The nonlinear partial differential equations of mathematical physics are major subjects in physical science [1] . Exact solutions for these equations play an important role in many phenomena in physics such as fluid mechanics, hydrodynamics, Optics, Plasma physics and so on. Recently many new approaches for finding these solutions have been proposed, for example, tanh-sech method [2] -[4] , extended tanh-method [5] , extended jacobain method [6] , modified simple equation method [7] [8] , sine-cosine method [9] [10] , homogeneous balance method [11] [12] , F-expansion method [13] -[15] , exp-function method [16] [17] , trigonometric function series method [18] , -expansion method [19] -[22] , Jacobi elliptic function method [23] -[26] , the

-expansion method [27] -[29] and so on.

The objective of this article is to investigate more applications than obtained in [27] -[29] to justify and

demonstrate the advantages of the method. Here, we apply this method to (2+1)-dimensional

soliton breaking equation [30] and (3+1)-dimensional Kadomstev-Petviash-vili.

Consider the following nonlinear evolution equation

where F is a polynomial in and its partial derivatives in which the highest order derivatives and nonlinear terms are involved. In the following,we give the main steps of this method

Step 1. We use the wave transformation

where c is a positive constant, to reduce Equation (1) to the following ODE:

where P is a polynomial in and its total derivatives,while.

Step 2. Suppose that the solution of ODE (3) can be expressed by a polynomial in as follows

where satisfies the ODE in the form

the solutions of ODE (5) are

when

when

when

when

when

where are constants to be determined later,

Step 3. Substitute Equation (4) along Equation (5) into Equation (3) and collecting all the terms of the same power, and equating them to zero, we obtain a system of algebraic equations, which can be solved by Maple or Mathematica to get the values of.

Step 4. substituting these values and the solutions of Equation (5) into Equation (3) we obtain the exact solutions of Equation (1).

Here, we will apply the method described in Section 2 to find the exact traveling wave solutions and then the solitary wave slutions for the following nonlinear systems of evolution evolution equations.

The method has been successfully used to find the exact traveling wave solutions of nonlinear evolution equations. As an application, the traveling wave solutions for (2+1)-dimensional soliton breaking equation and (3+1)-dimensional Kadomstev-Petviash-vili which have been constructed using the modified simple equation method. Let us compare between our results obtained in the present article with the well-known results obtained by other authors using different methods as follows: Our results of (2+1)-dimensional soliton breaking equation and (3+1)-dimensional Kadomstev-Petviash-viliare are new and different from those obtained in [38] [39] . Figures 1-4 show the solitary wave solutions of equations. It can be concluded that this method is

reliable and propose a variety of exact solutions NPDEs. The performance of this method is effective and can be applied to many other nonlinear evolution equations.

Maha S. M.Shehata, (2015) The exp(-j(x)) Method and Its Applications for Solving Some Nonlinear Evolution Equations in Mathematical Physics. American Journal of Computational Mathematics,05,468-480. doi: 10.4236/ajcm.2015.54041