Regarding on The Novel Forms of the ( 3 + 1 )-Dimensional Kadomstev-Petviashvili Equation

This article, we have applied the Bernoulli Sub-Equation method to the (3+1)-Dimensional Kademstev-Petviashvili equation. We have obtained some new analytical solutions such as exponential function and rational solutions by using this technique. We have observed that two analytical solutions have been verified the (3+1)-Dimentional KadomstevPetviashvili equations by using Wolfram Mathematica 9. At the end of this manuscript, we submitted a conclusion in a comprehensive manner.


Introduction
With the aid of powerful devices including computer technology and computational in nonlinear sciences, linear and nonlinear equations have been become an important area among scientists and engineers for obtaining various solutions.For example, a solitary wave has been firstly investigated by the Scottish engineer John Russel [1].He has followed a water wave travelling through a canal.The nonlinear partial differential equation systems give us the better physical explanations of significant mathematical models [2].Many powerful methods have been developed to find the exact solutions to the NLEEs.Some of these methods such as (G'/G)-expansion method [3,4], the improved (G'/G)-expansion method, the Sumudu transform method [5][6] have been submitted to the literature.
In the rest of this work, we apply the Bernoulli-Sub equation function method (BSFM) to the (3+1)-dimensional Kadomstev-Petviashvili equation [7]; Xie et al. [8] have investigated the some wave solutions of Eq. ( 1) by the improved tanh function method.We introduce the general properties of BSEFM in section 2. We apply BSEFM in section 3 for obtaining some exponential and rational function solutions.At the end of this study, we submit a conclusion in the comprehensive manner.

Fundamental Properties of Bernoulli Sub-Equation Function Method
Step 1.
We consider the following nonlinear partial differential equation (NLPDE) in two variables and a dependent variable u ;

 
, , , , 0, and take the travelling wave transformation where 0 c  .Substituting Eq.(3) in Eq.( 2) yields a nonlinear ordinary differential equation Take trial equation of solution for Eq.( 4) as following: where   F  is Bernoulli differential polynomial.Substituting Eq.( 5) along with Eq.( 6) in Eq.( 4), it yields an equation of polynomial ) according to the balance principle, we can obtain a relationship between n and M.

   
F   all be zero will yield an algebraic equations system 0, 0, , .
solving this system, we will specify the values of n a a , , 0  .

Step 4.
When we solve Eq.( 6), we obtain the following two situation according to b and d; Using a complete discrimination system for polynomial, we obtain the solutions to Eq.( 4) with the help of Wolfram Mathematica 9 programming and classify the exact solutions to Eq.( 4).For a better understanding of results obtained in this way, we can plot two-and three-dimensional surfaces of solutions by taking into consideration suitable values of parameters.

Implementation of the BSEFM
In this section, we have successfully considered the BSEFM to the (3+1)-Dimensional Kademstev-Petviashvili equation for getting some new travelling wave solutions.when we consider to Eq.( 5) and Eq.( 6) for balance principle between U  and 2 U , we obtain the following relationship between n and M; 2 2.

M n  
(11) this relationship gives us some new different analytical solutions for Eq. ( 1).

Case 1:
If we take as 4 n  and 3 M  in Eq. ( 11), we can write following equations; .
where 4 0, When we put Eq. (12-14) in Eq. ( 10), we obtain a system of algebraic equations.Therefore, we obtain a system of algebraic equations from these coefficients of polynomial of F .Solving this system with the help of wolfram Mathematica 9, we find the following coefficients and solutions;

Case 1a.
For b ≠ d, it can be considered the following coefficients;

Case 1b.
Another coefficients for Eq.( 1) and for b d  , it can be considered follows When we consider the travelling wave transformation and perform the transformation which c is constant non-zero, we obtain NLODE as following; integrating twice and setting the constants of integration to zero, substituting these coefficients in Eq.(12) along with Eq.(9), we obtain the following new exponential function solution for Eq.(1)
when we regulate Eq.(12) under the terms of Eq.(9,17), we find the other new exponential function solution for Eq.(1);