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

A Few Notes on Lax Integrability, Integrable Couplings and Computing Formula of the Constant γ

Journal of Applied Nonlinear Dynamics 1(4) (2012) 401--406 | DOI:10.5890/JAND.2012.06.003

F.K. Guo$^{1}$, B.L. Feng$^{2}$, T.T. Guo$^{3}$

$^{1}$ Information School, Shandong University of Science and Technology, Qingdao 266510, China

$^{2}$ School of Mathematics and Information Sciences, Weifang University, Weifang 261061, China

$^{3}$ Business College of Shanxi University, Taiyuan 030031, China

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Abstract

We point out in the paper that the current available definition on Lax integrability is necessary to be modified; the existed method for generating integrable couplings by the approach “original equations + symmetric equations” is wrong. Besides, a simple and efficient formula for calculating the constant γ appearing in the trace identity and the quadratic-form identity is proposed, which is universal for finding Hamiltonian structures of integrable dynamics.

References

1.  [1] Lax, P.D. (1968), Integrals of nonlinear equations of evolution and solitary waves, Commun. Pur. Appl. Math., 21, 467-490.
2.  [2] Fuchssteiner, B. (1993), Coupling of completely integrable system: the perturbation bundle, in: Applications of analytic and geometric methods to nonlinear differential equations (Exeter, 1992), (Clarkson PA, Ed.), Kluwer, Dordrecht, 125-138.
3.  [3] Guo, F.K. and Zhang, Y.F. (2002), A type of expanding integrable model of the AKNS hierarchy, Acta Physica Sinica, 51, 951-954.
4.  [4] Tu, G.Z.(1989), The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems, J. Math. Phys., 30, 330-338.
5.  [5] Guo, F.K. and Zhang, Y.F. (2005), The quadratic-form identity for constructing the hamiltonian structure of integrable systems, J. Phys. A, 38, 8537-8548.
6.  [6] Guo, F.K. and Zhang, Y.F. (2007), Two unified formulae, Phys. Lett. A, 366, 403-410.