Abstract and Applied Analysis

Volume 2015, Article ID 834521, 5 pages

http://dx.doi.org/10.1155/2015/834521

## Bilinear Form and Two Bäcklund Transformations for the (3+1)-Dimensional Jimbo-Miwa Equation

^{1}Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing 100191, China^{2}State Key Laboratory of Information Photonics and Optical Communications and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

Received 27 February 2015; Accepted 12 May 2015

Academic Editor: Luiz Duarte

Copyright © 2015 He Li and Yi-Tian Gao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

With Bell polynomials and symbolic computation, this paper investigates the (3+1)-dimensional Jimbo-Miwa equation, which is one of the equations in the Kadomtsev-Petviashvili hierarchy of integrable systems. We derive a bilinear form and construct a bilinear Bäcklund transformation (BT) for the (3+1)-dimensional Jimbo-Miwa equation, by virtue of which the soliton solutions are obtained. Bell-polynomial-typed BT is also constructed and cast into the bilinear BT.

#### 1. Introduction

Dynamical systems, such as those for the shallow waters [1, 2], plasmas and optical fiber communications [3–6], can often be described by the nonlinear evolution equations (NLEEs) [7–9] and studied by the relevant methods including the inverse scattering [1], Bäcklund transformation (BT) [10–13], and Hirota method [14–16]. Among them, the Hirota method [17, 18] is a direct tool for dealing with certain NLEEs and relevant soliton problems [19, 20]. Based on the bilinear form of a given NLEE, one can obtain the multisoliton solutions [21], bilinear auto-BTs [18], nonlinear superposition formulas, Lax pair, Wronskian formulation [22], and so on [23].

Reflecting the complex nonlinear phenomena in our real world [24–26], higher-dimensional NLEEs with their analytic solutions and integrable properties [27–29] have been of great interest. In fact, some -dimensional NLEEs have been investigated with different methods, for example, the -dimensional breaking soliton equation, Kadomtsev-Petviashvili equation, and -dimensional Kaup-Kupershmidt equation [30–32]. However, for some -dimensional NLEEs, the conventional integrability test fails [27], and then a natural problem is whether or not there exists BT for a given -dimensional NLEE. Moreover, for the higher-dimensional NLEEs, finding a bilinear BT via the exchange formula is often difficult, even if possible [18, 21].

In this paper, we will study the following -dimensional Jimbo-Miwa (JM) equation [32]:where is a real scalar function with four independent variables , , , and and the subscripts denote the corresponding partial derivatives. Seen as one of the equations in the Kadomtsev-Petviashvili hierarchy of integrable systems [32, 33], (1) describes certain -dimensional waves [13, 32] but does not have the Painlevé property [34] as defined in [35]. The soliton [36, 37], periodic [15], rational, and dromion solutions [38, 39] for (1) have been obtained. BTs and analytic solitonic solutions have been given in [13] with the truncated Painlevé expansion at the constant level term.

However, existing literature has not studied the bilinear BT and Bell-polynomial-typed BT of (1) as yet. Therefore, in this paper, by means of the Bell polynomials and Hirota bilinear method, we will obtain two BTs for (1), which are different from those in [13]. In Section 2, we will introduce some concepts on the Bell polynomials and their connection with the bilinear forms. In Section 3, using the Bell-polynomial expressions, we will derive a bilinear form of (1). In Section 4, based on this bilinear form, we will obtain a bilinear BT with soliton solutions and a Bell-polynomial-typed BT. Finally, our conclusions will be given in Section 5.

#### 2. Preliminaries

Suppose that is -function with respect to , and set . Then the Bell exponential polynomials are given as [40–43]where .

For example,Two-dimensional Bell polynomials are expressed as [40–43]with hereby being -function of and .

Based on the Bell polynomials given above, the binary Bell polynomials, namely, -polynomials, can be defined as [41]where the vertical line means that the elements on the left-hand side are chosen according to the rule on the right-hand side, while and are the functions that replace in the corresponding positions of the Bell polynomials. For simplicity, we denote as or if or , respectively.

As one special kind of -polynomials, -polynomials only possess the even-order partial differential terms and, with , are defined as [40, 41]which vanish unless is even.

According to the above, the lower-order -polynomials can be given as

For a given pair of exponentials,where and are -functions of and , while the Hirota -operators are defined as [17, 42, 43]where , are the formal variables.

It has been found that there exist some relations between the binary Bell polynomials and the Hirota -operators [40, 41]. When , , a binary Bell polynomial can be transformed into a bilinear term according to the identity [40, 41]Likewise, when , -polynomials can be associated with the Hirota -operators according to the identity [40, 41]

#### 3. Bilinear Form

We will next investigate (1), to be written in -polynomial form with one independent variable. Based on the relation between the binary Bell polynomials and Hirota bilinear operators, namely, identities (10) and (11), (1) can be translated into the corresponding bilinear forms.

