New exact solutions for the (2 + 1)-dimensional generalized Broer–Kaup system

doi:10.1016/j.amc.2007.10.012

Applied Mathematics and Computation

Volume 199, Issue 2, 1 June 2008, Pages 572-580

https://doi.org/10.1016/j.amc.2007.10.012 Get rights and content

Abstract

The (1 + 1)-dimensional Broer–Kaup system, which describes the propagation of shallow water waves, is extended to a generalized (2 + 1)-dimensional model with Painleve property. In this paper, based on the general variable separation approach and two extended Riccati equations, we first find several new families of exact soliton-like solutions and periodic-like wave solutions with arbitrary functions for the (2 + 1)-dimensional simplified generalized Broer–Kaup (GBK) system (B = 0). Abundant new localized excitations can be found by selecting appropriate functions. After that, we consider the conditions of (B ≠ 0) to the GBK, and several new results are obtained.

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 t $P (u, u_{t}, u_{x}, u_{y}, u_{tt}, u_{xt}, u_{yt}, u_{xx}, u_{yy}, \dots) = 0 .$ With the idea of general variable separation approach, by using the traveling wave transformation $u (x, y, t) = u (ξ), ξ = ξ (x, y, t) = l (y) x + k (y, t),$ where l(y), k(y, t) are arbitrary functions with the corresponding variables. Then Eq. (2.1) reduce to a nonlinear ordinary differential equations (ODEs): $O (u, u^{'}, u^{″}, u^{‴}, \dots) = 0,$ where “′” denotes $\frac{d}{d ξ}$ , 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] $\{\begin{matrix} u_{t} = {uu}_{x} + v_{x} - \frac{1}{2} u_{xx}, \\ v_{t} = (uv)_{x} + \frac{1}{2} v_{xx}, \end{matrix}$ 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: $\{\begin{matrix} H_{t} - H_{xx} + 2 {HH}_{x} + U_{x} + AU + BG = 0, \\ G_{t} + 2 (GH)_{x} + G_{xx} + 4 A (G_{x} - H_{xy}) + 4 B (G_{y} - H_{yy}) + C (G - 2 H_{y}) = 0, \\ U_{y} = G_{x}, \end{matrix}$ 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 to $H_{t} + 2 {HH}_{x} + H_{xx} + 2 {AH}_{x} + 2 {BH}_{y} = R_{1} (x) + R_{2} (t),$ we assume that $H = p (t) + a_{1} (y) f (ξ) + b_{1} (y) g (ξ) + a_{- 1} (y) f^{- 1} (ξ) + b_{- 1} (y) g^{- 1} (ξ),$ where 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: $\begin{matrix} H_{13} = p (t) + \frac{\pm al (y) q \sqrt{r^{2} + ε} + l (y) q [b \sinh (q ξ_{13}) + c \cosh (q ξ_{13})]}{2 b \cosh (q ξ_{13}) + 2 c \sinh (q ξ_{13}) + 2 ar}, \\ H_{23} = p (t) + \frac{l (y) q [b \sinh (q ξ_{23}) + c \cosh (q ξ_{23})]}{2 b \cosh (q ξ_{23}) + 2 c \sinh (q ξ_{23}) \pm 2 a \sqrt{- ε}}, \\ H_{33} = p (t) + l (y) q [b \sinh ( \end{matrix}$

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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¹: Supported by Nature Science Foundation of China (10420130638).

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New exact solutions for the (2 + 1)-dimensional generalized Broer–Kaup system

Abstract

Introduction

Section snippets

Summary of the general Riccati equation method

Exact solutions of the (2 + 1)-dimensional simplified GBK system

Exact solutions of the (2 + 1)-dimensional GBK system

Conclusion

Acknowledgement

Phys. Lett. A

Chaos Solitons Fract.

Phys. Lett. A

Commun. Theor. Phys.

Nonlinear Anal-Theor.

Chaos Solitons Fract.

Chaos Solitons Fract.

Chaos, Solitons Fract.

Chaos, Solitons Fract.

Comput. Math. Appl.

Solitons, Nonlinear Evolution Equations and Inverse Scattering

Backlund transformation and n-soliton-like solutions to the combined KdV-Burgers equation with variable coefficients

Int. J. Nonlinear Sci.

Darbooux Transformation and Soliton