Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-28T02:06:09.189Z Has data issue: false hasContentIssue false
  • English
  • Français

Integrable Couplings of the Boiti-Pempinelli-Tu Hierarchy and Their Hamiltonian Structures

Part of: Equations of mathematical physics and other areas of application Infinite-dimensional Hamiltonian systems

Published online by Cambridge University Press:  27 May 2016

Huiqun Zhang ,
Yubin Zhou  and
Junqin Xu
Huiqun Zhang*
Affiliation:
College of Mathematical Science, Qingdao University, Shandong 266071, China
Yubin Zhou*
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Gansu 712000, China
Junqin Xu*
Affiliation:
College of Mathematical Science, Qingdao University, Shandong 266071, China
*
*Corresponding author. Email:[email protected] (H. Zhang), [email protected] (Y. Zhou), [email protected] (J. Xu)
*Corresponding author. Email:[email protected] (H. Zhang), [email protected] (Y. Zhou), [email protected] (J. Xu)
*Corresponding author. Email:[email protected] (H. Zhang), [email protected] (Y. Zhou), [email protected] (J. Xu)
Get access

Abstract

Integrable couplings of the Boiti-Pempinelli-Tu hierarchy are constructed by a class of non-semisimple block matrix loop algebras. Further, through using the variational identity theory, the Hamiltonian structures of those integrable couplings are obtained. The method can be applied to obtain other integrable hierarchies.

Keywords

Integrable couplingbi-integrable couplingHamiltonian structureblock matrix loop algebra

MSC classification

Secondary: 37K05: Hamiltonian structures, symmetries, variational principles, conservation laws 35Q53: KdV-like equations (Korteweg-de Vries)
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 4 , August 2016 , pp. 588 - 598
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Arnold, V. I., Mathematical Methods of Classical Mechanics, Springer, Berlin, 1978.Google Scholar
[2]Tu, G. Z., The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems, J. Math. Phys., 30 (1989), pp. 330–339.CrossRefGoogle Scholar
[3]Tu, G. Z., On liouville integrability of zero curvature equations and Yang hierarchy, J. Phys. A Math. Gen., 22 (1990), pp. 2375–2392.Google Scholar
[4]Tu, G. Z. and Xu, B. Z., The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems (III), China Ann. Math. Ser. B, 17 (1996), pp. 497–506.Google Scholar
[5]Hu, X. B., A powerful approach to generate new integrable systems, J. Phys. A Math. Gen., 27 (1994), pp. 2497–2514.Google Scholar
[6]Hu, X. B., An approach to generate superextensions of integrable system, J. Phys. A Math. Gen., 30 (1997), pp. 619–632.CrossRefGoogle Scholar
[7]Ma, W. X. and Chen, M., Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras, J. Phys. A Math. Gen., 39 (2006), pp. 10787–10801.CrossRefGoogle Scholar
[8]Ma, W. X., A discrete variational identity on semi-direct sums of Lie algebras, J. Phys. A Math. Theory, 40 (2007), pp. 15055–15069.CrossRefGoogle Scholar
[9]Ma, W. X., A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction, Chin. Ann. Math. A, 13 (1992), pp. 115–123.Google Scholar
[10]Guo, F. K., Subalgebras of the loop algebra A1 and integrable Hamiltonian hierarchies of equations, Acta. Math. Phys., 19 (1999), pp. 507–512.Google Scholar
[11]Zhang, Y. F., A generalized Boite-Pempinelli-Tu (BPT) hierarchy and its bi-Hamiltonian structure, Phys. Lett. A, 317 (2003), pp. 280–286.CrossRefGoogle Scholar
[12]Xu, X. X. and Zhang, Y. F., A hierarchy of Lax integrable Lattice equations, Liouville integrability and a new intehrable symplectic map, Commun. Theor. Phys., 41 (2004), pp. 321–330.Google Scholar
[13]Xia, T. C., Chen, X. H. and Chen, D. Y., Two types of new Lie algebras and corresponding hierarchies of evolution equations, Chaos Solitons Fractals, 23 (2005), pp. 1033–1041.CrossRefGoogle Scholar
[14]Zhang, J. F. and Wei, X. L., A generalized Levi hierarchy and its integrable couplings, Int. J. Modern Phys. B, 24 (2010), pp. 3453–3460.CrossRefGoogle Scholar
[15]Ma, W. X. and Fuchssteiner, B., Integrable theory of the perturbation equations, Chaos Solitons Fractals, 7 (1996), pp. 1227–1250.Google Scholar
[16]Ma, W. X., Loop algebras and bi-integrable couplings, Chinese Ann. Math. Ser. B, 33 (2012), pp. 207–224.Google Scholar
[17]Boiti, M., Pempinelli, F. and Tu, G. Z., Canonical structure of soliton equations via isospectral eigenvalue problems, Nuovo. Cimento. B, 79 (1984), pp. 231–265.Google Scholar
[18]Ma, W. X., Variational identities and Hamiltonian structures, in: Nonlinear and Modern Mathematical Physics, eds. Ma, W. X., Hu, X. B., and Liu, Q. P., AIP Conference Proceedings, Vol. 1212, American Institute of Physics, Melville, NY, (2010), pp. 1–27.Google Scholar
[19]MA, W. X., Variational identities and applications to Hamiltonian structures of soliton equations, Nonlinear Anal. Theory Methods Appl., 71 (2009), pp. e1716-e1726.Google Scholar