Generalized and Improved (G′/G)-Expansion Method for (3+1)-Dimensional Modified KdV-Zakharov-Kuznetsev Equation

The generalized and improved -expansion method is a powerful and advantageous mathematical tool for establishing abundant new traveling wave solutions of nonlinear partial differential equations. In this article, we investigate the higher dimensional nonlinear evolution equation, namely, the (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation via this powerful method. The solutions are found in hyperbolic, trigonometric and rational function form involving more parameters and some of our constructed solutions are identical with results obtained by other authors if certain parameters take special values and some are new. The numerical results described in the figures were obtained with the aid of commercial software Maple.

Every method has some restrictions in their implementations. Basically, there is no integrated method which could be utilized to handle all types of nonlinear PDEs. Another powerful and effective method has been presented by Wang et al. [29] to construct exact traveling wave solutions and called the G'=G ð Þ-expansion method. In this method, they employed the second order linear ordinary differential equation (ODE) G''zlG'zmG~0 where l and m are arbitrary constants. Afterwards, several researchers applied this basic method to obtain traveling wave solutions for different nonlinear PDEs [30][31][32][33]. Recently, Zhang et al. [34]  or e m may be zero, but both e {m and e m cannot be zero together and is called the generalized and improved G'=G ð Þ-expansion method. This method could be applied for generating a rich class of traveling wave solutions because, in this method, an additional variable d is applied and it can produce more general and abundant solutions. If d~0, we can obtain the same solutions according to Zhang et al. [34].
The aim of this work is that, we concentrate to find more general and abundant traveling wave solutions of the (3+1)dimensional modified KdV-Zakharov-Kuznetsev equation by implementing the generalized and improved G 0 =G ð Þ-expansion method.

Description of the Method
Let us consider a general nonlinear PDE: A u,u t ,u x ,u y ,u z ,u xt ,u yt ,u zt ,u xy ,u t t ,u xx ,u yy ,::: where u~u x,y,z,t ð Þis an unknown function, A is a polynomial in its arguments and the subscripts stand for the partial derivatives.
The main steps of the method [39] are as follows: Step 1. We suppose the traveling wave variable: where V is the speed of the traveling wave. Using Eq. (2), Eq. (1) is converted into an ordinary differential equation for b j ð Þ : where the superscripts indicate the ordinary derivatives with respect to j: Step 2. According to possibility, Eq. (3) can be integrated term by term one or more times, yielding constant(s) of integration. The integral constant may be zero, for simplicity.
Step 3. Suppose that the traveling wave solution of Eq. (3) can be expressed by a polynomial in where either e {m or e m may be zero, but both e {m and e m cannot be zero simultaneously, e n n~0,+ 1,+ 2,:::,+ m ð Þ and d are arbitrary constants to be determined later, and G~G j ð Þsatisfies the following second order linear ODE: where l and m are constants.
Step 4. To determine the positive integer m, taking the homogeneous balance between the highest order nonlinear terms and the highest order derivatives appearing in Eq. (3). If the degree of b j ð Þ is D b j ð Þ ½ ~m, then the degree of the other expression would be as follows: Step 5. Substituting Eq. (4) and Eq. (5) into Eq. (3) together with the value of m obtained in Step 4 yields polynomials in dzG 0 =G ð Þ m and dzG 0 =G ð Þ {m , m~0, 1,2,::: ð Þ : Collecting each coefficient of the resulted polynomials to zero, we obtain a set of algebraic equations for e n n~0,+ 1,+ 2,:::,+ m ð Þ , d and V : Step 6. Suppose that the value of the constants e n n~0,+ 1,+ 2,:::,+m ð Þ , d and Vcan be found by solving the algebraic equations which are obtained in step 5. Since the general solution of Eq. (5) is well known to us, substituting the values of e n n~0,+ 1,+ 2,:::,+m ð Þ , d and V into Eq. (4), we can obtain more general type and new exact traveling wave solutions of the nonlinear partial differential equation (1).

Application of the Method
In this section, we apply the generalized and improved G 0 =G ð Þexpansion method to establish more general and some new exact traveling wave solutions of the well known (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation.
Let us consider the (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation followed by Zayed [40]: Now, we use the wave transformation Eq. (2) into the Eq. (7), which yields: Eq. (8) is integrable, therefore, integrating with respect j once yields: where C is an integral constant which is to be determined later.
Taking the homogeneous balance between b 3 and b'' in Eq. (9), we obtain m~1.
Therefore, the solution of Eq. (9) is of the form: where e {1 , e 0 and e 1 are constants to be determined. Substituting Eq. (10) together with Eq. (5) into Eq. (9), the lefthand side is converted into polynomials in dzG 0 =G ð Þ m and dzG 0 =G ð Þ {m , m~0, 1,2,::: ð Þ : We collect each coefficient of these resulting polynomials to zero, yielding a set of simultaneous algebraic equations (for simplicity, which are not presented) for e {1 , e 0 , e 1 , d, C and V : Solving these algebraic equations with the help of symbolic computation system Maple 13, we obtain following.
Case 1: where e 0 , l and m are free parameters. Case 2: where e 0 , d, l and m are free parameters. Case 3: where e 0 , l and m are free parameters.
When l 2 {4mw0, we obtain following hyperbolic function solution: where C and D are arbitrary constants, if C and D take particular values, various known solutions can be rediscovered. When l 2 {4mv0, we obtain the trigonometric function solution: where C and D are arbitrary constants, if C and D take particular values, various known solutions can be rediscovered.

Results and Discussion
The higher dimensional modified KdV-Zakharov-Kuznetsev equation has been solved by many authors by implementing different methods. For example, Zayed [40] executed the basic G 0 =G ð Þ-expansion method, Xu [41] used the elliptic equation method, Naher et al. [18] applied the Exp-function method furthermore, they [42] employed the improved G 0 =G ð Þ-expansion method to obtain traveling wave solutions of this mentioned equation. But in this article, we construct more general and new exact traveling wave solutions by applying the generalized and improved G 0 =G ð Þ-expansion method with an additional free parameter d: The obtained solutions would be useful to understand the mechanism of the complicated nonlinear physical phenomena in a wave interaction. Moreover, some solutions are identical with already published results which are described in table 1. Beyond this table, we obtain new exact solutions b2 1 ,b2 2 ,b2 3 ,b2 4 ,b2 5 ,b3 1 ,b3 2 ,b3 3 ,b4 1 ,b4 2 ,b4 3 ,b4 4 and b4 5 which are not established in the previous literature. Also, solutions b1 2 ,b1 4 ,b3 2 ,b3 3 ,b4 2 and b4 4 are depicted in Figures 1, 2, 3

Graphical Presentations of Some Solutions
The graphical presentations of some solutions are illustrated in Figures 1, 2, 3

Conclusions
In this article, the generalized and improved G 0 =G ð Þ-expansion method is implemented to produce plentiful new traveling wave solutions of the (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation. The used method has many advantages: it   is straightforward and concise. Further, the obtained solutions reveal that this method is a promising mathematical tool because it can furnish a different class of new traveling wave solutions with free parameter of distinct physical structures. Subsequently, this prominent method could be more effectively used to solve various nonlinear partial differential equations which regularly arise in science, engineering and other technical arenas.