Analytic Solutions of the Kadomtsev-Petviashvili Equation with Power Law Nonlinearity Using the Sine-Cosine Method

In this paper, a sine-cosine method is used to construct many periodic and solitary wave solutions to Kadomtsev-Petviashvili equation with power law nonlinearity. Many new families of exact traveling wave solutions of the Kadomtsev-Petviashvili equation with power law nonlinearity are successfully obtained.

In this paper, by means of the Sine-cosine method, we will obtain some analytic solutions for the Kadomtsev-Petviashvili equation with power law nonlinearity. In the following section we have a brief review on the Sine-cosine method and in Section 3 and 4 , we apply the Sine-cosine method to obtain analytic solutions of the Kadomtsev-Petviashvili equation with power law nonlinearity. Finally, the paper is concluded in Section 5.

The Sine-cosine method
One can immediately reduce the nonlinear PDE (1) into a nonlinear ODE ( , , , , ) = 0. Q u u u u ξ ξξ ξξξ  The ordinary differential equation (3) is then integrated as long as all terms contain derivatives, where we neglect integration constants.
2. The solutions of many nonlinear equations can be expressed in the form [8] ( ), , (4) or in the form ( ), , 2 ( ) = 0 , where , λ µ and 0 β ≠ are parameters that will be determined, µ and c are the wave number and the wave speed respectively. We use 3.We substitute (6) or (7) into the reduced equation obtained above in (3), balance the terms of the cosine functions when (7) is used, or balance the terms of the sine functions when (6) is used, and solving the resulting system of algebraic equations by using the computerized symbolic calculations. We next collect all terms whit same power in ( ) cos k µξ or ( ) sin k µξ and set to zero their coefficients to get a system of algebraic equations among the unknowns , µ β and λ . We obtained all possible value of the parameters , µ β and λ [7].

The (1+2)-Dimensional KP Equation with Power Law Nonlinearity
The dimensionless form of the (1+2)-dimensional KP equation, with power law nonlinearity, that is going to be studied in this paper is given by [31] ( ) = 0.
Equating the exponents and the coefficients of each pair of the sine functions we find the following system of algebraic equations: Solving the system (13) yields where c is a free parameter. Hence, for > b c , the following periodic solutions ( 2)( 1)( However, for > c b , the following periodic solutions

The (1+3)-Dimensional KP Equation with Power Law Nonlinearity
The dimensionless form of the (1+3)-dimensional KP equation, with power law nonlinearity, that is going to be studied in this paper is given by [32] ( ) = 0 n x xxx x yy zz t u au u u bu cu Here in Eq. (17) (20) substituting (4) into (20) gives ( 1) ( 1) Equating the exponents and the coefficients of each pair of the sine functions we find the following system of algebraic equations:

Conclusions
In this paper, by using the sine-cosine method, we obtained some new explicit formulas of solutions for the generalized (1+2)-dimensional and the generalized (1+3)dimensional KP equations. Those solutions were similar to the solutions obtained in other paper. The study reveals the power of the method.