In this work, an extended Jacobian elliptic function expansion method is proposed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to Dynamical system in a new Double-Chain Model of DNA and a diffusive predator-prey system which play an important role in biology.
The nonlinear partial differential equations of mathematical physics are major subjects in physical science [
metric function series method [
and so on.
The objective of this article is to apply the extended Jacobian elliptic function expansion method for finding the exact traveling 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 extended Jacobi elliptic function expansion 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 F is polynomial in
Step 1. Using the transformation
where k and c are the wave number and wave speed, to reduce Equation (1) to the following ODE:
where P is a polynomial in
Step 2. Making good use of ten Jacobian elliptic functions, we assume that (3) has the solutions in these forms:
With
where
That have the relations
With the modulus m
The derivatives of other Jacobian elliptic functions are obtained by using Equation (8). To balance the highest order linear term with nonlinear term we define the degree of u as
According the rules, we can balance the highest order linear term and nonlinear term in Equation (3) so that n in Equation (4) can be determined.
In addition we see that when
Therefore the extended Jacobian elliptic function expansion method is more general than sine-cosine method, the tan-function method and Jacobian elliptic function expansion method.
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 non linearity is respon- sible 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 a and b are constants, to reduce Equations (14) and (15) to the following system of equations:
and
Comparing Equations (18) and (19) and using (17) we deduce that
and (19) can be written as
where
The wave transformation
where
where
Substituting Equations (23) and (25) into Equation (22) and equating all coefficients of
Solving the above system with the aid of Maple or Mathematica, we have the following solitary wave solution:
Case 1.
Case 2.
Case 3.
Sothat solution of Equation (22) has the form
Case 1.
Case 2.
Case 3.
Consider a system of two coupled nonlinear partial differential equations describing the spatio-temporal dy- namics of a predator-prey system [
where
relations between the parameters, namely
can be rewritten in the form:
We use the wave transformation
where c is a nonzero constant.
In order to solve Equation (41), let us consider the following transformation
Substituting the transformation (42) into Equation (41), we get
Balancing
solving Equations (44)-(50) using the maple or mathematica program to get solitary wave solution of equations we get
So we get
when
We establish exact solutions for the dynamics of DNA molecules which 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 All effect. The extended Jacobian elliptic function 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 diffusive predator-prey system, which have been constructed using the extended Jacobian elliptic function 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 the system of shallow water wave equations and a diffusive predator-prey system, are new and different from those obtained in [