Journal of Applied Mathematics

1. Introduction

Nonlinear partial differential equations (NLPDEs) play a prominent role in different branches of the applied sciences. In recent time, many researchers investigated exact traveling wave solutions of NLPDEs which play a crucial role to reveal the insight of complex physical phenomena. In the past several decades, a variety of effective and powerful methods, such as variational iteration method [1–3], tanh-coth method [4], homotopy perturbation method [5–7], Fan subequation method [8], projective Riccati equation method [9], differential transform method [10], direct algebraic method [11], first integral method [12], Hirota’s bilinear method [13], modified extended direct algebraic method [14], extended tanh method [15], Backlund transformation [16], bifurcation method [17], Cole-Hopf transformation method [18], sech-tanh method [19], -expansion method [20–22], modified -expansion method [23], multiwave method [24], extended -expansion method [25, 26], and others [27–33] were used to seek exact traveling wave solutions of the nonlinear evolution equations (NLEEs).

Recently, He and Wu [34] have presented a novel method called the Exp-function method for searching traveling wave solutions of the nonlinear evolution equations arising in mathematical physics. The Exp-function method is widely used to many kinds of NLPDEs, such as good Boussinesq equations [35], nonlinear differential equations [36], higher-order boundary value problems [37], nonlinear problems [38], Calogero-Degasperis-Fokas equation [39], nonlinear reaction-diffusion equations [40], 2D Bratu type equation [41], nonlinear lattice differential equations [42], generalized-Zakharov equations [43], (3 + 1)-dimensional Jimbo-Miwa equation [44], modified Zakharov-Kuznetsov equation [45], Brusselator reaction diffusion model [46], nonlinear heat equation [47], and the other important NLPDEs [48–51].

In this article, we apply the Exp-function method [34] to obtain the analytical solutions of the nonlinear partial differential equation, namely, (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsev equation.

2. Description of the Exp-Function Method

Consider the general nonlinear partial differential equation The main steps of the Exp-function method [34] are as follows.

Step 1. Consider a complex variable as Now using (2.2), (2.1) converts to a nonlinear ordinary differential equation for where primes denote the ordinary derivative with respect to .

Step 2. We assume that the traveling wave solution of (2.3) can be expressed in the form [34] where , and are positive integers to be determined later, and and are unknown constants. Equation (2.4) can be rewritten in the following equivalent form:

Step 3. In order to determine the values of and , we balance the highest order linear term with the highest order nonlinear term, and, determining the values of and , we balance the lowest order linear term with the lowest order nonlinear term in (2.3). Thus, we obtain the values of , and .

Step 4. Substituting the values of , and into (2.5), and then substituted (2.5) into (2.3) and simplifying, we obtain Then each coefficient is to set, yields a system of algebraic equations for and .

Step 5. We assume that the unknown and can be determined by solving the system of algebraic equations obtained in Step 4. Putting these values into (2.5), we obtain exact traveling wave solutions of the (2.1).

3. Application of the Method

In this section, we apply the method to construct the traveling wave solutions of the (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsev equation. The obtained solutions will be displayed in Figures 1, 2, 3, 4, 5, and 6 by using the software Maple 13.

Figure 1: Periodic solution for , and .

Figure 2: Periodic solution for , and .

Figure 3: Solitons solution for and .

Figure 4: Solitons solution for and .

Figure 5: Solitons solution for and .

Figure 6: Solitons solution for and .

We consider the (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsev equation where is a nonzero constant.

Zayed [52] solved (3.1) using the -expansion method. Later, in article [53], he solved same equation by the generalized -expansion method.

Here, we will solve this equation by the Exp-function method.

Now, we use the transformation (2.2) into (3.1), which yields where primes denote the derivatives with respect to .

