_{1}

^{*}

The exp(-j(x)) method is employed to find the exact traveling wave solutions involving parameters for nonlinear evolution equations. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact traveling wave solutions. It is shown that the exp(-j(x)) method provides an effective and a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. Comparison between our results and the well-known results will be presented.

The nonlinear partial differential equations of mathematical physics are major subjects in physical science [

The objective of this article is to investigate more applications than obtained in [

demonstrate the advantages of the

soliton breaking equation [

Consider the following nonlinear evolution equation

where F is a polynomial in

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

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

where

the solutions of ODE (5) are

when

when

when

when

when

where

Step 3. Substitute Equation (4) along Equation (5) into Equation (3) and collecting all the terms of the same power

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

Let us consider the (2+1)-dimensional breaking soliton equations [

where

Using the wave variable

Integrating the second equation in the system and neglecting constant of integration we find

Substituting (13) into the first equation of the system and integration we find

Balancing

where

Substituting (15) and (17) into Equation (14) and equating all the coefficients of

From Equations (18)-(22), we have the following results:

Case 1.

Case 2.

So that the exact solution of Equation (14)

Case 1.

when

when

when

when

when

Case 2.

when

when

when

when

when

We next consider the (3+1)-dimensional KP equation

Xie et al. [

Using the wave variable

Integrating twice and setting the constants of integration to zero, we obtain

Balancing

Substituting (15)-(17) into Equation (35) and equating the coefficients of

From Equations (36)-(40), we have the following results:

Case 1.

Case 2.

So that the exact solution of equation

Case 1.

when

when

when

when

when

Case 2.

when

when

when

when

when

The

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