General Solution of Generalized ( 2 + 1 )-Dimensional Kadomtsev-Petviashvili ( KP ) Equation by Using the G G-Expansion Method

In this work, the G G   -expansion method is proposed for constructing more general exact solutions of the (2 + 1)-dimensional Kadomtsev-Petviashvili (KP) equation and its generalized forms. Our work is motivated by the fact that the G G   -expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.


Introduction
Nonlinear evolution equations (NLEEs) have been the subject of study in various branches of mathematicalphysical sciences such as physics, biology, chemistry, etc.The analytical solutions of such equations are of fundamental importance since a lot of mathematical-physical models are described by NLEEs.Among the possible solutions to NLEEs, certain special form solutions may depend only on a single combination of variables such as traveling wave variables.In the literature, there is a wide variety of approaches to nonlinear problems for constructing traveling wave solutions.Some of these approaches are the Jacobi elliptic function method [1], inverse scattering method [2], Hirotas bilinear method [3], homogeneous balance method [4], homotopy perturbation method [5], Weierstrass function method [6], symmetry method [7], Adomian decomposition method [8], sine/cosine method [9], tanh/coth method [10], the Exp-function method [11][12][13][14][15][16] and so on.But, most of the methods may sometimes fail or can only lead to a kind of special solution and the solution procedures become very complex as the degree of nonlinearity increases.
Recently, the G G   troduced by Wang et al. [17], has become widely used to -expansion method, firstly in-search for various exact solutions of NLEEs [17][18][19][20][21][22][23][24][25][26][27].The value of the   G G  -expansion method is that one treats nonlinear pro by essentially linear methods.The method is based on the explicit linearization of NLEEs for traveling waves with a certain substitution which leads to a second-order differential equation with constant coefficients.Moreover, it transforms a nonlinear equation to a simple algebraic computation.
The generalized (2 + 1)-dimensional Ka blems domtsov-Petviashivilli (gKP) equation given by The objectives of this work are twofold.First, we describe the   G G  -expansion method.Second, we aim to implemen resent method to obtain general exact travelling wave solutions of governing equation.

Description of the    G G -Expansion
The objective of this section is to outline the use of the

 
where  and  are arbitrary constants.Using the general solutions of Equation ( 4), we have and it follows, from ( 3) and ( 4), that and so on, here the prime denotes the derivative with spective to To determine u explicitly, we take the following four steps: Step 1. Determin the integer n by substituting Equation (3) alo e ng with Equation (4) into Equation (2), and ba nl Equation ( 4) into Equ tio lancing the highest order no inear term(s) and the highest order partial derivative.
Step 2. Substitute Equation (3) give the value of n determined in Step 1, along with af n (2) and collect all terms with the same order o Then set each ent of this polynomial to zero to derive a set of algebraic equations for 0 , , k coeffici   and i  .
Step 3. Solve the system of algebraic equations obtained in Step 2, for

Application
In this section, we will demonstrate th   -expansion method on the generalized (2 + 1) al Kaation given by domtsev-Petviashvili (KP) equ where ,  and  are constants.Using the wave vari- (7) and integrating the resulting equation and neglecting the constant of i we find ntegration, our goal, we use the transformation To achieve According to Step 1, we get 2 , hence We then suppose that Equatio e follo wing formal solutions: where 2 1 , ,   and 0 ,  are constants which known to be determined later.tituting on me order of are un-Subs Equati (10) into Equation ( 9) and collecting all terms with the sa Substitute the above general case in (10), we get

 
together, the left-hand sides of Equation ( 9 then use the transformation the hyperbolic function solutions of Equation (7), becomes: and when , the trigonometric function solutions of Equation (7), will be: , , , C C  and  are arbitrary constants.
In particular, when then the general solutions and 2 = 0, C (13) and ( 14) reduces, respectively, and when then we deduce from general solutions ( 13) and ( 14) that, where , , , k 


and  are arbitrary constants.
where (19) and  ,  are constants, then according to results in (11), the general hyperbolic and trigonometric function solution of (19) will be the KP Equation (7) and when then the general solution ( 20)-( 21) reduce to We would like to note that the obtained solutions with an explicit linear function in  have been checked with Maple by putting them back into the original Equations (7).

Conclusions and Future Work
This study shows that the   G G  -expansion method is the equations considered, they might serve as seeding quite efficient and practically well suited for use in finding exact solutions for the ge sional Kadomtsev-Petviashvili (gKP) equation.The reliability of the method and the reduction in the size of computational domain give this method a wider applicability.Though the obtained solutions represent only a small part of the large variety of possible solutions for neralized (2 + 1)-dimen-ical systems.Furthermore, our solutions are in m general forms, and many known solutions to these equahe aid of Maple, we have assured the correctness of the obtained sok into solutions for a class of localized structures existing in the phys ore tions are only special cases of them.With t lutions by putting them bac the original equation.We hope that they will be useful for further studies in applied sciences.According to Case 5, present method failed to obtain the general solution of gKP for = 1, n  and = 2, n  therefore the authors hope to extend the

 
G G  -expansion method to solve these especial type of gKP.
  a nd i  by use of M obtained e eries of fundamental solutions aple.Step 4. Use the results in abov steps to derive a s   u  of Equa- tion (2) depending on   G G  , since the solutions of Equation (4) have b n for us, then we can ob e een well know tain exact solutions of Equation (1).
) are converted into a polynomial in   G G  .Setting each coefficient of each polynomial to zero, we derive a set of algebraic equations use of Maple, we get the following general result: