Skip to main content
SpringerLink
Log in

An so (3, ℝ) Counterpart of the Dirac Soliton Hierarchy and its Bi-Integrable Couplings

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

We derive a counterpart hierarchy of the Dirac soliton hierarchy from zero curvature equations associated with a matrix spectral problem from so (3, ℝ). Inspired by a special class of non-semisimple loop algebras, we construct a hierarchy of bi-integrable couplings for the counterpart soliton hierarchy. By applying the variational identities which cope with the enlarged Lax pairs, we generate the corresponding Hamiltonian structure for the hierarchy of the resulting bi-integrable couplings. To show Liouville integrability, infinitely many commuting symmetries and conserved densities are presented for the counterpart soliton hierarchy and its hierarchy of bi-integrable couplings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ma, W.X., Fuchssteiner, B.: Integrable theory of the perturbation equations. Chaos Soliton Fract. 7(8), 1227–1250 (1996)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  2. Fuchssteiner, B.: Coupling of completely integrable system: the perturbation bundle. In: Clarkson, P.A. (ed.) Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, Vol. 413, pp 125–138. Springer, Netherlands (1993)

    Chapter  Google Scholar 

  3. Ma, W.X.: Variational identities and applications to Hamiltonian structures of soliton equations. Nonlinear Anal. 71(12), 1716–1726 (2009)

    Article  MathSciNet  Google Scholar 

  4. Ma, W.X.: Integrable couplings and matrix loop algebras. In: Ma, W.X., Kaup, D. (eds.) Proceedings of the 2nd International Workshop of Nonlinear and Modern Mathematial Physics, Vol. 1562, pp 105–122 (2013)

  5. Ma, W.X., Strampp, W.: Bilnear forms and Bäcklund transformations of the perturbation systems. Phys. Lett. A 341(5-6), 441–449 (2005)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  6. Ma, W.X., Fuchssteiner B.: The bi-Hamiltonian structure of the perturbation equations of the KdV hierarchy. J. Phys. A 213(1-2), 49–55 (1996)

    MathSciNet  MATH  Google Scholar 

  7. Frappat, L., Sciarrino, A., Sorba, P.: Dictionary on Lie Algebras and Superalgebras. Academic, San Diego, CA (2000)

    Google Scholar 

  8. Ma, W.X., Xu, X.X., Zhang, Y.F.: Semi-direct sums of Lie algebras and continuous integrable couplings. Phys. Lett. A 351(3), 125–130 (2006)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  9. Ma, W.X., Xu, X.X., Zhang, Y.F.: Semi-direct sums of Lie algebras and discrete integrable couplings. J. Math. Phys. 47(5), 053501 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  10. Ma, W.X., Meng, J.H., Zhang, H.Q.: Integrable couplings, variational identities and Hamiltonian formulations. Global J. Math. Sci. 01, 1–17 (2012)

    Google Scholar 

  11. Ma, W.X., Chen, M.: Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras. J. Phys. A: Math. Gen. 39, 10787–10801 (2006)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  12. Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series. Cambridge Univ. Press, Cambridge (1991)

    Book  Google Scholar 

  13. Ma, W.X.: Loop algebras and bi-integrable couplings. Chine. Ann. Math. B 33(2) (2012)

  14. Ma, W.X., Zhang, H.Q., Meng, J.H.: A block mattrix loop algebra and bi-intergrable couplings of the Dirac equations. East Asian J. Appl. Math. 3(3), 171–189 (2013)

    MathSciNet  MATH  Google Scholar 

  15. Grosse, H.: New solitons connected to the Dirac equation. Phys. Rep. 134(5-6), 297–304 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  16. Hinton, D.B., Jordan, A.K., Klaus, M., Shaw, J.K.: Inverse scattering on the line for a Dirac system. J. Math. Phys. 32, 3015–3030 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  17. Ma, W.X., Li, K.S.: Virasoro symmetry algebra of Dirac soliton hierarchy. Inverse Probl. 12, 12–25 (1996)

    Article  Google Scholar 

  18. Ma, W.X.: Binary nonlinearization for the Dirac systems. Chin. Ann. Math. Ser. B 01, 79–88 (1997)

    Google Scholar 

  19. Olver, P.J.: Applications of Lie Groups to Differential Equations (1986)

Download references

Acknowledgments

The work was supported in part by NSF under the grant DMS-1301675, NSFC under the grants 11371241, 11371326, 11271008 and 61072147), Zhejiang Innovation Project of China (Grant no. T200905), and the First-class Discipline of Universities in Shanghai and the Shanghai Univ. Leading Academic Discipline Project (No. A.13-0101-12-004). This paper was also supported by the China Scholarship Council (File No. 201306890028). The authors are also grateful to D. J. Zhang for reading the paper, Y.Q.Yao, E. A. Appiah, C. X. Li, S. Manukure, M. Mcanally, S. F. Shen and S. M. Yu for their stimulating discussions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Wen-Ying Zhang

  2. Department of Mathematics and Statistics, University of South Florida, Tampa, FL, 33620-5700, USA

    Wen-Xiu Ma

Authors
  1. Wen-Ying Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wen-Xiu Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wen-Ying Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, WY., Ma, WX. An so (3, ℝ) Counterpart of the Dirac Soliton Hierarchy and its Bi-Integrable Couplings. Int J Theor Phys 53, 4211–4222 (2014). https://doi.org/10.1007/s10773-014-2172-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-014-2172-z

Keywords

Search

Navigation