Extended Mapping Method and Its Applications to Nonlinear Evolution Equations

We use extended mapping method and auxiliary equation method for finding new periodic wave solutions of nonlinear evolution equations in mathematical physics, and we obtain some new periodic wave solution for the Boussinesq system and the coupled KdV equations. This method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.


Introduction
The effort in finding exact solutions to nonlinear equations is important for the understanding of most nonlinear physical phenomena. For instance, the nonlinear wave phenomena observed in fluid dynamics, plasma, and optical fibers are often modeled by the bellshaped sech solutions and the kink-shaped tanh solutions. Many effective methods have been presented, such as inverse scattering transform method 1 , Bäcklund transformation 2 , Darboux transformation 3 , Hirota bilinear method 4 , variable separation approach 5 , various tanh methods 6-9 , homogeneous balance method 10 , similarity reductions method 11, 12 , G /G -expansion method 13 , the reduction mKdV equation method 14 , the trifunction method 15, 16 , the projective Riccati equation method 17 , the Weierstrass elliptic function method 18 , the Sine-Cosine method 19, 20 , the Jacobi elliptic function expansion 21, 22 , the complex hyperbolic function method 23 , the truncated Painlevé expansion 24 , the F-expansion method 25 , the rank analysis method 26 , the ansatz method 27, 28 , the exp-function expansion method 29 , and the sub-ODE method 30 .
The main objective of this paper is using the extended mapping method to construct the exact solutions for nonlinear evolution equations in the mathematical physics via the Boussinesq system and the coupled KdV equations.

Description of the Extended Mapping Method
Suppose we have the following nonlinear PDE: where u u x, t is an unknown function, F is a polynomial in u u x, t and its various partial derivatives in which the highest order derivatives and nonlinear terms are involved. In the following we give the main steps of a deformation method.
Step Step 2. If all the terms of 2.3 contain derivatives in ζ, then by integrating this equation and taking the constant of integration to be zero, we obtain a simplified ODE.
Step 3. Suppose that the solution 2.3 has the following form: where p, q, and r are constants.
Step 4. The positive integer "n" can be determined by considering the homogeneous balance between the highest derivative term and the nonlinear terms appearing in 2.3 . Therefore, we can get the value of n in 2.4 .

Applications of the Method
In this section, we apply the extended mapping method to construct the exact solutions for the Boussinesq system and the coupled KdV equations, which are very important nonlinear evolution equations in mathematical physics and have been paid attention by many researchers.
Example 3.1 the Boussinesq system . We start the Boussinesq system 32 in the following form:

3.1
The traveling wave variable 2.2 permits us converting 3.1 into the following ODE: ωu v 0.

3.2
Integrating 3.2 with respect to ξ once and taking the constant of integration to be zero, we obtain Suppose that the solutions of 3.3 and 3.4 can be expressed by and H i are constants to be determined later.

Journal of Applied Mathematics
Considering the homogeneous balance between the highest order derivative u and the nonlinear term u 2 in 3.3 , the order of u and v in 3.4 , then we can obtain n m 2, hence the exact solutions of 3.5 can be rewritten as,

3.10
Journal of Applied Mathematics

5
Note that there are other cases which are omitted here. Since the solutions obtained here are so many, we just list some of the exact solutions corresponding to Case 4 to illustrate the effectiveness of the extended mapping method. Substituting 3.10 into 3.6 yields According to Appendix A, we have the following families of exact solutions.
Journal of Applied Mathematics

3.23
Journal of Applied Mathematics Similarly, we can write down the other families of exact solutions of 3.1 which are omitted for convenience.

3.28
Integrating 3.2 with respect to ξ once and taking the constant of integration to be zero, we obtain , and e i are constants to be determined later.
Balancing the order of u and v 2 in 3.29 , the order of v and uv in 3.30 , then we can obtain n m 2, so 3.31 can be rewritten as

3.36
Note that there are other cases which are omitted here. Since the solutions obtained here are so many, we just list some of the exact solutions corresponding to Case 4 to illustrate the effectiveness of the extended mapping method.

A. The Jacobi Elliptic Functions
The general solutions to the Jacobi elliptic equation 2.3 and its derivatives 31 are listed in Table 1, where 0 < m < 1 is the modulus of the Jacobi elliptic functions and i √ −1.

C. Relations between the Jacobi Elliptic Functions
See Table 3.