^{1}

^{1}

^{1}

^{1}

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.

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 [

In 1992, Vakhnenko [

In 1998, Vakhnenko and Parkes [

Substituting traveling wave transform

If a meromorphic function

If

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

Let

Consider the following differential equation:

Let

By substituting the Laurent series

Given two complex numbers

A meromorphic function

In 2009, Eremenko et al. [

Let

Each rational function solution has

Each simply periodic solution has

Weierstrass elliptic functions

Substituting (

Multiplying (

Therefore, (

By (

Substituting

Therefore, we can determine that

So the rational solutions of (

To obtain simply periodic solutions, let

Substituting

So simply periodic solutions of (

From (

Putting

Therefore, the elliptic solutions of (

Applying the addition formula, we can rewrite it as

In this section, we illustrate our main results by some computer simulations. We carry out further analysis to the properties of simply periodic solutions

For

For

The solution of the Vakhnenko-Parkes equation corresponding to

The solution of the Vakhnenko-Parkes equation corresponding to

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

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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.

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.

^{'}/