Applications of an Extended G ′ / G-Expansion Method to Find Exact Solutions of Nonlinear PDEs in Mathematical Physics

We construct the traveling wave solutions of the 1 1 -dimensional modified Benjamin-BonaMahony equation, the 2 1 -dimensional typical breaking soliton equation, the 1 1 -dimensional classical Boussinesq equations, and the 2 1 -dimensional Broer-Kaup-Kuperschmidt equations by using an extended G′/G -expansion method, where G satisfies the second-order linear ordinary differential equation. By using this method, new exact solutions involving parameters, expressed by three types of functions which are hyperbolic, trigonometric and rational function solutions, are obtained. When the parameters are taken as special values, some solitary wave solutions are derived from the hyperbolic function solutions.


Introduction
The investigation of the traveling wave solutions of nonlinear partial differential equations NPDEs plays an important role in the study of nonlinear physical phenomena.In recent years, new exact solutions may help to find new phenomena.The exact solutions have been investigated by many authors see, e.g.,  who are interested in nonlinear physical phenomena.Many powerful methods have been presented such as the homogeneous balance method 13 , the tanh method 4, 15, 24 , the inverse scattering transform 1 , the expfunction expansion method 2, 6, 20 , the Jacobi elliptic function expansion 17 , the Backlund transform 8, 9 , the generalized Riccati equation 18 , the modified extended Fan subequation method 19 , the truncated Painlevé expansion 27 , and the auxiliary equation method 10, 11 .More recently, the G /G -expansion method 14, 22, 23, 26 has been proposed to obtain traveling wave solutions.This method is firstly proposed by the Chinese Mathematical Problems in Engineering mathematicians Wang et al. 14 for which the traveling wave solutions of the nonlinear evolution equations are obtained.This method has been extended to solve differencedifferential equations 28, 29 .The improved G /G -expansion method has been used in 21, 25 .Recently, Gou and Zhou 5 have obtained the exact traveling wave solutions of some nonlinear PDEs using an extended G /G -expansion method.
In the present article, we use the extended G /G -expansion method which is proposed in 5 to derive traveling wave solutions for some nonlinear PDEs in mathematical physics namely; the 1 1 -dimensional modified Benjamin-Bona-Mahony equation, the 2 1 -dimensional typical breaking soliton equation, the 1 1 -dimensional classical Boussinesq equations, and the 2 1 -dimensional Broer-Kaup-Kuperschmidt equations.The extended G /G -expansion method used in this article can be applied to further equations such as difference-differential equations which can be done in forthcoming articles.

Description of an Extended G /G -Expansion Method
Consider the nonlinear partial differential equation in the following form: where u u x, t is unknown functions, and F is a polynomial in u x, t and its partial derivatives.In the following, we give the main steps for solving 2.1 using an extended G /G -expansion method 5 .
Step 1.The traveling wave variable where V is a constant to be determined latter, permits us reducing 2.1 to an ODE in the form where P is a polynomial in u ξ and its total derivatives.
Step 2. Suppose the solution of 2.3 can be expressed in G /G as follows: where G G ξ satisfies the following second-order linear ODE: G ξ μG ξ 0, 2.5 while a i , b i i 1, . . ., n and a 0 are constants to be determined, such that σ ±1 and μ / 0. The positive integer "n can be determined by balancing the highest-order derivatives with the nonlinear terms appearing in 2.3 .
Step 3. Substituting 2.4 into 2.3 and using 2.5 , collecting all terms with the same powers of G /G k and G /G k σ 1 1/μ G /G 2 together, and equating each coefficient of them to zero, yield a set of algebraic equations for a 0 , a i , b i and V .
Step 4. Since the general solution of 2.5 has been well known for us, then substituting a i , b i , V and the general solution of 2.5 into 2.4 , we have the traveling wave solutions of the nonlinear partial differential equation 2.1 .
Remark 2.1.It is necessary to point out that by adding the term G /G k σ 1 1/μ G /G 2 into 2.4 , the ansätz proposed here is more general than the ansätz in the original G /G -expansion method 14 .Therefore, the extended G /G -expansion method is more powerful than the original G /G -expansion method 14 and some new types of traveling wave solutions and solitary wave solutions would be expected for some NPDEs.If we choose the parameters in 2.4 and 2.5 to take special values, the G /G -expansion method can be recovered by our proposed method.

Applications
In this section, we will apply the extended G /G -expansion method to some nonlinear PDEs in mathematical physics as follows.

