Some New Traveling Wave Exact Solutions of the (2+1)-Dimensional Boiti-Leon-Pempinelli Equations

We employ the complex method to obtain all meromorphic exact solutions of complex (2+1)-dimensional Boiti-Leon-Pempinelli equations (BLP system of equations). The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all rational and simply periodic traveling wave exact solutions of the equations (BLP) are solitary wave solutions, the complex method is simpler than other methods, and there exist some rational solutions u r,2 (z) and simply periodic solutions u s,2–6(z) which are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role for finding exact solutions in the mathematical physics. For these new traveling wave solutions, we give some computer simulations to illustrate our main results.


Introduction
Boiti et al. [1] introduced the Boiti-Leon-Pempinelli equations (BLP system of equations) A considerable research work has been invested in [2][3][4][5] to study the BLP system (1) and (2). The integrability of this system was studied in [1] by using the Sine-Gordon and the Sinh-Gordon equations. Other works have been conducted by using other methods such as Jacobi elliptic methods and balance methods [3][4][5].
In 2010, Wazwaz and Mehanna [17] used the tanh-coth method and Exp-function method to the BLP equations to derive many new varieties of travelling wave solutions with distinct physical structures. Substituting the traveling wave transformation into (1) and (2), one carries out the system of nonlinear ordinary differential equations as follows: where is the wave velocity and is a nonzero constant (see [17,23]). Afterwards we integrated (1) twice with respect to and considered the constants of integration to be zero and obtained Furthermore substituting (6) into (5) yields 2 The Scientific World Journal In 2011, Kudryashov [23] got the general solutions of (7) via Jacobi elliptic functions and analyzed the application of the tanh-coth method for finding exact solutions of (7) and showed that all the solutions which are presented by Wazwaz and Mehanna can be reduced to a single one and so on.
In this paper, we employ the complex method which was introduced by Yuan et al. [24][25][26] to obtain the general solutions and some new solutions of (7). In order to state our results, we need some concepts and notations.
A meromorphic function ( ) means that ( ) is holomorphic in the complex plane C except for poles. ℘( ; 2 , 3 ) is the Weierstrass elliptic function with invariants 2 and 3 . We say that a meromorphic function belongs to the class if is an elliptic function, or a rational function of , ∈ C, or a rational function of .
Our main result is the following theorem.

Theorem 1.
All meromorphic solutions of (7) belong to the class . Furthermore, (7) has the following three forms of solutions.
We will consider the following complex ordinary differential equations: where ̸ = 0, are constants, ∈ N. Let , ∈ N. Suppose that (18) has a meromorphic solution with at least one pole; we say that (18) satisfies weak ⟨ , ⟩ condition if, substituting Laurent series into (18), we can determine distinct Laurent singular parts below The Scientific World Journal 3 Lemma 2. Let , , , ∈ N, deg ( , ( ) ) < . Suppose that an order Briot-Bouquet equation satisfies weak ⟨ , ⟩ condition, then whose all meromorphic solutions belong to the class . If for some values of parameters such solution exists, then other meromorphic solutions form a one parametric family ( − 0 ), 0 ∈ C. Furthermore each elliptic solution with pole at = 0 can be written as where − are given by (19) Each rational function solution := ( ) is of the form with (≤ ) distinct poles of multiplicity . Each simply periodic solution is a rational function ( ) of = ( ∈ C). ( ) has (≤ ) distinct poles of multiplicity and is of the form In order to give the representations of elliptic solutions, we need some notations and results concerning elliptic function [27].
Let 1 , 2 be two given complex numbers such that 1 .
(I) Degeneracy to simply periodic functions (i.e., rational functions of one exponential ) according to By the above lemma and results, we can give a new method below, say complex method, to find exact solutions of some PDEs.
Step 5. Substituting the inverse transform −1 into these meromorphic solutions ( − 0 ), then we get all exact solutions ( , ) of the original given PDE.

Computer Simulations for New Solutions
In this section, we give some computer simulations to illustrate our main results. Here we take the new rational 6 The Scientific World Journal solutions ,2 ( ) and simply periodic solutions u s,2−6 (z) to further analyze their properties by Figures 1, 2, 3, 4, 5, and 6.

Conclusions
Complex method is a very important tool in finding the exact solutions of nonlinear evolution equations, and the (2+1)dimensional Boiti-Leon-Pempinelli equation is classic and simply periodic traveling wave exact solutions of the equations (BLP) are solitary wave solutions, the complex method is simpler than other methods, and there exist some rational solutions ,2 ( ) and simply periodic solutions u s,2−6 (z) which are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role for finding exact solutions in the mathematical physics. For these new traveling wave solutions, we give some computer simulations to illustrate our main results.