Jacobi Elliptic Function Expansion Method for the Nonlinear Vakhnenko Equation

By using Jacobi elliptic function expansion method, several kinds of travelling wave solutions of Nonlinear Vakhnenko equation are obtained in this paper. As a result, some new forms of traveling wave solutions of the equation are shown, and the numerical simulation with different parameters for the new forms solutions are given.


Introduction
The nature of the world is known by people step by step with many powerful methods. Solving linear equation is the beginning of the process, meanwhile, the results do not well agree with the solution of the linear equation. People overcome the difference and find the nonlinear equation, which is used to describe many phenomena in various physics fields. Many different nonlinear equations appear gradually. Nonlinear Vakhnenko equation is a kind of nonlinear partial differential equation, which is proposed to describe long waves of small amplitudes broadcasting in nonlinear dispersive media. The equation and its variable coefficients play an important role in further study mathematics and knowing the physics of nature. However, obtaining the solutions of the nonlinear equation is a difficult story. With deeply studying the relation of nature, many powerful methods appear [1]- [7]. The nonlinear Vakhnenko equation is investigated in many papers [8] [9] [10] [11]; some new solutions in the form of Jacobi elliptic function are given in this paper, which enrich the kinds of the solution for the equation.
Generally, the nonlinear Vakhnenko equation [12] is read as where the subscripts denote the partial derivatives of x and t. The remaining structure of this paper is organized as follows: Section 2 is a brief introduction to the Jacobi elliptic function and its properties. In Section 3, by implementing the Jacobi elliptic function expansion method, some new traveling wave solutions for nonlinear Vakhnenko are reported. The conclusion is summarized in the last section.

The Jacobi Elliptic Function Expansion Method and Its Properties
The nonlinear partial differential equation with independent variables x and t is generally in the following form ( ) , , , , , , 0 The above equation is a function about ( ) , u x t , the subscripts denote the partial derivatives with x and t, respectively. The wave variable x wt ξ = + is applied to the Equation (2), which is changed into the following ordinary differential equation where , , u u ξ ξξ  denotes the derivative with respect to the same variable ξ . Generally, the function u in terms of Jacobi elliptic function expansion method can be expressed with the first kind of Jacobi elliptic function ( ) where a i is constant parameter; m is determined by balancing the linear term of the highest order derivative with nonlinear term of Equation (3). The three kinds of Jacobi elliptic functions have useful properties [12] [13] [14] [15] [16], which are given as follow: where m (0 < m < 1) is the modulus of the elliptic function. The modulus substantially affects the Jacobi elliptic solutions, which will asymptotically go into hyperbolic functions and trigonometric functions when the modulus 1 m → and 0 m → , respectively. The asymptotical functions are listed in the following

Jacobi Elliptic Function Expansion Method for Nonlinear Vakhnenko Equation
Consider the Nonlinear Vakhnenko equation in the form of Equation (1) with By using m to balance the highest order derivative term and the nonlinear term of Equation (9), we have m = 2 obtained from Equation (9). Then, Equation (4) reduced as is shown in Figure 5.

Conclusion
The nonlinear Vakhnenko equation in the form Equation (1)

Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.