Analytical Treatment of the Evolutionary (1 + 1)-Dimensional Combined KdV-mKdV Equation via the Novel (G'/G)-Expansion Method

The novel (G'/G)-expansion method is a powerful and simple technique for finding exact traveling wave solutions to nonlinear evolution equations (NLEEs). In this article, we study explicit exact traveling wave solutions for the (1 + 1)-dimensional combined KdV-mKdV equation by using the novel (G'/G)-expansion method. Consequently, various traveling wave solutions patterns including solitary wave solutions, periodic solutions, and kinks are detected and exhibited.


Introduction
NLEEs arise in a wide variety of disciplines physical problems such as in physics, biology, fluid mechanics, solid-state physics, biophysics, solid mechanics, condensed matter physics, plasma physics, quantum mechanics, optical fibers, elastic media, reaction-diffusion models, and quantum field theory.Recently, many kinds of powerful methods have been proposed to find exact traveling wave solutions of NLEEs e.g., the ( ) ( ) [1]- [3], the (G'/G)-expansion method [4]- [7], the wave translation method [8], the Ansatz method [9] [10], the Darboux transformation method [11], the Hopf-Coletrans formation [12], the Miura transformation [13], the Jacobi elliptic function method [14], the A domian decomposition method [15] [16], the method of bifurcation of planar dynamical systems [17] [18], the inverse scattering transform method [19], the multipleexpansion method [20], Homotopy analysis method [21] [22], three-wave method [23], extended homoclinic test approach [24], the improved F-expansion method [25], the projective Riccati equation method [26], and the Weirstrass elliptic function method [27] to name a few.The novel (G'/G)-expansion method is beginning to find a pragmatic ever increasing use as can be seen in [28]- [33].Worthy is it to note perhaps that rudiments of the (G'/G)-expansion method was used by Eckstein and Belgacem, as early as the late 80's, to describe the platelet transport behavior in blood vessels, [34]- [36].Recently, Alam and Belgacem in their study appearing in the Waves, Wavelets and Fractals-Abstract Analysis Journal, applied the novel method to the long wave equation, [37].The aim of this paper is to find exact and solitary wave solutions of the (1 + 1)-dimensional combined KdV-mKdV equation by the novel (G'/G)-expansion method.

Description of the Method
For a given nonlinear wave equation with one physical field ( ) , u x t in two variables x and t ( ) , , , , , , 0 where ( ) , u x t and P is a polynomial about u in and its derivatives.Let us consider that the traveling wave variable is The traveling wave variable (2), transforms (1) into a nonlinear ODE for ( ) We seek for the solution of Equation (3) in the following generalized ansatze where Herein N α − or α N might be zero, but both of them could not be zero simultaneously.α j ( ) 0, 1, 2, , j N = ± ± ±  and d are constants to be determined later and ( ) satisfies the second order nonlinear ODE: ( ) where prime denotes the derivative with respect to ξ and λ , µ , ν are real parameters.
The value of the positive integer N can be determined by balancing the highest order linear terms with the nonlinear terms of the highest order come out in Equation (3).If the degree of ( ) , then the degree of the other expressions will be as follows: Substituting Equation (4) including Equations ( 5) and ( 6) into Equation (3), we obtain polynomials in . Collecting all coefficients of identical power of the resulted polynomials to zero, yields an over-determined set of algebraic equations for ( ) Suppose the value of the constants can be obtained by solving the algebraic equations obtained in Step 4. Substituting the values of the constants together with the solutions of Equation ( 6), we will obtain some new and comprehensive exact traveling wave solutions to the nonlinear evolution Equation (1).

The (1 + 1)-Dimensional Combined KdV-mKdV Equation
In this section, we will employ the novel (G'/G)-expansion method to get several novel and further wide-ranging exact traveling wave solutions to the famous (1 + 1)-dimensional combined KdV-mKdV equation.
Let us consider the (1 + 1)-dimensional combined KdV-mKdV equation Using the traveling wave transformation x Vt ξ = − , Equation ( 8) is converted into the following ODE: Integrating Equation ( 9), we obtain where C is a constant of integration.Inserting Equation (4) into Equation ( 10) and balancing the highest order derivative u ′ ′ with the nonlinear term of the highest order 3 u , we obtain 1 M = .Therefore, the solution of Equation ( 10) takes the form, ( ( ) Substituting Equation (11) into Equation (10), the left hand side is transformed into polynomials of . Equating the coefficients of like power of these polynomials to zero, we obtain an over-determine set of algebraic equations (for simplicity we leave out to display the equations) for 0 α , 1 α , 1 α − , d, C and V. Solving the over-determined set of algebraic equations by using the symbolic computation software, such as Maple 13, we obtain where d, λ , µ and υ are arbitrary constants.
For Set, substituting Equation ( 12) and the values of ) (ξ ψ into Equation (11) and simplifying, we obtain the following: When λ υ − ≠ (or ( ) where ) where A and B are real constants.λ υ − ≠ (or ( ) ) ( ) where A and B are arbitrary constants such that 2 ( ) When 0 µ = and ( )   36) where k is an arbitrary constant.
When ( ) where 1 c is an arbitrary constant.

Conclusion
In this letter, the novel (G'/G)-expansion method has been successfully applied to find the exact solution for the (1 + 1)-dimensional combined KdV-mKdV equation.The novel (G'/G)-expansion method is used to find a new exact traveling wave solution.The results show that the novel (G'/G)-expansion method is reliable and effective tool to solve the (1 + 1)-dimensional combined KdV-mKdV equation.Thus the novel (G'/G)-expansion method could be a powerful mathematical tool for solving NLEEs.