Abstract
We derive exact solutions to the Vakhnenko-Parkes equation by means of the complex method, and then we illustrate our main results by some computer simulations. We can apply the idea of this study to related nonlinear differential equation.
1. Introduction and Main Results
Nonlinear differential equations are widely used as models to 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 extensive interest in the study of nonlinear differential equations [1–15].
In 1992, Vakhnenko [16] first presented the nonlinear differential equation and obtained solitary wave solutions to (1). The equation above gives a description of high-frequency waves in the relaxation medium [17].
In 1998, Vakhnenko and Parkes [18] found the soliton solution to the transformed form of (1) as follows: Hereafter (2) is called the Vakhnenko-Parkes equation (VPE). In recent years, many powerful methods for constructing the solutions of VPE have been used, for instance, the Hirota-Backlund transformations method [19], the inverse scattering method [20, 21], the exp-function method [22], and the ()-expansion method [23]. In this article, we would like to use the complex method [24–26] to obtain traveling wave solutions of VPE.
Substituting traveling wave transform into Vakhnenko-Parkes (2), we get Integrating (4) yields where and are constants.
If a meromorphic function is a rational function of , or a rational function of , or an elliptic function, then we say that belongs to the class [27].
Theorem 1. If , then the meromorphic solutions of (5) belong to the class . In addition, (5) has the following classes of solutions.
(I) The Rational Function Solutions where .
(II) The Simply Periodic Solutions where , .
(III) The Elliptic Function Solutions where , , and .
2. Preliminaries
At first, we give some notations and definitions, and then we introduce some lemmas.
Let , , , and then 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,
Let , and assume that the meromorphic solutions of (11) have at least one pole. If (11) has exactly distinct meromorphic solutions, and their multiplicity of the pole at is , then (11) is said to satisfy the condition. It is not easy to verify that the condition of (11) holds, so we need the weak condition as follows.
By substituting the Laurent series into (11), we determine exactly different Laurent singular parts: then (11) is said to satisfy the weak condition.
Given two complex numbers , such that , and let be the discrete subset , and is isomorphic to . Let the discriminant and
A meromorphic function with double periods , , which satisfies the equation where , , and , is called the Weierstrass elliptic function.
In 2009, Eremenko et al. [28] studied the -order Briot-Bouquet equation (BBEq) where are constant coefficients polynomials, . For the -order BBEq, we have the following lemma.
Lemma 2 (see [26, 29, 30]). Let , . If a -order BBEq satisfies the weak condition, then the meromorphic solutions belong to the class . Suppose that 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 as where are determined by (12), , and
Each rational function solution has ≤p distinct poles of multiplicity and is expressed as
Each simply periodic solution has ≤p distinct poles of multiplicity and is expressed as which is a rational function of .
Lemma 3 (see [27, 30]). Weierstrass elliptic functions have an addition formula as below: When , Weierstrass elliptic functions can be degenerated to rational functions according to When , Weierstrass elliptic functions can be degenerated to simple periodic functions according to
3. Proof of Theorem 1
Substituting (12) into (5) we obtain , is an arbitrary constant, , and .
Multiplying (5) by , we get Integrating (24) yields where is an arbitrary constant and is the integrable constant.
Therefore, (25) is a first-order BBEq and satisfies the weak condition. Hence, by Lemma 2, the meromorphic solutions of (25) . It means that the meromorphic solutions of (5) . The forms of the meromorphic solutions to (5) will be given in the following.
By (19), we infer that the indeterminate rational solutions of (5) are with pole at .
Substituting into (5), we have then we get .
Therefore, we can determine that where .
So the rational solutions of (5) are where .
To obtain simply periodic solutions, let , and substitute into (5); then Substituting into (30), we obtain that where .
Substituting into (32), we can get simply periodic solutions to (5) with pole at where .
So simply periodic solutions of (5) are where .
From (18) of Lemma 2, we can express the elliptic solutions of (5) as with pole at .
Putting into (5), we obtain that where and .
Therefore, the elliptic solutions of (5) are where .
Applying the addition formula, we can rewrite it as where , and .
4. Computer Simulations
In this section, we illustrate our main results by some computer simulations. We carry out further analysis to the properties of simply periodic solutions and the rational solutions as in Figures 1 and 2.(1)For , take , and .(2)For , take and .
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5. Conclusions
Employing the complex method, we can easily find exact solutions to some nonlinear differential equation. By this method, we get the meromorphic exact solutions of VPE, and then we obtain the traveling wave solutions to VPE. In of our solutions, let and ; then it will be equivalent to Eq. () in [23]. Simply periodic solutions are new and cannot be degenerated through elliptic function solutions. Our results demonstrate that the complex method is more simpler, and we can apply the idea of this study to related nonlinear evolution equation.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
Yongyi Gu and Wenjun Yuan carried out the design of this paper and performed the analysis. Najva Aminakbari and Qinghua Jiang participated in the calculations and computer simulations. All authors typed, read, and approved the final manuscript.
Acknowledgments
This work was supported by the NSF of China (11271090), the NSF of Guangdong Province (2016A030310257), Young Talents Innovation Project of Guangdong Province (2015KQNCX116), and Joint PHD Program of Guangzhou University and Curtin University.