Abstract
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 and simply periodic solutions 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.
1. Introduction
Boiti et al. [1] introduced the Boiti-Leon-Pempinelli equations (BLP system of equations)
A considerable research work has been invested in [2–5] 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 [3–5].
For finding exact solutions of the BLP system, many authors applied the tanh-coth method and the Exp-function method to derive them [6–17] and [8–22], respectively.
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
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. [24–26] 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 except for poles. is the Weierstrass elliptic function with invariants and . We say that a meromorphic function belongs to the class if is an elliptic function, or a rational function of , 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.
(I) The elliptic general solutions
where , , and and are arbitrary constants.
(II) The simply periodic solutions, where are obtained, for ,
where ;
where ;
where ;
where ;
where ;
where .
(III) All rational function solutions are of the following two distinct forms. For any ,
where .
2. Preliminary Lemmas and the Complex Method
In order to give complex method and the proof of Theorem 1, we need some notations and results.
Set , , , . We define a differential monomial denoted by and are called the weight and degree of , respectively.
A differential polynomial is defined as follows: where are constants and is a finite index set. The total weight and degree of are defined by and , respectively.
We will consider the following complex ordinary differential equations: where are constants, .
Let . 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
Lemma 2. Let , . 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 . Furthermore each elliptic solution with pole at can be written as
where are given by (19), , and .
Each rational function solution is of the form
with distinct poles of multiplicity .
Each simply periodic solution is a rational function of . 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 be two given complex numbers such that be discrete subset , which is isomorphic to . The discriminant and
Weierstrass elliptic function is a meromorphic function with double periods and satisfying the equation where and .
If we change (24) to the form we have .
Inversely, given two complex numbers and such that , then there exists double periods Weierstrass elliptic function such that the above results hold.
Lemma 3 (see [27, 28]). Weierstrass elliptic functions have two successive degeneracies and addition formula.
(I) Degeneracy to simply periodic functions (i.e., rational functions of one exponential ) according to
if one root is double ().
(II) Degeneracy to rational functions of according to
if one root is triple ().
(III) Addition formula
By the above lemma and results, we can give a new method below, say complex method, to find exact solutions of some PDEs.
Step 1. Substituting the transform into a given PDE gives a nonlinear ordinary differential equation (18) or (21).
Step 2. Substitute (19) into (18) or (21) to determine that weak condition holds.
Step 3. By determinant relation (22)–(24) we find the elliptic, rational, and simply periodic solutions of (18) or (21) with pole at , respectively.
Step 4. By Lemmas 2 and 3 we obtain all the meromorphic solutions .
Step 5. Substituting the inverse transform into these meromorphic solutions , then we get all exact solutions of the original given PDE.
3. Proof of Theorem 1
Substituting (19) into (7) we have . , . Hence, (7) satisfies weak condition and is a 2nd order Briot-Bouquet differential equation. Obviously, (7) satisfies the dominant condition. So, by Lemma 2, we know that all meromorphic solutions of (7) belong to . Now we will give the forms of all meromorphic solutions of (7).
By (22), we infer the indeterminant rational solutions of (7) with pole at that Substituting into (7), we get two distinct forms. One of them is where . The other is where .
Thus all rational solutions of (7) are where .
In order to have simply periodic solutions, set , put into (7), and then Substituting into (7), we obtain the indeterminant simply periodic solutions of (35) with pole at that where ; where ; where ; where ; where ; where .
Substitute into the above six relations, and then we get all simply periodic solutions of (7) with pole at where ; where ; where ; where ; where ; where .
So all simply periodic solutions of (7) are obtained, for , by where ; where ; where ; where ; where ; where .
From (21) in Lemma 2, we have indeterminant relations of elliptic solutions of (7) with pole at where . Applying conclusion II of Lemma 2 to and noting the results of rational solutions obtained above, we deduce that , , and . Then we get that Therefore, all elliptic function solutions of (7) Here . Making use of the addition of Lemma 3, we rewrite it to the form Here , , , and are arbitrary constants.
This completes the proof of Theorem 1.
4. Computer Simulations for New Solutions
In this section, we give some computer simulations to illustrate our main results. Here we take the new rational solutions and simply periodic solutions us,2–6(z) to further analyze their properties by Figures 1, 2, 3, 4, 5, and 6.
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5. 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 simplest case of the nonlinear reaction-diffusion equation. In this paper, we employ the complex method to obtain the general meromorphic solutions of the (2+1)-dimensional Boiti-Leon-Pempinelli equation, which improves the corresponding result obtained by Kudryashov [23] and Wazwaz and Mehanna [17]. 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 and simply periodic solutions us,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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the NSF of China (11271090), Tianyuan Youth Fund of the NSF of China (11326083), Shanghai University Young Teacher Training Program (ZZSDJ12020), Innovation Progrom of Shanghai Municipal Education Commission (14YZ164), the NSF of Guangdong Province (S2012010010121), and Projects 13XKJC01 and 10XKJ01 from the Leading Academic Discipline Project of Shanghai Dianji University. The first and third authors would like to express their hearty thanks to Professor R. Conte, T. W. Ng, Fang Mingliang, and Liao Liangwen for their helpful discussions and suggestions. This work was supported by the Visiting Scholar Program of Chern Institute of Mathematics at Nankai University when the first and third authors worked as visiting scholars.