Study of Nonlinear Evolution Equations to Construct Traveling Wave Solutions via the New Approach of the Generalized ) / ( GG ′-Expansion Method

Abstract Exact solutions of nonlinear evolution equations (NLEEs) play very important role to make known the inner mechanism of compound physical phenomena. In this paper, the new generalized ( G G / ′ )-expansion method is used for constructing the new exact traveling wave solutions for some nonlinear evolution equations arising in mathematical physics namely, the (3+1)-dimensional Zakharov-Kuznetsov equation and the Burgers equation. As a result, the traveling wave solutions are expressed in terms of hyperbolic, trigonometric and rational functions. This method is very easy, direct, concise and simple to implement as compared with other existing methods. This method presents a wider applicability for handling nonlinear wave equations. Moreover, this procedure reduces the large volume of calculations.


Introduction
The investigation of the travelling wave solutions for nonlinear partial differential equations plays an important role in the study of nonlinear physical phenomena.
Nonlinear wave phenomena appears in various scientific and engineering fields, such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, chemical physics and geochemistry. Nonlinear wave phenomena of dispersion, dissipation, diffusion, reaction and convection are very important in nonlinear wave equations. In the past several decades, new exact solutions may help to find new phenomena. A variety of powerfull methods, such as the ansatz method [1,2], the Adomian decomposition method [3], the Darboux transformation method [4], the Backlund transformation method [5], the inverse scattering transform [6], the wave of translation method [7], the Jacobi elliptic function method [8][9][10][11], the Exp-function method [12][13][14][15][16][17], the extended tanh method [18,19], the sine-cosine method [20], the Cole-Hopf transformation [21] The rest of the article is organized as follows: In Section 2, the description of the new generalized ) / ( G G′ expansion method is given. In Section 3, we apply the method to obtain the traveling wave solution of the (3+1)-dimensional Zakharov-Kuznetsov equation and the Burgers equation and also give some discussion, Graphical representation and Table. In Sections 4, we give some conclusions.

Materials and Methods
Let us consider a general nonlinear PDE in the form 0 ) , , , , , , ( =  xx tx tt x t u u u u u u P (1) where ) , ( t x u u = is an unknown function, P is a polynomial in ) , ( t x u and its derivatives in which highest order derivatives and nonlinear terms are involved and the subscripts stand for the partial derivatives.
Step 1: We combine the real variables x and t by a where V is the speed of the traveling wave. The traveling wave transformation (2) converts Eq. (1) into an ordinary differential equation (ODE) for ) (η u u = : where Q is a polynomial of u and it derivatives and the superscripts indicate the ordinary derivatives with respect to η . Step 2: According to possibility, Eq. (3) can be integrated term by term one or more times, yields constant(s) of integration. The integral constant may be zero for simplicity.
Step 3: Suppose the traveling wave solution of Eq. (3) can be expressed as follows: and d are arbitrary constants to be determined later and ) (η H is given by satisfies the following auxiliary nonlinear ordinary differential equation: (6) where the prime stands for derivative with respect to η ; A , B , C and E are real parameters.
Step 4: To determine the positive integer N , taking the homogeneous balance between the highest order nonlinear terms and the derivatives of the highest order appearing in Eq. (3).
Step 5: Substitute Eq. (4) and Eq. (6) including Eq. (5) into Eq. (3) with the value of N obtained in Step 4, we obtain polynomials in . Then, we collect each coefficient of the resulted polynomials to zero yields a set of algebraic equations for Step 6: Suppose that the value of the constants , d and V can be found by solving the algebraic equations obtained in Step 5.
Since the general solution of Eq. (6) is well known to us, inserting the values of , d and V into Eq. (4), we obtain more general type and new exact traveling wave solutions of the nonlinear partial differential equation (1).
Using the general solution of Eq. (6), we have the following solutions of Eq. (5): Family 4: When

Applications of the Method
In this section, the method is used to construct some new traveling wave solutions for the (3+1)-dimensional Eq. (13) is integrable, therefore, integrating with respect to η once yields: , 0 3 2 where P is an integration constant which is to be determined.
Taking the homogeneous balance between highest order nonlinear term 2 S and linear term of the highest order S ′ in Eq. (14), we obtain 1 = N . Therefore, the solution of Eq. (14) is of the form: We collect each coefficient of these resulted polynomials to zero yields a set of simultaneous algebraic equations (for simplicity, the equations are not presented) for 0 α , 1 α , 1 β , d , P and V . Solving these algebraic equations with the help of computer algebra, we obtain following: Set 1: , C E are free parameters.
Set 2: Set 3: , C E are free parameters.
For set 1, substituting Eq. (16) into Eq. (15), along with Eq. (7) and simplifying, yields following traveling wave solutions, if  (15), together with Eq. (9) and simplifying, our obtained solution becomes: (15), along with Eq. (10) and simplifying, we obtain following traveling wave solutions, if Substituting Eq. (16) into Eq. (15), together with Eq. (11) and simplifying, our obtained exact solutions become, if Again for set 2, substituting Eq. (17) into Eq. (15), along with Eq. (7) and simplifying, our traveling wave solutions become, if (15), along with Eq. (8) and simplifying yields exact solutions, if (15), along with Eq. (9) and simplifying, our obtained solution becomes: (15), together with Eq. (10) and simplifying, yields following traveling wave solutions, if Substituting Eq. (17) into Eq. (15), along with Eq. (11) and simplifying, our exact solutions become, if Similarly, for set 3, substituting Eq. (18) into Eq. (15), together with Eq. (7) and simplifying, yields following traveling wave solutions, if  (15), along with Eq. (9) and simplifying, our obtained solution becomes: (4 ) ( tan( )) 2 2 The advantages and validity of the method over the modified simple equation method have been discussed in the following: Advantages: The crucial advantage of the new approach against the modified simple equation method is that the method provides more general and large amount of new exact traveling wave solutions with several free parameters. The exact solutions have its great importance to expose the inner mechanism of the physical phenomena. Apart from the physical application, the close-form solutions of nonlinear evolution equations assist the numerical solvers to compare the accuracy of their results and help them in the stability analysis.
Comparison: In Ref.

Application of the Method
In this section, we will put forth the new generalized ) / ( G G′ expansion method to construct many new and more general traveling wave solutions of the Burgers equation. Let us consider the Burgers equation, (19) We utilize the traveling wave variable ) , ( (20) Eq. (20) is integrable, therefore, integrating with respect to η once yields: , 0 2 (21) where P is an integration constant which is to be determined.

Results and Discussion
It is significant to state that one of our obtained solutions is in good agreement with the existing results which are shown in the Table 2. Beside this table, we obtain further new exact traveling wave      We can randomly choose the parameters 1 a and 2 a .