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
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
References
- Ablowitz, MJ, Clarkson, PA: Soliton, Nonlinear Evolution Equations and Inverse Scattering. Cambridge Universty Press, New York (1991) View ArticleGoogle Scholar
- Matveev, VB, Salle, MA: Darboux Transformation and Solitons. Springer, Berlin (1991) View ArticleGoogle Scholar
- Miura, MR: Backlund Transformation. Springer, Berlin (1978) Google Scholar
- Hirota, R: Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192-1194 (1971) View ArticleMATHGoogle Scholar
- Parkes, EJ, Duffy, BR: An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. Comput. Phys. Commun. 98, 288-300 (1996) View ArticleMATHGoogle Scholar
- Fan, EG: Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A 277, 212-218 (2000) View ArticleMATHMathSciNetGoogle Scholar
- Yan, CT: A simple transformation for nonlinear waves. Phys. Lett. A 224, 77-84 (1996) View ArticleMATHMathSciNetGoogle Scholar
- Wang, ML: Exact solutions for a compound KdV-Burgers equation. Phys. Lett. A 213, 279-287 (1996) View ArticleMATHMathSciNetGoogle Scholar
- Chun, C, Sakthivel, R: Homotopy perturbation technique for solving two-point boundary value problems-comparison with other methods. Comput. Phys. Commun. 181, 1021-1024 (2010) View ArticleMATHMathSciNetGoogle Scholar
- Sakthivel, R, Chun, C, Lee, J: New travelling wave solutions of Burgers equation with finite transport memory. Z. Naturforsch. A 65, 633-640 (2010) Google Scholar
- Abdou, MA: The extended F-expansion method and its application for a class of nonlinear evolution equations. Chaos Solitons Fractals 31, 95-104 (2007) View ArticleMATHMathSciNetGoogle Scholar
- He, JH: Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 30, 700-708 (2006) View ArticleMATHMathSciNetGoogle Scholar
- Sakthivel, R, Chun, C: New soliton solutions of Chaffee-Infante equations using the Exp-function method. Z. Naturforsch. A 65, 197-202 (2010) Google Scholar
- Zayed, EME, Gepreel, KA: The \((G'/G)\)-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics. J. Math. Phys. 50, 013502 (2009) View ArticleMathSciNetGoogle Scholar
- Kim, H, Sakthivel, R: New exact traveling wave solutions of some nonlinear higher-dimensional physical models. Rep. Math. Phys. 70, 39-50 (2012) View ArticleMATHMathSciNetGoogle Scholar
- Kim, H, Bae, JH, Sakthivel, R: Exact travelling wave solutions of two important nonlinear partial differential equations. Z. Naturforsch. A 69, 155-162 (2014) View ArticleGoogle Scholar
- Lou, SY, Ni, GJ: The relations among a special type of solitons in some \((D+1)\) dimensional nonlinear equations. J. Math. Phys. 30, 1614-1620 (1989) View ArticleMATHMathSciNetGoogle Scholar
- Bai, CJ, Zhao, H, Xu, HY, Zhang, X: New traveling wave solutions for a class of nonlinear evolution equations. Int. J. Mod. Phys. B 25, 319-327 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Hirota, R, Ito, M: Resonance of solitons in one dimension. J. Phys. Soc. Jpn. 52, 744-748 (1983) View ArticleMathSciNetGoogle Scholar
- Hirota, R: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004) View ArticleMATHGoogle Scholar
- Hirota, R: Exact solutions of the modified Korteweg-de Vries equation for multiple collisions of solitons. J. Phys. Soc. Jpn. 33, 1456-1458 (1972) View ArticleGoogle Scholar
- Hirota, R: Exact solutions of the Sine-Gordon equation for multiple collisions of solitons. J. Phys. Soc. Jpn. 33, 1459-1463 (1972) View ArticleGoogle Scholar
- Hietarinta, J: A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations. J. Math. Phys. 28, 1732-1742 (1987) View ArticleMATHMathSciNetGoogle Scholar
- Hietarinta, J: A search for bilinear equations passing Hirota’s three-soliton condition. II. mKdV-type bilinear equations. J. Math. Phys. 28, 2094-2101 (1987) View ArticleMATHMathSciNetGoogle Scholar
- Wazwaz, AM: Partial Differential Equations and Solitary Waves Theory. Higher Education Press, Beijing (2009) View ArticleMATHGoogle Scholar
- Sachs, RL: On the integrable variant of the Boussinesq system: Painlevé property, rational solutions, a related many-body system, and equivalence with the AKNS hierarchy. Physica D 30, 1-27 (1988) View ArticleMATHMathSciNetGoogle Scholar
- Wang, ML: Solitary wave solutions for variant Boussinesq equations. Phys. Lett. A 199, 169-172 (1995) View ArticleMathSciNetGoogle Scholar
- Yan, ZY, Zhang, HQ: New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics. Phys. Lett. A 252, 291-296 (1999) View ArticleMATHMathSciNetGoogle Scholar
- Naz, R, Mahomed, FM, Hayat, T: Conservation laws for third-order variant Boussinesq system. Appl. Math. Lett. 23, 883-886 (2010) View ArticleMATHMathSciNetGoogle Scholar
- Fan, EG, Hon, YC: A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves. Chaos Solitons Fractals 15, 559-566 (2003) View ArticleMATHMathSciNetGoogle Scholar
- Lü, DZ: Jacobi elliptic function solutions for two variant Boussinesq equations. Chaos Solitons Fractals 24, 1373-1385 (2005) View ArticleMATHMathSciNetGoogle Scholar
- Yuan, YB, Pu, DM, Li, SM: Bifurcations of travelling wave solutions in variant Boussinesq equations. Appl. Math. Mech. 27, 811-822 (2006) View ArticleMATHMathSciNetGoogle Scholar
- Li, H, Ma, LL, Feng, DH: Single-peak solitary wave solutions for the variant Boussinesq equations. Pramana 80, 933-944 (2013) View ArticleGoogle Scholar