The G ′ / G-Expansion Method for Abundant Traveling Wave Solutions of Caudrey-Dodd-Gibbon Equation

We construct the traveling wave solutions of the fifth-order Caudrey-Dodd-Gibbon CDG equation by the G′/G -expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, the trigonometric, and the rational functions. It is shown that the G′/G -expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations.


Introduction
The investigation of exact traveling wave solutions of nonlinear partial differential equations NLPDEs plays an important role in the analysis of complex physical phenomena.The NLPDEs appear in physical sciences, various scientific and engineering problems, such as, fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, chemical physics, chemistry and many others.In recent years, to obtain exact traveling wave solutions of NLPDEs, many effective and powerful methods have been presented in the literature, such as the Backlund transformation 1 , the tanh function method 2 , the extended tanh function method 3 , the variational iteration method 4 , the Adomian decomposition method 5, 6 , the homotopy perturbation method 7 , the F-expansion method 8 , the Hirota's bilinear method 9 , the exp-function method 10 , the Cole-Hopf transformation 11 , the general projective Riccati equation method 12 and others 13-20 , Nowaday, searching analytical solutions of the NLPDEs has become more lucrative partly due to the accessibility computer symbolic systems, like Maple, Mathematica,

Description of the G /G -Expansion Method
Suppose that the nonlinear partial differential equation is of the form P u, u t , u x , u tt , u xt , u xx , ... 0, 2.1 where u u x, t is an unknown function and P is a polynomial in u u x, t which has various partial derivatives, and the highest order derivatives and nonlinear terms are involved.
The main steps of the G /G -expansion method 21 are conveyed in the following.
Step 1.The traveling wave transformation: where s is the wave speed, ξ is the combination of two independent variables x and t, transform 2.1 into an ordinary differential equation for v ξ : where primes denote the ordinary derivatives with respect to ξ.
Step 2. If possible, integrate 2.3 term by term for one or more times, yields constant s of integration.For simplicity, the integration constant s may be zero.
Step 3. Suppose that the solution of 2.3 can be expressed by a polynomial in G /G : where α n n 0, 1, 2, 3, . . ., m are constants and α m / 0, and G G ξ satisfies the second order linear ordinary differential equation LODE : where λ and μ are arbitrary constants.The positive integer m can be determined by considering the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in 2.3 .
Step 4. Substituting 2.4 together with 2.5 into 2.3 yields an algebraic equation involving powers of G /G .Then, equating coefficients of each power of G /G to zero yields a set of algebraic equations for α n n 0, 1, 2, 3, . . ., m , s, λ, and μ.Since the general solution of 2.5 is known for us, then substituting α n n 0, 1, 2, 3, . . ., m and s together with general solution of 2.5 into 2.4 , we obtain exact traveling wave solutions of the nonlinear partial differential equation 2.1 .

Application of the Method
In this section, we apply the method to construct the hyperbolic, the trigonometric, and the rational function solutions of the fifth-order Caudrey-Dodd-Gibbon equation, and the solutions are shown in graphs.

The Caudrey-Dodd-Gibbon Equation
We consider the fifth-order Caudrey-Dodd-Gibbon equation: Now, we use the transformation equation 2.2 into 3.1 , which yields, where primes denote the derivative with respect to ξ. Equation.3.2 is integrable, therefore, integrating once with respect to ξ yields where C is an integral constant, that is, to be determined later.

3.5
Solving the system of algebraic equations with the aid of Maple 13, we obtain two different sets of solution.
Case 1.One has where λ and μ are arbitrary constants.
Case 1. Substituting 3.6 into 3.4 yields Substituting the general solution of 2.5 into 3.8 , we obtain three types of traveling wave solutions of 3.3 as follows.

Hyperbolic Function Solutions
When λ 2 − 4μ > 0, substituting the general solution of 2.5 into 3.8 , we obtain the following traveling wave solution of 3.3 : where ξ x −λ 4 8μλ 2 − 16μ 2 t, C 1 , and C 2 are arbitrary constants.The various known results can be rediscovered, if C 1 and C 2 are taken as special values.For example: where
Case 2. Substituting 3.7 into 3.4 yields Substituting the general solution of 2.5 into 3.15 , we obtain three types of traveling wave solutions of 3.3 as follows.

Discussion
The solutions of the CDG 3.1 are investigated by different methods, such as Jin 32 investigated solutions by the variational iteration method, Salas 33 by the projective Riccati equation method, and Wazwaz 34 by using the tanh method.To the best of our awareness the CDG equation is not solved by the prominent G /G -expansion method.In this paper, we solve this equation by the G /G -expansion method.It is noteworthy to point out that our attained solutions are new and cannot be found from above author's solutions by any choice of arbitrary constants.

Graphical Representations of the Solutions
The solutions are shown in the graphs with the aid of Maple 13 in Figures 1-5.

Conclusions
In this paper, three types of traveling wave solutions, such as the hyperbolic, the trigonometric, and the rational functions of the Caudrey-Dodd-Gibbon equation are successfully obtained by using the G /G -expansion method.Exact traveling wave solutions of this equation have many potential applications in engineering and mathematical physics.
The obtained solutions also show that the method is effective, more powerful, and simple for searching exact traveling wave solutions of the NLPDEs.The method can be applied in different types of NLEEs and it is our task in the future.
C 1 and C 2 are arbitrary constants.The various known results can be rediscovered, if C 1 and C 2 are taken as special values.