Traveling Wave Solutions For The Couple Boiti-Leon-Pempinelli System By Using Extended Jacobian Elliptic Function Expansion Method

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 Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics.

The objective of this article is to apply the extended Jacobian elliptic function expansion method for finding the exact traveling wave solution the Couple Boiti-Leon-Pempinelli System 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.

2-Description of method
Consider the following nonlinear evolution equation Since, P is a polynomial in and its partial derivatives.In the following, we give the main steps of this method Step 1.We use the traveling wave solution in the form where c is a positive constant, to reduce Eq.(2.1) to the following ODE: where P is a polynomial in u (ξ) and its total derivatives, while Step 2. Making good use of ten Jacobian elliptic functions we assume that (2.3) have the solutions in these forms: those have the relations with the modulus m (0 < m < 1).In addition we know that The derivatives of other Jacobian elliptic functions are obtained by using Eq.(2.8).To balance the highest order linear with nonlinear term we define the degree of u as D[u] = n gives rise to the degrees of other expressions as   ,.
According the rules, we can balance the highest order linear term and nonlinear term in Eq. ( 2.3) so that n in Eq. (2.4) can be determined.
In addition we see that when  respectively, while when therefore Eq. (2.5) degenerate as the following forms Therefore the extended Jacobian elliptic function expansion method is more general than sine-cosine method, the tanfunction method and Jacobian elliptic function expansion method.

3-The Couple Boiti-Leon-Pempinelli System
Consider the Couple Boiti-Leon-Pempinelli System [24] is in the form D e c e m b e r 07, 2 0 1 5 are the Jacobian elliptic sine function, The jacobian elliptic cosine function and the Jacobian elliptic function of the third kind and other Jacobian functions which is denoted by Glaisher's symbols and are generated by these three kinds of functions, namely D e c e m b e r 07, 2 0 1 5

(3. 1 )
Using the transformation since and integrate the first equation of the system (3.1)three times with zero constant of integration and substitute it into the second equation of system (3.1)we obtained (3.2) Balancing and So that, we assume the solution of Eq. (3.2) be in the form (3.3)