New exact solutions for the (2 + 1)-dimensional generalized Broer–Kaup system
Introduction
As we all know, nonlinear evolution equations (NEEs) are widely used to describe complex phenomena in various fields of science, such as fluid mechanics, plasma physics, hydro-dynamics, solid state physics and optical fibres. To better understand these nonlinear phenomena as well as to further them in practical life, it is important to seek more exact solutions to them. Because of the complexity of NEEs, there does not exist a uniform method to find all solutions of all NEEs. Over the past years, many powerful methods had been developed such as inverse scattering transform [1], homogeneous balance method [2], Backlund transformation [3], and Darboux transformation [4].
In 1992, Conte and Musette [5] presented a projective Riccati equation method to seek more new solitary wave solutions to NEEs that can be expressed as polynomial in two elementary functions which satisfy a projective Riccati equation [6]. The method had been applied to find many solitary wave solutions of many equations. In Ref. [7], Huang and Zhang developed a variable-coefficient projective Riccati equation method for certain nonlinear evolution equations. But the result they obtained is simple. In this paper, we improve this method by constructing two new Riccati equations which is more general than the method in [7]. Based on this method and general variable separation approach, several new families of exact solutions for the GBK are obtained.
The paper is arranged as follows. In Section 2, we briefly describe the general Riccati method. In Section 3, several types of exact soliton-like solutions and periodic-like wave solutions of the (2 + 1)-dimensional simplified generalized Broer–Kaup (GBK) system (B = 0) are obtained. In Section 4, we discuss some new solutions of the (2 + 1)-dimensional generalized Broer–Kaup (GBK) system (B ≠ 0). In Section 5, some conclusions are given.
Section snippets
Summary of the general Riccati equation method
For a given (2 + 1)-dimensional nonlinear evolution equations (NEEs) in three variables x, y and tWith the idea of general variable separation approach, by using the traveling wave transformationwhere l(y), k(y, t) are arbitrary functions with the corresponding variables. Then Eq. (2.1) reduce to a nonlinear ordinary differential equations (ODEs):where “′” denotes , so do in the following.
In order to
Exact solutions of the (2 + 1)-dimensional simplified GBK system
In Ref. [9], Zhang et al. investigated the (1 + 1)-dimensional Broer–Kaup system [10], [11]which describes the bi-directional propagation of long waves in shallow water. Then by means of the Painleve analysis, they derived the following (2 + 1)-dimensional generalized Broer–Kaup (GBK) system:where A, B, C are constant parameters of the system. Obviously, when A = B = C = 0, the GBK system will be
Exact solutions of the (2 + 1)-dimensional GBK system
When B ≠ 0, let R(x, t) = R1(x) + R2(t), Eq. (3.5) turns towe assume thatwhere ξ = lx + ny + k(t), l, n are arbitrary constants, k(t) is arbitrary function of t. Similarly, we can obtain new exact solutions of system (3.2) from Eq. (4.1) as the following:
Conclusion
In this paper, based on two general Riccati equations, we find many new exact soliton-like solutions and periodic-like solutions for the GBK system, and it is easy to see that the results in [7] are only the special case of H1, G1. By selecting some different functions of l(y), C(y), p1(y), R2(t), we can obtain rich structures of system (3.2). Of course, this method is sufficient to seek more new exact solutions to other nonlinear evolution equations [23], [24], [25], [26].
Acknowledgement
The authors express their sincere thanks to the referee for his (or her) useful suggestion.
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Supported by Nature Science Foundation of China (10420130638).