Traveling Wave Solutions of the Nonlinear 3 1-Dimensional Kadomtsev-Petviashvili Equation Using the Two Variables G ′ / G , 1 / G-Expansion Method

The two variables G′/G, 1/G -expansion method is proposed in this paper to construct new exact traveling wave solutions with parameters of the nonlinear 3 1 -dimensional KadomtsevPetviashvili equation. This method can be considered as an extension of the basic G′/G expansion method obtained recently by Wang et al. When the parameters are replaced by special values, the well-known solitary wave solutions and the trigonometric periodic solutions of this equationwere rediscovered from the traveling waves.


Introduction
In the recent years, investigations of exact solutions to nonlinear PDEs play an important role in the study of nonlinear physical phenomena.Many powerful methods have been presented, such as the inverse scattering transform method 1 , the Hirota method 2 , the truncated Painlevé expansion method 3-6 , the Backlund transform method 7, 8 , the exp-function method 9-14 , the tanh function method 15-18 , the Jacobi elliptic function expansion method 19-21 , the original G /G -expansion method 22-33 , the two variables G /G, 1/G -expansion method 34, 35 , and the first integral method 36 .The key idea of the original G /G -expansion method is that the exact solutions of nonlinear PDEs can be expressed by a polynomial in one variable G /G in which G G ξ satisfies the second ordinary differential equation G ξ λG ξ μG ξ 0, where λ and μ are constants.In this paper, we will use the two variables G /G, 1/G -expansion method, which can be considered as an extension of the original G /G -expansion method.The key idea of the two variables G /G, 1/G -expansion method is that the exact traveling wave solutions of nonlinear PDEs can be expressed by a polynomial in the two variables G /G and 1/G , in which G G ξ satisfies a second order linear ODE, namely, G ξ λG ξ μ, where λ and μ are constants.The degree of this polynomial can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms in the given nonlinear PDEs, while the coefficients of this polynomial can be obtained by solving a set of algebraic equations resulted from the process of using the method.According to Aslan 29 , the two variables G /G, 1/G -expansion method becomes the basic G /G -expansion method if μ 0 in 2.1 and b i 0 in 2.12 .Recently,Li et al. 34 have applied the two variables G /G, 1/G -expansion method and determined the exact solutions of the Zakharov equations, while Zayed andabdelaziz 35 have applied this method to determine the exact solutions of the nonlinear KdV-mKdV equation.
The objective of this paper is to apply the two variables G /G, 1/G -expansion method to find the exact traveling wave solutions of the following nonlinear 3 1 -dimensional Kadomtsev-Petviashvili equation: This equation describes the dynamics of solitons and nonlinear wave in plasma and superfluids.Recently, Zayed 24 has found the exact solutions of 1.1 using the original G /G -expansion method, while Aslan 14 has discussed 1.1 using the exp-function method.Comparison between our results and that obtained in 14, 24 will be discussed later.The rest of this paper is organized as follows.In Section 2, the description of the two variables G /G, 1/G -expansion method is given.In Section 3, we apply this method to 1.1 .In Section 4, conclusions are obtained.

Description of the Two Variables G /G, 1/G -Expansion Method
Before we describe the main steps of this method, we need the following remarks see where A 1 and A 2 are arbitrary constants.Consequently, we have Remark 2.3.If λ > 0, then the general solutions of 2.1 is and hence Remark 2.4.If λ 0, then the general solutions of 2.1 is and hence Suppose we have the following NLPDEs in the form: where F is a polynomial in u and its partial derivatives.In the following, we give the main steps of the two variables G /G, 1/G -expansion method 34, 35 .
Step 1.The traveling wave variable u x, t u ξ , ξ x − V t 2.10 reduces 2.9 to an ODE in the form where V is a constant and P is a polynomial in u and its total derivatives, while { } d/dξ.
Step 2. Suppose that the solutions of 2.11 can be expressed by a polynomial in the two variables φ and ψ as follows: where a i i 0, 1, . . ., N and b i i 1, . . ., N are constants to be determined later.
Step 3. Determine the positive integer N in 2.12 by using the homogeneous balance between the highest order derivatives and the nonlinear terms in 2.11 .
Step 4. Substitute 2.12 into 2.11 along with 2.2 and 2.4 , the left-hand side of 2.11 can be converted into a polynomial in φ and ψ, in which the degree of ψ is not longer than 1.Equating each coefficients of this polynomial to zero yields a system of algebraic equations which can be solved by using the Maple or Mathematica to get the values of a i , b i , V, μ, A 1 , A 2 , and λ where λ < 0. Thus, we get the exact solutions in terms of the hyperbolic functions.
Step 5. Similar to Step 4, substitute 2.12 into 2.11 along with 2.2 and 2.6 for λ > 0 or 2.2 and 2.8 for λ 0 , we obtain the exact solutions of 2.11 expressed by trigonometric functions or by rational functions , respectively.

An Application
In this section, we apply the method described in Section 2, to find the exact traveling wave solutions of the nonlinear 3 1 -dimensional Kadomtsev-Petviashvili equation 1.1 .To this end, we see that the traveling wave variables ξ x y z − V t reduce 1.1 to the following ODE: Balancing u with uu in 3.1 we get N 2. Consequently, we get where a 0 , a 1 , a 2 , b 1 , and b 2 are constants to be determined later.There are three cases to be discussed as follows.
Case 1. Hyperbolic function solutions λ < 0 .If λ < 0, substituting 3.2 into 3.1 and using 2.2 and 2.4 , the left-hand side of 3.1 becomes a polynomial in φ and ψ.Setting the coefficients of this polynomial to zero yields a system of algebraic equations in a 0 , a 1 , a 2 , b 1 , b 2 , μ, σ, and λ which can be solved by using the Maple or Mathematica to find the following results:

3.3
From 2.3 , 3.2 , and 3.3 , we deduce the traveling wave solution of 1.1 as follows:
If λ > 0, substituting 3.2 into 3.1 and using 2.2 and 2.6 , we get a polynomial in φ and ψ.Vanishing each coefficient of this polynomial to get the algebraic equations which can be solved by using the Maple or Mathematica to find the following results:

3.8
From 2.5 , 3.2 , and 3.8 , we deduce the traveling wave solution of 1.1 as follows: 3.9 where ξ has the same form 3.5 .

3.11
Case 3. Rational function solutions λ 0 .If λ 0, substituting 3.2 into 3.1 and using 2.2 and 2.8 , we get a polynomial in φ and ψ.Setting each coefficients of this polynomial to be zero to get the algebraic equations which can be solved by using the Maple or Mathematica to find the following results:

3.12
From 2.7 , 3.2 , and 3.12 , we deduce the traveling wave solution of 1.1 as follows:

2 , 3 . 13 where ξ x y z − 6a 0 − 2 t. 3 . 14 Remark 3 . 1 .
All solutions of this paper have been checked with Maple by putting them back into the original equation 1.1 .