New Approach of Generalized exp (−φ (ξ)) Expansion Method and Its Application to Som Nonlinear Partial Differential Equations

In this article, the new approach of generalized exp (−φ (ξ))expansion method has been successfully implemented to seek traveling wave solutions of the Korteweg-de vries equation and the modified Zakharov-Kuznetsov equation. The result reveals that the method together with the new ordinary differential equation is a very influential and effective tool for solving nonlinear partial differential equations in mathematical physics and engineering. The obtained solutions have been articulated by the hyperbolic functions, trigonometric functions and rational functions with arbitrary constants.


Introduction
In recent years, the nonlinear partial deferential equations (NPDEs) are widely used to describe many important phenomena and dynamic processes in physics, mechanics, chemistry and biology, etc.With the development of soliton theory, many powerful methods have been presented, such as the Jacobi elliptic function expansion method (M.Inc & M. Ergut, 2005), The ( Ǵ

Description of New Approach of Generalized exp (−ϕ (ξ)) Expansion Method
Suppose that we have a nonlinear PDE in the following form : where u = u(x, t) is an unknown function F is a polynomial in u = u(x, t) and its partial derivatives, in which the highest order derivatives and nonlinear terms are involved.The main steps of this method are as follows Step 1: Use the traveling wave transformation : where k 1 , k 2 are a constants to be determined latter, permits us reducing (2.1) to an ODE for u = u (ξ) in the form where P is a polynomial of u = u(ξ) and its total derivatives.
Stepe 2: Balancing the highest derivative term with the nonlinear terms in (2.3),we find the value of the positive integer (m).
Step 5 : Suppose that the value of the constants k 1 , k 2 , λ, µ and α i (i = 0, 1, 2, ...m) can be found by solving the algebraic equations which are obtained in step 4. Since the general solutions of (5) have been well known for us, substituting k 1 , k 2 , λ, µ, α i and the general solutions of (2.5) into (2.4),we have the exact solutions of the nonlinear PDEs (2.1).

Korteweg -De Vries Equation
In this section, we will apply the The Generalized of exp (−ϕ (ξ)) expansion method to find the exact solutions of the Kdv equation.Let us consider Kdv equation : We may choose the following traveling wave transformation where k 1 , k 2 are costants to be determined later.Eq. (3.3) becomes By balancing the height order derivative term ) with the nonlinear term 3), gives (m = 2).Therefore, the generalized of exp (−ϕ (ξ)) expansion method allows us to use the solution in the following form: Substituting (3.4)and(2.5)into(3.3), the left-hand side is converted into polynomials in ( exp )) j ,( j = 0, 1, 2, ....).We collect each coefficient of these resulted polynomials to zero, yields a set of simultaneous algebraic equations (for simplicity,which are not presented) for α 0 , α 1 , , α 2 , k 1 , k 2 , λ, µ and δ.Solving these algebraic equations with the help of algebraic software Maple, we obtain Substituting (3.5) into(3.4),we have : Consequently,the exact solution of the Kdv equation (3.1) with the help of Eq. (2.6) to Eq. (2.15) are obtained in the following form: In this section, we will apply the The Generalized of exp (−ϕ (ξ)) expansion method to find the exact solutions of the modified ZK equation, Let us consider modified ZK equation : We may choose the following traveling wave transformation where k 1 , k 2 and k 3 are costants to be determined later.

Conclusion
In this paper, we have proposed the new approach of exp (−ϕ (ξ)) expansion method to construct more general exact solutions of NLEEs.With the help of Maple, the method provides a straightforward mathematical tool for obtaining more general exact solutions of NLEEs in mathematical physics.Applying this method to the Kdv Equation and modified ZK Equation, we have successfully obtained many new and more general exact solutions, these solutions have rich local structures, It may be important to explain some physical phenomena .This work shows that, the new approach of exp (−ϕ (ξ)) expansion method is direct, effective and can be used for many other NLPDEs in mathematical physics.