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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 the system of shallow water wave equations and modified Liouville equation which play an important role in mathematical physics.

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

[

method [

The objective of this article is to apply the extended Jacobian elliptic function expansion method for finding the exact traveling wave solution the system of shallow water wave equations and modified Liouville equation which play an important role in 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 c is 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.

Noticed that

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.

We first consider the system of the shallow water wave equation [

We use the wave transformation

where by integrating once the second equation with zero constant of integration, we find

substituting Equation (16) into the first equation of Equation (15) we obtain

Integrating Equation (17) with zero constant of integration, we find

Balancing

where

Substituting Equations (19) and (21) into Equation (18) and equating all coefficients of

Solving the above system with the aid of Maple or Mathematica, we have the following solution:

Case 1.

So that the solution of Equation (18) can be written as

when

Case 2.

So that the solution of Equation (18) can be written as

when

Case 3.

So that the solution of Equation (18) can be written as

when

Now, let us consider the modified Liouville equation [

respectively, where

Balancing

where

Substituting (37) and (39) into Equation (36) and equating all the coefficients of

Solving the above system with the aid of Maple or Mathematica, we have the following solution:

So that the solve of Equation (36) can be written in the form

When

We establish exact solutions for the system of shallow water wave equations and modified Liouville equation which are two of the most fascinating problems of modern mathematical physics.

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 the system of shallow water wave equations and modified Liouville equation, 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 modified Liouville equation are new and different from those obtained in [

(30) and (56). 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.