Consider the following scale transformations:where , , , , , and are the real constants. Invariance of (1) under such transformations requires that , , and .

Notice that if we require that , we have to set in (1) and obtainwhere is an arbitrary constant.

In order to express (13) with -polynomials, we choose . Thenwhose corresponding bilinear form is

Therefore, we get the bilinear form of (1), which is (14) with -polynomials or (15) with the bilinear operators. We note that (15) is the same as that in [15], but the method that we used is different from that in [15].

#### 4. Bell-Polynomial-Typed BT and Bilinear BT with Soliton Solutions

To construct a BT, we express (1) with -polynomials:

Based on we will derive the Bell-polynomial-typed BT under the homogenous constraints between the primary and replica fields instead of using exchange formulae.

Using the Bell polynomials, we have

Note that Therefore, substituting (18) into (17), we havewhereand is an arbitrary constant.

Further computation shows thatHereby, if we choose and set , thenHence, Moreover, a decomposition of (23) leads to the following Bell-polynomial-typed BT:where , , , and are the arbitrary constants.

Using the connection between the Bell polynomials and bilinear operators, we give a bilinear BT between and as

As an application, we derive the one-soliton solutions from a trivial solution, by virtue of the bilinear BT, that is, (25). Taking , , and in (25), we getSubstituting (26) into (28), we haveand hence we take where and is a nonzero constant. On the other hand, we can choose and substitute it into (26) and (29), which implies thatSimilarly, substituting (26) into (27) leads toSolving (32), we obtain Finally, we can present the one-soliton solutions of (1) as where , while the parameters , , , and are all arbitrary constants.

#### 5. Discussions and Conclusions

We have investigated the -dimensional Jimbo-Miwa equation, that is, (1). With the aid of the Bell polynomials and Hirota bilinear operators, we have derived bilinear form (16) of (1) and then constructed a new BT, that is, (25), with the Bell polynomials and symbolic computation. The bilinear form and BT are important integrable property for the nonlinear evolution equations. Moreover, a BT often can be cast into the Lax pair for integrable equations. It may be possible to construct the bilinear BTs for the -dimensional Jimbo-Miwa equation via the exchange formulae; however, the computation is tedious. Bell-polynomial-typed BTs (24) have been constructed hereby and then cast into bilinear BTs (25), which help us avoid the difficulties in using the exchange formulae. As an application, one-soliton solutions (34) have been obtained via BT (25). The existence of solution obtained via solving this BT indicates that (24) or (25) are genuine ones.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors would like to express their sincere thanks to Mr. Q. X. Qu and Mr. K. Sun for their helpful suggestions. This work has been supported by the National Natural Science Foundation of China under Grant no. 11272023, by the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), and by the Fundamental Research Funds for the Central Universities of China under Grant no. 2011BUPTYB02.

