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Advances in Mathematical Physics
Volume 2018, Article ID 5971646, 9 pages
https://doi.org/10.1155/2018/5971646
Research Article

Exact Traveling Wave Solutions to the (2 + 1)-Dimensional Jaulent-Miodek Equation

1School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou 510320, China
2Institute of Applied Mathematics, South China Agricultural University, Guangzhou 510642, China
3School of Economic and Management, Guangzhou University of Chinese Medicine, Guangzhou 510006, China

Correspondence should be addressed to Yongyi Gu; moc.361@iygnoyugdg and Jianming Lin; moc.621@ilnaugmjl

Received 9 September 2017; Revised 31 March 2018; Accepted 23 April 2018; Published 19 June 2018

Academic Editor: Stephen C. Anco

Copyright © 2018 Yongyi Gu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We derive exact traveling wave solutions to the (2 + 1)-dimensional Jaulent-Miodek equation by means of the complex method, and then we illustrate our main result by some computer simulations. It has presented that the applied method is very efficient and is practically well suited for the nonlinear differential equations that arise in mathematical physics.

1. Introduction and Main Results

Nonlinear differential equations widely describe many important dynamical systems in various fields of science, especially in nonlinear optics, plasma physics, solid state physics, and fluid mechanics. It has aroused widespread attention in the study of nonlinear differential equations [128]. Exact solutions of nonlinear differential equations play an important role in the study of mathematical physics phenomena. Hence, seeking explicit solutions of physics equations is an interesting and significant subject.

In 2001, Geng et al. [29] developed some (2 + 1)-dimensional models from the Jaulent-Miodek hierarchy [30]. Over the past few years, many research results for the (2 + 1)-dimensional Jaulent-Miodek equations have been generated [3134], such as the algebraic-geometrical solutions, the bifurcation and exact solutions, the N-soliton solution, and Multiple kink solutions for the (2 + 1)-dimensional Jaulent-Miodek equations.

In 2012, Zhang et al. [35] studied the following (2 + 1)-dimensional Jaulent-Miodek equation: where are constants, .

Substituting traveling wave transform into (1), and then integrating it we getwhere , and are constants, and is the integration constant. Setting , (3) becomes

We say that a meromorphic function belongs to the class if is an elliptic function, or a rational function of , or a rational function of , . Only these functions can satisfy an algebraic addition theorem which was proved by Weierstrass, so the letter was utilized [36]. In 2006, Eremenko [36] proved that all meromorphic solutions of the Kuramoto-Sivashinsky algebraic differential equation belong to the class . Recently, Kudryashov et al. [37, 38] used Laurent series to seek meromorphic exact solutions of some nonlinear differential equations. Following their work, the complex method was proposed by Yuan et al. [39, 40]. They employed the Nevanlinna value distribution theory to investigate the existence of meromorphic solutions to some differential equations and then obtain the representations of meromorphic solutions to these differential equations [41, 42]. It shows that the complex method has a strong theoretical basis which can proof that meromorphic solutions of certain differential equations belong to the class and obtain exact solutions by the indeterminate forms of the solutions. Besides, this method can be applied to get all traveling wave exact solutions or general solutions of related differential equations [43, 44]. In this article, we would like to use the complex method to obtain exact traveling wave solutions to the (2 + 1)-dimensional Jaulent-Miodek equation.

Theorem 1. If , then the meromorphic solutions of (4) belong to the class . In addition, (4) has the following classes of solutions.
(i) The rational function solutionswhere , . , in the former case, or , in the latter case.
(ii) The simply periodic solutionswhere , . , in the former case, or , in the latter case.
(iii) The elliptic function solutions where , , and are arbitrary.

Theorem 2. If , then traveling wave exact solutions of (3) have the following forms.
(i) The rational function solutions where , , and are integral constants.
(ii) The simply periodic solutionswhere , , and are integral constants.
(iii) The elliptic function solutionswhere is the integral constant, , .

The rest of this paper is organized as follows. Section 2 introduces some preliminary theory and the complex method. In Section 3, we will give the proof of Theorems 1 and 2. Some computer simulations will be given to illustrate our main results in Section 4. Conclusions are presented at the end of the paper.

2. Preliminary Theory and the Complex Method

At first, we give some notations and definitions, and then we introduce some lemmas and the complex method.

Let , , , , andthen is the degree of . Let the differential polynomial be defined by where is a finite index set, and are constants, then is the degree of .

Consider the following differential equation: where , are constants.

Set , and meromorphic solutions of (13) have at least one pole. If (13) has exactly distinct meromorphic solutions, and their multiplicity of the pole at is , then (13) is said to satisfy the condition. It could be not easy to show that the condition of (13) holds, so we need the weak condition as follows.

Inserting the Laurent series into (13), we can determine exactly different Laurent singular parts:then (13) is said to satisfy the weak condition.

Given two complex numbers , , , and let be the discrete subset , and is isomorphic to . Let the discriminant and

A meromorphic function with double periods , , which satisfies the equationin which , , and , is called the Weierstrass elliptic function.

In 2009, Eremenko et al. [45] studied the -order Briot-Bouquet equation (BBEq)where are constant coefficient polynomials, . For the -order BBEq, we have the following lemma.

