutions obtained above, we deduce that , , and . Then we get that in (64), .
So all elliptic function solutions of (2) are in (65), . Making use the addition of Lemma 5, we can rewrite it to the form In (66), , , and and are arbitrary constants.
3.2. Proof of Theorem 2
Taking the homogeneous balance between and in (2) we getin (67), and are constants which need to be determined, and satisfies equation , whereas and are arbitrary constants.
From (67), we insert , , into (2) and sort out the coefficients of to zero, and then we obtain in (68), and .
By putting (68) into (67), we getor
We apply (43)–(49) into (69) and (70), respectively, to obtain traveling wave solutions of the mKdV equation as follows.
If , ,
If , ,
If , ,
If , ,
If , ,
4. Computer Simulations
In this section, we will show some computer simulation pictures to illustrate some results. Considering , , , the simply periodic solutions are shown in Figure 1, and the rational function solutions are shown in Figure 2. And through the exp(-)-expansion method, we get some other simply periodic solutions. We take the solutions to further analyze their properties by Figure 3.
4.1. The Physical Significance of the Figures
Figures 1, 2, and 3 shows Waveform Graphs of several functions at different times. We can see that there will be some distinct generation poles shown in Figures.
5. Conclusions
It can be seen from the above analysis that the complex method and exp-expansion method are powerful tools for solving the exact solutions of nonlinear evolution equations. The general meromorphic solutions of (2+1)-dimensional mKdV equation are obtained by the complex method, and we found eight solutions of (2+1)-dimensional mKdV equation. Using exp-expansion method, we also find fourteen solutions of (2+1)-dimensional mKdV equation. By comparing with the two methods, we find more solutions by exp-expansion method, while we can say that the solutions of the elliptic function are only obtained by the complex method.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors typed, read, and approved the final manuscript.
Acknowledgments
The work presented in this paper is supported by the Plateau Disciplines in Shanghai. Also this work was supported by Leading Academic Applied Mathematical of Shanghai Dianji University (16JCXK02) and Humanity and Social Science Youth foundation of Ministry of Education (18YJC630120).