#### References

- M. J. Ablowitz and P. A. Clarkson,
*Solitons, Nonlinear Evolution Equations and Inverse Scattering*, Cambridge University Press, Cambridge, UK, 1991. View at Publisher · View at Google Scholar · View at MathSciNet - M. P. Barnett, J. F. Capitani, J. von zur Gathen, and J. Gerhard, “Symbolic calculation in chemistry: selected examples,”
*International Journal of Quantum Chemistry*, vol. 100, no. 2, pp. 80–104, 2004. View at Publisher · View at Google Scholar · View at Scopus - X. Lü, H.-W. Zhu, X.-H. Meng, Z.-C. Yang, and B. Tian, “Soliton solutions and a Bäcklund transformation for a generalized nonlinear Schrödinger equation with variable coefficients from optical fiber communications,”
*Journal of Mathematical Analysis and Applications*, vol. 336, no. 2, pp. 1305–1315, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - X. Lü, “Bright-soliton collisions with shape change by intensity redistribution for the coupled Sasa-Satsuma system in the optical fiber communications,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 19, no. 11, pp. 3969–3987, 2014. View at Publisher · View at Google Scholar - B. Tian, Y. T. Gao, and H. W. Zhu, “Variable-coefficient higher-order nonlinear Schrödinger model in optical fibers: variable-coefficient bilinear form, Bäcklund transformation, brightons and symbolic computation,”
*Physics Letters A*, vol. 366, pp. 223–229, 2007. View at Publisher · View at Google Scholar - B. Tian and Y. T. Gao, “Symbolic computation on cylindrical-modified dust-ion-acoustic nebulons in dusty plasmas,”
*Physics Letters A*, vol. 362, no. 4, pp. 283–288, 2007. View at Publisher · View at Google Scholar - Z.-Y. Yan and H.-q. Zhang, “Symbolic computation and new families of exact soliton-like solutions to the integrable Broer-Kaup (BK) equations in $(2+1)$-dimensional spaces,”
*Journal of Physics A: Mathematical and General*, vol. 34, no. 8, pp. 1785–1792, 2001. View at Publisher · View at Google Scholar · View at MathSciNet - X. Lü and M. Peng, “Systematic construction of infinitely many conservation laws for certain nonlinear evolution equations in mathematical physics,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 18, no. 9, pp. 2304–2312, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - X. Lü and M. Peng, “Nonautonomous motion study on accelerated and decelerated solitons for the variable-coefficient Lenells-Fokas model,”
*Chaos*, vol. 23, no. 1, Article ID 013122, 2013. View at Publisher · View at Google Scholar - C. Rogers and W. F. Shadwick,
*Bäacklund Transformations and Their Applications*, Academic Press, New York, NY, USA, 1982. - X. Lü, “New bilinear Bäcklund transformation with multisoliton solutions for the (2+1)-dimensional Sawada–Kotera model,”
*Nonlinear Dynamics*, vol. 76, no. 1, pp. 161–168, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - X. Lü and J. Li, “Integrability with symbolic computation on the Bogoyavlensky-Konoplechenko model: BELl-polynomial manipulation, bilinear representation, and Wronskian solution,”
*Nonlinear Dynamics*, vol. 77, no. 1-2, pp. 135–143, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - B. Tian, Y.-T. Gao, and W. Hong, “The solitonic features of a nonintegrable $(3+1)$-dimensional Jimbo-Miwa equation,”
*Computers & Mathematics with Applications*, vol. 44, no. 3-4, pp. 525–528, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - J. F. Zhang and F. M. Wu, “Bäcklund transformation and multiple soliton solutions for the $(3+1)$-dimensional Jimbo-Miwa equation,”
*Chinese Physics*, vol. 11, no. 5, p. 425, 2002. View at Publisher · View at Google Scholar - Q. L. Zha and Z. B. Li, “Multiple periodic-soliton solutions for (3+1)-dimensional Jimbo-Miwa equation,”
*Communications in Theoretical Physics*, vol. 50, no. 5, pp. 1036–1040, 2008. View at Publisher · View at Google Scholar - X. Lü, “Soliton behavior for a generalized mixed nonlinear Schrödinger model with
*N*-fold Darboux transformation,”*Chaos*, vol. 23, Article ID 033137, 2013. View at Publisher · View at Google Scholar - R. Hirota, “Exact solution of the korteweg—de Vries equation for multiple collisions of solitons,”
*Physical Review Letters*, vol. 27, no. 18, pp. 1192–1194, 1971. View at Publisher · View at Google Scholar - R. Hirota, “A new form of Bäcklund transformations and its relation to the inverse scattering problem,”
*Progress of Theoretical Physics*, vol. 52, no. 5, pp. 1498–1512, 1974. View at Publisher · View at Google Scholar - R. Hirota and J. Satsuma, “A variety of nonlinear network equations generated from the bäcklund transformation for the toda lattice,”
*Progress of Theoretical Physics Supplements*, vol. 59, pp. 64–100, 1976. View at Publisher · View at Google Scholar - R. Hirota, X.-B. Hu, and X.-Y. Tang, “A vector potential KdV equation and vector Ito equation: soliton solutions, bilinear Bäcklund transformations and Lax pairs,”
*Journal of Mathematical Analysis and Applications*, vol. 288, no. 1, pp. 326–348, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - R. Hirota,
*The Direct Method in Soliton Theory*, Cambridge University Press, Cambridge, UK, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Matsuno,
