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The exp(-φ(ξ)) -expansion method is used as the first time to investigate the wave solution of a nonlinear dynamical system in a new double-Chain model of DNA and a diffusive predator-prey system. The proposed method also can be used for many other nonlinear evolution equations.

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

trigonometric function series method [

The objective of this article is to apply the exp

wave solution of dynamical system in a new double-Chain model of DNA and a diffusive predator-prey system which play an important role in biology and mathematical physics.

The rest of this paper is organized as follows: In Section 2, we give the description of the exp

pansion method. In Section 3, we use this method to find the exact solutions of the nonlinear evolution equations pointed out above. In Section 4, conclusions are given.

Consider the following nonlinear evolution equation

where

Step 1. We use the wave transformation

where

where

Step 2. Suppose that the solution of ODE (2.3) can be expressed by a polynomial in

where

the solutions of ODE (2.5) are when

when

when

when

when

where

Step 3. Substitute Equation (2.4) along Equation (2.5) into Equation (2.3) and collecting all the terms of the same power

tions, which can be solved by Maple or Mathematica to get the values of

Step 4. substituting these values and the solutions of Equation (2.5) into Equation (2.3) we obtain the exact solutions of Equation (2.3).

An attractive nonlinear model for the nonlinear science in the deoxyribonucleic acid (DNA). The dynamics of DNA molecules is one of the most fascinating problems of modern biophysics because it is at the basis of life. The DNA structure has been studied during last decades. The investigation of DNA dynamics has successfully predicted the appearance of important nonlinear structures. It has been shown that the nonlinearity is responsible for forming localized waves. These localized waves are interesting because they have the capability to transport energy without dissipation [

where

where

we first introduce the transformation

where

and

Comparing Equations (3.5) and (3.6) and using (3.4) we deduce that

where

The wave transformation

where

where

substituting Equation (3.10) and its derivatives in Equation (3.9) and equating the coefficient of different power’s of

Equations (3.12)-(3.15) yields

Thus the solution is

Let us now discuse the following case:

Case 1. if

Case 2. if

Case 3. if

Case 4. if

Case 5. if

Consider a system of two coupled nonlinear partial differential equations describing the spatio-temporal dyna- mics of a predator-prey system [

where

relations between the parameters, namely

We use the wave transformation

where

In order to solve Equation (3.24), let us consider the following transformation

Substituting the transformation (3.25) into Equation (3.24), we get

Balancing

Equations (3.27)-(3.30) yields

Case 1.

Case 2.

Thus the solution is

Case 1.

Case 2.

Let us now discuss the following cases:

Case 1.

Case (1.1). if

Case (1.2). if

Case (1.3). if

Case (1.4). if

Case (1.5). if

Case 2.

Case (2.1). if

Case (2.2). if

Case (2.3). if

Case (2.4). if

Case (2.5). if

We establish exact solutions for the dynamics of DNA molecules is one of the most fascinating problems of modern biophysics because it is at the basis of life. The DNA structure has been studied during last decades. The investigation of DNA dynamics has successfully predicted the appearance of important nonlinear structures and a system of two coupled nonlinear partial differential equations describing the spatio-temporal dynamics of a

predator-prey system where the prey per capita growth rate is subject to the Allee effect. The

expansion method has been successfully used to find the exact traveling wave solutions of some nonlinear evolution equations. As an application, the traveling wave solutions for Dynamical system in a new Double- Chain Model of DNA and a diffuusive predator-prey system, which have been constructed using the

expansion 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 Dynamical system in a new Double-Chain Model of DNA and a diffusive predator-prey system, are new and different from those obtained in [

Double-Chain Model of DNA and a diffusive predator-prey system. It can be concluded that this method is reliable and proposes a variety of exact solutions NPDEs. The performance of this method is effective and can be applied to many other nonlinear evolution equations.

Mahmoud A. E.Abdelrahman,Emad H. M.Zahran,Mostafa M. A.Khater, (2015) The exp(-φ(ξ))-Expansion Method and Its Application for Solving Nonlinear Evolution Equations. International Journal of Modern Nonlinear Theory and Application,04,37-47. doi: 10.4236/ijmnta.2015.41004