- Research
- Open Access
Multiple soliton solutions for the variant Boussinesq equations
- Peng Guo^{1, 2, 3}Email author,
- Xiang Wu^{2, 3} and
- Liang-bi Wang^{2, 3}
https://doi.org/10.1186/s13662-015-0371-4
© Guo et al.; licensee Springer. 2015
- Received: 28 September 2014
- Accepted: 13 January 2015
- Published: 31 January 2015
Abstract
The Hirota bilinear method is used to handle the variant Boussinesq equations. Multiple soliton solutions and multiple singular soliton solutions are formally established. It is shown that the Hirota bilinear method may provide us with a straightforward and effective mathematic tool for generating multiple soliton solutions of nonlinear wave equations in fluid mechanics.
Keywords
- Hirota bilinear method
- the variant Boussinesq equations
- multiple soliton solutions
- multiple singular soliton solutions
1 Introduction
Many phenomena in physics, biology, chemistry, mechanics, etc. are described by nonlinear partial differential equations. Nonlinear wave phenomena of dissipation, diffusion, reaction, and convection are very important, and they can be represented with a variety of nonlinear wave equations. The investigation of exact solutions of these equations will help ones to understand these phenomena better.
During the past several decades, many powerful and efficient methods have been proposed to obtain the exact solutions of nonlinear wave equations, such as inverse scattering method [1], Darboux and Bäcklund transformations [2, 3], the Hirota bilinear method [4], the tanh method [5], the extended tanh method [6], the sine-cosine method [7], the homogeneous balance method [8], the homotopy perturbation method [9, 10], the F-expansion method [11], the Exp-function expansion method [12, 13], the \((G'/G)\)-expansion method [14, 15], the Kudryashov method [16], the mapping method [17], the extended mapping method [18], and so on.
The above methods can be used to handle the nonlinear wave equations for single soliton solutions, but the multiple soliton solutions of the nonlinear wave equations can be obtained only by three different methods: the inverse scattering method, the Bäcklund transformation method, and the Hirota bilinear method. However, the Hirota bilinear method is rather heuristic and possesses significant features that make it be ideal for the determination of multiple soliton solutions for a wide class of the nonlinear wave equations [19–25]. When the Hirota bilinear method is used, computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations.
The above review shows that many works to obtain the exact solutions of Eq. (1) have been carried out in recent years, but the multiple soliton solutions for Eq. (1) have not been obtained. The existence of multiple soliton solutions often implies the integrability of the considered equations. The objectives of this paper are twofold. First, we aim to apply the Hirota bilinear method to handle Eq. (1). Second, we seek to establish multiple soliton solutions and multiple singular soliton solutions to confirm that Eq. (1) is completely integrable. The rest of this paper is organized as follows. In Section 2, the Hirota bilinear method for finding the multiple soliton solutions of the nonlinear wave equations is described. In Sections 3 and 4, the method to solve Eq. (1) is illustrated in detail. Multiple soliton solutions and multiple singular soliton solutions are obtained. In Section 5, some conclusions are given.
2 The Hirota bilinear method
The Hirota direct method is well known, and it gives soliton solutions by polynomials of exponentials. We only summarize the main steps as follows.
However, for multiple singular soliton solutions, we apply the following steps:
3 Multiple soliton solutions of the variant Boussinesq equations
4 Multiple singular soliton solutions of the variant Boussinesq equations
5 Conclusions
The Hirota bilinear method is applied to emphasize the integrability of the variant Boussinesq equations. Multiple soliton solutions and multiple singular soliton solutions are formally derived. The analysis confirms the fact that the variant Boussinesq equations have N soliton solutions, and have N singular soliton solutions simultaneously. The results obtained for the phase shift \(a_{ij}\) show that the system is resonance free. The method used here is standard and direct, so we believe that multiple soliton solutions and multiple singular soliton solutions may exist for other classes of nonlinear mathematic physics models, such as the coupled Kadomtsev-Petviashvili system, the Davey-Stewartson system, the generalized Hirota-Satsuma coupled KdV system, and so on. Further work on these aspects is worthy of performing.
Declarations
Acknowledgements
The authors would like to express their sincere thanks to editors and referees for their valuable suggestions and comments. This work was supported by the National Natural Science Foundation of China under Grant no. 11464027, the Scientific Research Foundation of the Higher Education Institutions of Gansu Province under Grant no. 2014A-053, and the Young Scholars Science Foundation of Lanzhou Jiaotong University under Grant no. 2013026.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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