ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Lie Algebraic Approach to Nonlinear Integrable Couplings of Evolution Type

Journal of Applied Nonlinear Dynamics 1(1) (2011) 1--28 | DOI:10.5890/JAND.2011.12.001

Yufeng Zhang$^{1}$; Wen-XiuMa$^{2}$

$^{1}$ College of Science, China University of Mining and Technology, Xuzhou 221116, P. R. China

$^{2}$ Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA

Abstract

Based on two higher-dimensional extensions of Lie algebras, three kinds of specific Lie algebras are introduced. Upon constructing proper loop algebras, six isospectral matrix spectral problems are presented and they yield nonlinear integrable couplings of the Ablowitz- Kaup-Newell-Segur hierarchy, the Broer-Kaup hierarchy and the Kaup-Newell hierarchy. Especially, the reduced cases of the resulting integrable couplings give nonlinear integrable couplings of the nonlinear Schrödinger equation and the classical Boussinesq equation. Two linear functionals are introduced on two loop algebras of dimension 6 and Hamiltonian structures of the obtained nonlinear integrable couplings are worked out by employing the associated variational identity. The proposed approach can also be used to generate nonlinear integrable couplings for other integrable hierarchies.

Acknowledgments

The work was supported in part by the State Administration of Foreign Experts Affairs of China, the National Natural Science Foundation of China (Nos.6172147 and 11071159), Chunhui Plan of the Ministry of Education of China, the Natural Science Foundation of Shanghai, Shanghai Leading Academic Discipline Project (No. J50101) and the fundamental Research Funds of the Central University (2010LKSX08) and the Natural Science Foundation of Liaoning Province(20092171).

References

1.  [1] Ma, W.X., and Fuchssteiner B. (1996), Integrable theory of the perturbation equations, Chaos, Solitons and Fractals, 7, 1227-1250.
2.  [2] Ma W.X.(2000), Integrable couplings of soliton equations by perturbation I. A general theory and application to the KdV hierarchy,Meth. Appl. Anal., 7, 21-56.
3.  [3] Ma,W.X., Xu, X.X. and Zhang,Y.F. (2006), Semi-direct sums of Lie algebras and comtinuous integrable couplings, Phys. Lett. A, 351, 125-130.
4.  [4] Zhang,Y.F. and Zhang, H.Q. (2002), A direct method for integrable couplings of TD hierarchy, J. Math. Phys., 43(1), 466-472.
5.  [5] Ma, W.X.(2003), Enlarging spectral problems to construct integrable couplings, Phys Lett A, 316, 72-76.
6.  [6] Zhang, Y.F. and Guo, F.K.(2006), Matrix Lie algebras and integrable couplings, Commun. Theor. Phys., 46(5), 812-818.
7.  [7] Fan, E.G. and Zhang, Y.F.(2006), Vector loop algebra and its applications to integrable system, Chaos, Solitons and Fractals, 28(6), 966-971.
8.  [8] Zhang, Y.F. and Liu, J. (2008), Induced Lie algebras of a six-dimensional matrix Lie algebra,Commun. Theor. Phys., 50, 289-294.
9.  [9] Ma, W.X.(2009), Variational identities and applications to Hamiltonian structures of soliton equations,Nonl. Anal., 71, e1716-e1726.
10.  [10] Guo,F.K. and Zhang, Y.F.(2005), The quadratic-form identity for constructing the Hamiltonian structure of integrable system, J. Phys. A, 38, 8537-8546.
11.  [11] Ma, W.X. and Chen,M. (2006), Hamiltonian and qusi-Hamiltonian structures associated with semi-direct sums of Lie algebras, J. Phys. A, 39, 10787-10801.
12.  [12] Ma,W.X., He, J.S.and Qin,Z.Y. (2008), A supertrace identity and its applications to superintegrable systems, J. Math. Phys, 49., 033511.
13.  [13] Ma, W.X. and Gao, L. (2009), Coupling integrable couplings, Mod. Phys. Lett. B, 23(15), 1847-1860.
14.  [14] Zhang,Y.F. and Tam, W.H. (2010), Four Lie algebras associated with R6 and their applications, J. Math. Phys., 51, 043510.
15.  [15] Zhang, Y.F. and Fan,E.G. (2010), Coupling integrable couplings and bi-Hamiltonian structure associated with Boiti-Pempinelli-Tu hierarchy, J. Math. Phys., 51, 083506.
16.  [16] Ma,W.X. and Zhu, Z.N. (2010), Constructing nonlinear discrete integrable Hamiltonian couplings,Comput. Math. Appl., 60, 2601-2608.
17.  [17] Ma,W.X. (2011), Nonlinear continuous integrable Hamiltonian couplings, Appl. Math. Comput., 217, 7238- 7244.
18.  [18] Ma,W.X.(2011), Variational identities and Hamiltonian structures, in:Nonlinear and Modern Mathematical Physics, pp.1-27, edited byMa,W.X., Hu, X.B. and Liu, Q.P., AIP Conference. Proceedings, Vol.1212(American Institute of Physics, Melville, NY).
19.  [19] Tu, G.Z.(1989), The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems, J. Math. Phys., 30, 330-338.
20.  [20] Ma, W.X.(1992), A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reductions, Chin. J. Contemp. Math., 13, 79-89.
21.  [21] Ma, W.X., Xu,X.X. and Zhang, Y.F.(2006), Semidirect sums of Lie algebras and discrete integrable couplings, J. Math. Phys. , 47, 053501.
22.  [22] Li,Y.S., Ma, W.X. and Zhang, J.E.(2000), Darboux transformations of classical Boussinesq system and its new solutions, Phys. Lett. A, 275, 60-66.
23.  [23] Li Y.S.,and Zhang, J.E. (2001), Darboux transformations of classical Boussinesq system and its multi-soliton solutions, Phys. Lett. A, 284, 253-258.
24.  [24] Ablowitz, M.J., Chakravarty, S.and Halburd, R.G. (2003), Integrable systems and reductions of the self-dual mYang-Mills equations, J. Math. Phys., 44, 3147-3153.
25.  [25] Ma, W.X. and Fan, E.G.(2011), Linear superposition principle applying to Hirota bilinear equations, Comput. Math. Appl., 61, 950-959.