*Bilinear Transformation Method*, vol. 174 of*Mathematics in Science and Engineering*, Academic Press, London, UK, 1984. View at MathSciNet - C.-X. Li, W.-X. Ma, X.-J. Liu, and Y.-B. Zeng, “Wronskian solutions of the Boussinesq equation—solitons, negatons, positons and complexitons,”
*Inverse Problems*, vol. 23, no. 1, pp. 279–296, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - H. Aratyn, L. A. Ferreira, and A. H. Zimerman, “Exact static soliton solutions of (3+1)-dimensional integrable theory with nonzero Hopf numbers,”
*Physical Review Letters*, vol. 83, no. 9, pp. 1723–1726, 1999. View at Publisher · View at Google Scholar · View at Scopus - A. Wazwaz, “Integrable (2+1)-dimensional and (3+1)-dimensional breaking soliton equations,”
*Physica Scripta*, vol. 81, no. 3, Article ID 035005, 2010. View at Publisher · View at Google Scholar - Y. T. Gao and B. Tian, “New family of overturning soliton solutions for a typical breaking soliton equation,”
*Computers & Mathematics with Applications*, vol. 30, no. 12, pp. 97–100, 1995. View at Publisher · View at Google Scholar · View at MathSciNet - J. Yu and Z. Lou, “A $(3+1)$-dimensional Painlevé integrable model obtained by deformation,”
*Mathematical Methods in the Applied Sciences*, vol. 25, no. 2, pp. 141–148, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - J. B. Chen, “Finite-gap solutions of $2+1$
dimensional integrable nonlinear evolution equations generated by the Neumann systems,”
*Journal of Mathematical Physics*, vol. 51, no. 8, Article ID 083514, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - R. S. Ward, “Nontrivial scattering of localized solitons in a (2+1)-dimensional integrable system,”
*Physics Letters A*, vol. 208, no. 3, pp. 203–208, 1995. View at Publisher · View at Google Scholar · View at MathSciNet - J. F. Zhang, “Multiple soliton-like solutions for $(2+1)$-dimensional dispersive Long-Wave equations,”
*International Journal of Theoretical Physics*, vol. 37, no. 9, pp. 2449–2455, 1998. View at Publisher · View at Google Scholar - A. V. Mikhailov and R. I. Yamilov, “Towards classification of $(2+1)$-dimensional integrable equations,”
*Journal of Physics A*, vol. 31, Article ID 6707, 1998. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Jimbo and T. Miwa, “Solitons and infinite dimensional Lie algebras,”
*Publications of the Research Institute for Mathematical Sciences*, vol. 19, pp. 943–1001, 1983. View at Publisher · View at Google Scholar - X.-Y. Tang and J. Lin, “Conditional similarity reductions of Jimbo-Miwa equations via the classical Lie group approach,”
*Communications in Theoretical Physics*, vol. 39, no. 1, pp. 6–8, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - B. Dorrizzi, B. Grammaticos, A. Ramani, and P. Winternitz, “Are all the equations of the KP hierarchy integrable?”
*Journal of Mathematical Physics*, vol. 27, pp. 2848–2852, 1986. View at Google Scholar - J. Weiss, M. Tabor, and G. Carnevale, “The Painlevé property for partial differential equations,”
*Journal of Mathematical Physics*, vol. 24, no. 3, pp. 522–526, 1983. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - C.-L. Bai and H. Zhao, “Some special types of solitary wave solutions for $(3+1)$-dimensional Jimbo-Miwa equation,”
*Communications in Theoretical Physics*, vol. 41, no. 6, pp. 875–877, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - A.-M. Wazwaz, “Multiple-soliton solutions for the Calogero-Bogoyavlenskii-Schiff, Jimbo-Miwa and {YTSF} equations,”
*Applied Mathematics and Computation*, vol. 203, no. 2, pp. 592–597, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - G. Q. Xu, “The soliton solutions, dromions of the Kadomtsev-Petviashvili and Jimbo-Miwa equations in $(3+1)$-dimensions,”
*Chaos, Solitons and Fractals*, vol. 30, no. 1, pp. 71–76, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - S.-H. Ma, J.-P. Fang, B.-H. Hong, and C.-L. Zheng, “New exact solutions and interactions between two solitary waves for (3+1)-dimensional jimbo–miwa system,”
*Communications in Theoretical Physics*, vol. 49, no. 5, pp. 1245–1248, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - C. Gilson, F. Lambert, J. J. Nimmo, and R. Willox, “On the combinatorics of the Hirota
*D*-operators,”*Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences*, vol. 452, no. 1945, pp. 223–234, 1996. View at Publisher · View at Google Scholar · View at MathSciNet - F. Lambert and J. Springael, “On a direct procedure for the disclosure of Lax pairs and Bäcklund transformations,”
*Chaos, Solitons & Fractals*, vol. 12, no. 14-15, pp. 2821–2832, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - X. Lü, B. Tian, K. Sun, and P. Wang, “Bell-polynomial manipulations on the Bäcklund transformations and Lax pairs for some soliton equations with one tau-function,”
*Journal of Mathematical Physics*, vol. 51, no. 11, Article ID 113506, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - X. Lü, F. H. Lin, and F. H. Qi, “Analytical study on a two-dimensional Korteweg-de Vries model with bilinear representation, Bäcklund transformation and soliton solutions,”
*Applied Mathematical Modelling*, vol. 39, pp. 3221–3226, 2015. View at Publisher · View at Google Scholar