Example 1: The (1+1)-Dimensional Modified Benjamin-Bona-Mahony Equation
We start with the following 1 1 -dimensional nonlinear dispersive modified Benjamin-Bona Mahony equation 20 written in the following form: where α is a nonzero positive constant.This equation was first derived to describe an approximation for surface long waves in nonlinear dispersive media.It can also characterize the hydromagnetic waves in cold plasma, acoustic waves in inharmonic crystals and acousticgravity waves in compressible fluids.Yusufoglu 20 has used the Exp-function method to find the traveling wave solutions of 3.1 .Let us now solve 3.1 by the proposed method.
To this end, we see that the traveling wave variable 2.2 permits us converting 3.1 into the following ODE: Integrating 3.2 with respect to ξ once, we get where K is an integration constant.Considering the homogeneous balance between the highest-order derivatives and nonlinear terms in 3.3 , we get n 1. Hence we suppose that the solution u ξ of 3.3 has the form where G G ξ satisfies 2.5 .Substituting 3.4 along with 2.5 into 3.3 , collecting all terms with the same powers of and setting them to zero, we have the following algebraic equations:

3.5
Solving these algebraic equations by Maple or Mathematica, we obtain the following results.
Case 1.One has Case 3. One has where σ ±1.From 3.4 and the general solution of 2.5 , we deduce the traveling wave solutions of 3.1 as follows.
When μ < 0, then Case 1 gives the exact traveling wave solution: where ξ x − 1 2μ t and A, B are arbitrary constants while σ ±1.Case 2 gives the exact traveling wave solutions: where ξ x − 1 − μ t.Case 3 gives the exact traveling wave solutions: where ξ x − 2 μ /2 t.When μ > 0, then Case 1 gives the exact traveling wave solutions: Case 2 gives the exact traveling wave solutions Case 3 gives the exact traveling wave solutions:

3.14
In particular, we deduce from 3.9 that the solitary wave solutions of 3.1 are derived as follows.
If A 0, B / 0 and μ < 0, then we obtain , and μ < 0, then we obtain where ξ 0 tanh −1 B/A .Similarly, we can find more solitary wave solutions of 3.1 using 3.10 -3.11 but we omitted them for simplicity.

Example 2: The (2+1)-Dimensional Typical Breaking Soliton Equation
In this subsection, we study the following the 2 1 -dimensional typical breaking soliton equation 3 in the following form: Let us now solve 3.17 by the proposed method.To this end, we see that the traveling wave variable u x, y, t u ξ , ξ x y − V t 3.18 permits us to convert 3.17 into the following ODE: Integrating 3.19 with respect to ξ once yields where K is an integration constant.Considering the homogeneous balance between the highest-order derivatives and nonlinear terms in 3.20 we deduce that the solution u ξ of 3.20 has the same form of 3.4 .Substituting 3.4 along with 2.5 into 3.20 , collecting all terms with the same powers of G /G k , G /G k σ 1 1/μ G /G 2 and setting them to zero, we have the following algebraic equations:

3.21
Solving these algebraic equations by Maple or Mathematica, we obtain the following results.

Case 1. One has
Case 2. One has From 3.4 and the general solution of 2.5 , we deduce the traveling wave solutions of 3.17 as follows.
When μ < 0, then Case 1 gives the exact traveling wave solution: where ξ x 4μt.Case 2 gives the exact traveling wave solution:

Mathematical Problems in Engineering
Case 2 gives the exact traveling wave solution: In particular, we deduce from 3.24 that the solitary wave solutions of 3.17 are derived as follows.
If A 0, B / 0, and μ < 0, then we obtain , and μ < 0, then we obtain where ξ 0 tanh −1 B/A .Similarly, we can find more solitary solutions of 3.17 using 3.25 but we omitted them for simplicity.

3.30
This system has been derived by Wu and Zhang 16 for modelling nonlinear and dispersive long gravity wave traveling in two horizontal directions on shallow water of uniform depth.
Let us now solve 3.30 by the proposed method.To this end, we see that the traveling wave variables 2.2 permit us converting 3.30 into the following ODEs:

3.31
Integrating 3.31 with respect to ξ once yields where K 1 and K 2 are integration constants.Considering the homogeneous balance between highest order derivatives and nonlinear terms in Equations 3.32 , 3.33 we deduce that the solution u ξ which has the same form of 3.4 while, v ξ has the following form:

3.34
Substituting 3.4 and 3.34 along with 2.5 into 3.32 , collecting all terms with the same and setting them to zero, we have the following algebraic equations:

3.35
Similarly, substituting 3.4 and 3.34 along with 2.5 into 3.33 , collecting all terms with the same powers of G /G k , G /G k σ 1 1/μ G /G 2 and setting them to zero, we have the following algebraic equations:

Mathematical Problems in Engineering
Solving these algebraic equations by Maple or Mathematica, we obtain the following results.

