Analytical Treatment of the Evolutionary (1 + 1)-Dimensional Combined KdV-mKdV Equation via the Novel (G'/G)-Expansion Method ()
Received 24 October 2015; accepted 5 December 2015; published 8 December 2015
1. 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 
- expansion [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 multiple- expansion 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.
2. Description of the Method
For a given nonlinear wave equation with one physical field 
in two variables x and t
(1)
where 
and P is a polynomial about u in and its derivatives.
Let us consider that the traveling wave variable is
,. (2)
The traveling wave variable (2), transforms (1) into a nonlinear ODE for
:
. (3)
We seek for the solution of Equation (3) in the following generalized ansatze
, (4)
where
. (5)
Herein 
or αN might be zero, but both of them could not be zero simultaneously. αj 
and d are constants to be determined later and 
satisfies the second order nonlinear ODE:
(6)
where prime denotes the derivative with respect to 
and
, ![]()
, ![]()
are real parameters.
The Cole-Hopf transformation ![]()
reduces Equation (6) into Riccati equation:
![]()
. (7)
Equation (7) has individual twenty five solutions (see Zhu, [38] for details).
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 ![]()
is![]()
, 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
![]()
and![]()
,![]()
. Collecting all coefficients of identical power of the re-
sulted polynomials to zero, yields an over-determined set of algebraic equations for![]()
, d and V.
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).
3. 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
![]()
. (8)
Using the traveling wave transformation![]()
, Equation (8) is converted into the following ODE:
![]()
. (9)
Integrating Equation (9), we obtain
![]()
(10)
where C is a constant of integration. Inserting Equation (4) into Equation (10) and balancing the highest order derivative ![]()
with the nonlinear term of the highest order![]()
, we obtain![]()
.
Therefore, the solution of Equation (10) takes the form,
![]()
. (11)
Substituting Equation (11) into Equation (10), the left hand side is transformed into polynomials of
![]()
and![]()
,![]()
. Equating the coefficients of like power of these poly-
nomials to zero, we obtain an over-determine set of algebraic equations (for simplicity we leave out to display the equations) for![]()
, ![]()
, ![]()
, d, C and V. Solving the over-determined set of algebraic equations by using the symbolic computation software, such as Maple 13, we obtain
![]()
(12)
where d, ![]()
, ![]()
and ![]()
are arbitrary constants.
For Set, substituting Equation (12) and the values of ![]()
into Equation (11) and simplifying, we obtain the following:
When ![]()
and ![]()
(or![]()
),
![]()
(13)
where![]()
, and d, ![]()
, ![]()
and ![]()
are arbitrary constants.
![]()
(14)
![]()
(15)
![]()
(16)
![]()
(17)
![]()
(18)
![]()
(19)
where A and B are real constants.
![]()
(20)
![]()
(21)
![]()
(22)
![]()
(23)
When ![]()
and ![]()
(or![]()
),
![]()
(24)
![]()
(25)
![]()
(26)
![]()
(27)
![]()
(28)
![]()
(29)
![]()
(30)
where A and B are arbitrary constants such that![]()
.
![]()
(31)
![]()
(32)
![]()
(33)
![]()
(34)
When ![]()
and![]()
,
![]()
(35)
![]()
(36)
where k is an arbitrary constant.
When ![]()
and![]()
, the solution of Equation (8) is,
![]()
(37)
where ![]()
is an arbitrary constant.
4. 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.
Acknowledgements
The Authors offer sincere thanks to the referees and editorial board of JAMP for their helpful and kind support. Furthermore, Fethi Bin Muhammad Belgacem is pleased to acknowledge the continued support of the Public Authority for Applied Education and Training (PAAET RD) for their continued support.
NOTES
![]()
*Corresponding author.