Lemma 3 (see [37, 40, 46]). Let , and a -order BBEq satisfies the weak condition; then the meromorphic solutions belong to the class . Suppose for some values of parameters such solution exists; then other meromorphic solutions form a one-parametric family , . Furthermore, each elliptic solution with pole at can be written aswhere are determined by (14), , and .
Each rational function solution is expressed aswhich has distinct poles of multiplicity .
Each simply periodic solution is a rational function of and is expressed as which has distinct poles of multiplicity .

Lemma 4 (see [46, 47]). Weierstrass elliptic functions have an addition formula as below: When , Weierstrass elliptic functions can be degenerated to rational functions according to When , it can also be degenerated to simple periodic functions according to

By the above definitions and lemmas, we now present the complex method as below for the convenience of readers.

Step 1. Substitute the transformation defined by into a given partial differential equation (PDE) to yield a nonlinear ordinary differential equation (ODE).

Step 2. Substitute (14) into the ODE to determine whether the weak condition holds.

Step 3. Find out meromorphic solutions of the ODE with a pole at , in which we have integral constants.

Step 4. Obtain meromorphic solutions by Lemmas 3 and 4.

Step 5. Substituting the inverse transformation into the meromorphic solutions, we get the exact solutions for the original PDE.

3. Proof of Main Results

Proof of Theorem 1. Substituting (14) into (4) we have , , , , , and is an arbitrary constant.
Therefore, (4) is a second-order BBEq and satisfies weak condition. Hence, by Lemma 3, we obtain that meromorphic solutions of (4) belong to . We will show meromorphic solutions of (4) in the following.
By (21), we infer that the indeterminate rational solutions of (4) are with pole at .
Substituting into (4), we havewhere and . where and .
So the rational solutions of (4) are where , . , in the former case, or , in the latter case.
To obtain simply periodic solutions, let , and substitute into Eq. (4), then Substitutinginto (30), we obtain that where in the former case, or , in the latter case.
Inserting into (32), we can get simply periodic solutions to (4) with pole at where , in the former case, or , in the latter case.
So simply periodic solutions of (4) are where , . , in the former case, or , in the latter case.
From (20), we have the indeterminate relations to elliptic solutions of (4) with pole at where . Making use of Lemma 4 to , and considering the results obtained above, we infer that , , . So we obtain where .
Thus, the elliptic function solutions of (4) arewhere , , is arbitrary. Applying the addition formula, we can rewrite it aswhere , , and are arbitrary.

Proof of Theorem 2. By Theorem 1, we can obtain the rational function solutions of (3) which are where , , and are integral constants.
The simply periodic solutions of (3) arewhere , , and are integral constants.
The elliptic function solutions of (3) arewhere is the integral constant, , .

4. Computer Simulations

In this section, we illustrate our main results by some computer simulations. We carry out further analysis to the properties of the new solutions as in the following figures.

(1) By employing the complex method, we are able to obtain the rational solutions and of (4). Figure 1 shows shape of solutions for , , , , , and within the interval . Note that they have one generation pole which are showed by Figure 1.

Figure 1: The solution of the (2 + 1)-dimensional Jaulent-Miodek equation corresponding to , (a) , (b) , and (c) .

(2) By applying the complex method, we achieve the simply periodic solutions and of (4). The solutions and come from the hyperbolic function. Figure 2 shows the shape of solutions for , , , , , and within the interval .

Figure 2: The solution of the (2 + 1)-dimensional Jaulent-Miodek equation corresponding to , (a) , (b) , and (c) .

(3) By using the complex method, we are able to get the rational solutions and of (3). Figure 3 shows the shape of solutions for , , , , , , and within the interval . Note that they have one generation pole which are showed by Figure 3.

Figure 3: The solution of the (2 + 1)-dimensional Jaulent-Miodek equation corresponding to , (a) , (b) , and (c) .

(4) By employing the complex method, we obtain the simply periodic solutions and of (3). The solutions and are in terms of the hyperbolic function solution. Figure 4 shows the shape of solutions for , , , , , , and within the interval . It may be observed from Figure 4 that when increases, there would be a delay for the appearance of the peak within the natural topology of traveling wave solution.

Figure 4: The solution of the (2 + 1)-dimensional Jaulent-Miodek equation corresponding to , (a) , (b) , and (c) .

5. Conclusions

In summary, we have utilized the complex method to construct exact solutions of the nonlinear evolution equation. We first show that meromorphic solutions of the (2 + 1)-dimensional Jaulent-Miodek equation belong to the class , and then we obtain the exact traveling wave solutions for this equation. To our knowledge, the solutions in this paper have not been reported in former literature. The simply periodic solutions , and the rational solutions , are not only new but also not degenerated successively by the elliptic function solutions. We expand the results in [32, 33].

Based on the previous works [32, 33], the complex method allows us to confirm that meromorphic solutions of the differential equation belong to the class easily. By the indeterminate forms of the solutions, we can find meromorphic solutions for the differential equation with a pole at ; then we are able to obtain all meromorphic solutions , for the differential equation with an arbitrary pole. The results demonstrate that the applied method is direct and efficient method, which allows us to do tedious and complicated algebraic calculation. We can apply the idea of this study to other nonlinear evolution equations.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the NSF of China (11271090); the NSF of Guangdong Province (2016A030310257); and the Foundation for Young Talents in Educational Commission of Guangdong Province (2015KQNCX116).

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