3.37
Case 2. One has

3.38
Case 3. One has

3.39
where a 0 is an arbitrary constant.From 3.4 , 3.34 and the general solution of 2.5 , we deduce the traveling wave solutions of 3.30 as follows.
When μ < 0, then Case 1 gives the exact traveling wave solution:

Mathematical Problems in Engineering 11
Case 2 gives the exact traveling wave solution:

3.41
Case 3 gives the exact traveling wave solution

3.43
where ξ x − a 0 t.When μ > 0, then Case 1 gives the exact traveling wave solution:

Mathematical Problems in Engineering
Case 2 gives the exact traveling wave solution

3.45
Case 3 gives the exact traveling wave solution:

3.46
where ξ x − a 0 t.In particular, we deduce from 3.40 that the solitary wave solutions of 3.30 are derived as follows: If A 0, B / 0 and μ < 0, then we obtain

3.48
where ξ 0 tanh −1 B/A .Similarly, we can find more solitary solutions of 3.30 using 3.41 -3.43 but we omitted them for simplicity.

3.49
This system has been widely applied to many branches of physics like plasma physics, fluid dynamics, nonlinear optics and so on.Yomba 19 has obtained new and more general solutions of 3.49 including a series of nontraveling wave and coefficient function solutions using the modified extended Fan subequation method.Let us now solve 3.49 by the proposed method.To this end, we see that the traveling wave variables 3.18 permit us converting 3.49 into the following ODEs:

3.50
Integrating Equation 3.50 with respect to ξ once, yields where K 1 and K 2 are integration constants.Considering the homogeneous balance between the highest-order derivatives and nonlinear terms in 3.51 and 3.52 , we deduce the solution u ξ which has the same form of 3.4 while, v ξ has the same form of 3.34 .Substituting 3.4 and 3.34 along with 2.5 into 3.51 , collecting all terms with the same powers of G /G k , G /G k σ 1 1/μ G /G 2 , and setting them to zero, we have the following algebraic equations:

3.53
Similarly, substituting 3.4 and 3.34 along with 2.5 into 3.52 , collecting all terms with the same powers of G /G k , G /G k σ 1 1/μ G /G 2 , and setting them to zero, we have the following algebraic equations:

3.54
Solving these algebraic equations by Maple or Mathematica, we obtain the following results.
Case 1.One has

3.56
Case 3. One has where a 0 is an arbitrary constant.From 3.4 , 3.34 , and the general solution of 2.5 , we deduce the traveling wave solutions of 3.49 as follows.
When μ < 0, then Case 1 gives the exact traveling wave solution:

3.58
Case 2 gives the exact traveling wave solution:

3.59
Case 3 gives the exact traveling wave solution:

3.63
where ξ x − 2a 0 t.In particular, we deduce from 3.58 that the solitary wave solutions of 3.49 are derived as follows.
If A 0, B / 0, and μ < 0,then one obtain where ξ 0 tanh −1 B/A .Similarly, we can find more solitary solutions of 3.49 using 3.59 -3.60 but we omitted them for simplicity.

Conclusion
In this paper, we have seen that three types of traveling wave solutions in terms of hyperbolic, trigonometric, and rational functions for the 1 1 -dimensional modified Benjamin-Bona-Mahony equation, the 2 1 -dimensional typical breaking soliton equation, the 1 1 -dimensional classical Boussinesq equations, and the 2 1 -dimensional Broer-Kaup-Kuperschmidt equations are successfully found out by using the extended G /Gexpansion method.The performance of this method is reliable, effective, and giving many which was first introduced by Calogero and Degasperis 3 .Tian et al. 12 have reduced new families of soliton-like solutions via the generalized tanh method which are of important significance in explaining some physical phenomena.Mei and Zhang 7 have reduced more families of new exact solutions which contain soliton-like solutions and periodic solution based on a newly generally projective Riccati equation expansion method and its algorithm.