According to Step 2, the solution of (3.2) can be written in the form of (2.5). To determine the values of and , according to Step 3, we balance the highest order linear term of with the highest order nonlinear term of in (3.2), that is, and . Therefore, we have the following: where are coefficients only for simplicity; from (3.3), we obtain that To determine the values of and , we balance the lowest order linear term of with the lowest order nonlinear term of in (3.2). We have where are determined coefficients only for simplicity; from (3.5), we obtain Any real values can be considered for and , since they are free parameters. But the final solutions of (3.1) do not depend upon the choice of and .

Case 1. We set and .
For this case, the trial solution (2.5) reduces to Since, , (3.7) can be simplified By substituting (3.8) into (3.2) and equating the coefficients of , with the aid of Maple 13, we obtain a set of algebraic equations in terms of , and And, setting each coefficient of , to zero, we obtain For determining unknowns, we solve the obtained system of algebraic (3.10) with the aid of Maple 13, and we obtain four different sets of solutions.Set 1. We obtain that where is free parameter.Set 2. We obtain that where and are free parameters.Set 3. We obtain that where and are free parameters.Set 4. We obtain that where is free parameter.
Now, substituting (3.11) into (3.8), we obtain traveling wave solution Equation (3.15) can be simplified as where .
If from (3.16), we obtain Substituting (3.12) into (3.8) and simplifying, we get traveling wave solution where .
If is negative, that is, , , and , then from (3.18), we obtain Substituting (3.13) into (3.8) and simplifying, we obtain where .
If , and , (3.20) becomes Substituting (3.14) into (3.8) and simplifying, we obtain where .
If and , (3.22) becomes

Case 2. We set and .
For this case, the trial solution (2.5) reduces to Since, there are some free parameters in (3.24), for simplicity, we may consider that and . Then the solution (3.24) is simplified as Performing the same procedure as described in Case 1, we obtain four sets of solutions.

Set 1. We obtain that where is free parameter.

Set 2. We obtain that where and are free parameters.

Set 3. We obtain that where and are free parameters.

Set 4. We obtain that where is a free parameter.
Using (3.26) into (3.25) and simplifying, we obtain that If , from (3.30), we obtain that where .
Substituting (3.27) into (3.25) and simplifying, we obtain that If is negative, that is, , , and , (3.32) can be simplified as where .
Substituting (3.28) into (3.25) and simplifying, we obtain that If , and , (3.34) becomes where .
Using (3.29) into (3.25) and simplifying, we obtain that If , and , (3.36) becomes where .

Case 3. We set and .
For this case, the trial solution (2.5) reduces to Since, there are some free parameters in (3.38), we may consider , and so that the (3.38) reduces to the (3.25). This indicates that the Case 3 is equivalent to the Case 2. Equation (3.38) can be rewritten as
If we put and into (3.39), we obtain the solution form as (3.8). This implies that the Case 3 is equivalent to the Case 1.
Also, if we consider and , it can be shown that this Case is also equivalent to the Cases 1 and 2.
Therefore, we think that no need to find the solutions again.

It is noted that the solution (3.17) and (3.31) are identical, solution (3.19) and (3.33) are identical, solution (3.21) and (3.35) are identical, and solution (3.23) and (3.37) are identical.

Beyond Table 1, Zayed [52] obtained another solution (3.39). But, we obtain two more new solutions (3.21) and (3.23).

Table 1: Comparison between Zayed [52] solutions and our solutions.

Graphical Representations of the Solutions
The above solutions are shown with the aid of Maple 13 in the graphs.

4. Conclusions

Using the Exp-function method, with the aid of symbolic computation software Maple 13, new exact traveling wave solutions of the (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsev equation are constructed. It is important that some of the obtained solutions are identical to the solutions available in the literature and some are new. These solutions can be used to describe the insight of the complex physical phenomena.

Acknowledgment

This paper is supported by USM short-term Grant no. 304/PMATHS/6310072, and the authors would like to express their thanks to the School of Mathematical Sciences, USM for providing related research facilities. The authors are also grateful to the referee(s) for their valuable comments and suggestions.