Skip to main content
SpringerLink
Log in

Noncommutative Constrained KP Hierarchy and Multi-component Noncommutative Constrained KP Hierarchy

  • Published:
Chinese Annals of Mathematics, Series B Aims and scope Submit manuscript

Abstract

In this paper, the authors define the noncommutative constrained Kadomtsev-Petviashvili (KP) hierarchy and multi-component noncommutative constrained KP hierarchy. Then they give the recursion operators for the noncommutative constrained KP (NcKP) hierarchy and multi-component noncommutative constrained KP (NmcKP) hierarchy. The authors hope these studies might be useful in the study of D-brane dynamics whose noncommutative coordinates emerge from limits of the M theory and string theory.

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. Fokas, A. S. and Santini, P. M., The recursion operator of the Kadomtsev-Petviashvili equation and the squared eigenfunctions of the Schrödinger operator, Stud. Appl. Math., 75, 1986, 179–185.

    Article  MathSciNet  MATH  Google Scholar 

  2. Fokas, A. S. and Santini, P. M., Recursion operators and bi-Hamiltonian structures in multidimensions, II, Commun. Math. Phys., 116, 1988, 449–474.

    Article  MathSciNet  MATH  Google Scholar 

  3. Matsukidaira, J., Satsuma, J. and Strampp, W., Conserved quantities and symmetries of KP hierarchy, J. Math. Phys., 31, 1990, 1426–1434.

    Article  MathSciNet  MATH  Google Scholar 

  4. Fuchssteiner, B. and Fokas, A. S., Symplectic structures, their Bäcklund transformation and hereditary symmetries, Phys. D., 4, 1981, 47–66.

    Article  MathSciNet  MATH  Google Scholar 

  5. Olver, P. J., Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986.

    Book  MATH  Google Scholar 

  6. Santini, P. M. and Fokas, A. S., Recursion operators and bi-Hamiltonian structures in multidimensions, I, Commun. Math. Phys., 115, 1988, 375–419.

    Article  MathSciNet  MATH  Google Scholar 

  7. Strampp, W. and Oevel, W., Recursion operators and Hamitonian structures in Sato theory, Lett. Math. Phys., 20, 1990, 195–210.

    Article  MathSciNet  MATH  Google Scholar 

  8. Stephen, C. A., Bi-Hamiltonian operators, integrable flows of curves using moving frames and geometric map equations, J. Phys. A: Math. Gen., 39, 2006, 2043–2072.

    Article  MathSciNet  MATH  Google Scholar 

  9. Douglas, M. R. and Nekrasov, N. A., Noncommutative field theory, Rev. Mod. Phys., 73, 2002, p.977.

    Article  MathSciNet  MATH  Google Scholar 

  10. Dickey, L. A., Soliton Equations and Hamiltonian Systems, 2nd ed., World Scintific, Singapore, 2003.

    Book  MATH  Google Scholar 

  11. Date, E., Kashiwara, M., Jimbo, M., et al., Transformation groups for soliton equations, Nonlinear Integrable Systems-Classical and Quantum Theory, Jimbo, M. and Miwa, T. (eds.), World Scientific, Singapore, 1983, 39–119.

    Google Scholar 

  12. Cheng, Y., Constraints of the Kadomtsev-Petviashvili hierarchy, J. Math. Phys., 33, 1992, 3774–3782.

    Article  MathSciNet  MATH  Google Scholar 

  13. Gurses, M., Karasu, A. and Sokolov, V. V., On construction of recursion operators from Lax representation, J. Math. Phys., 40, 1999, 6473–6490.

    Article  MathSciNet  MATH  Google Scholar 

  14. Loris, I., Recursion operator for a constrained BKP system, Proceedings of the Workshop on Nonlinearity, Integrability and All That Twenty years after NEEDS’79, Boiti, M., Martina, L., Pempinelli, F., et al., (eds.), World Scientific, Singapore, 1999, 325–330.

    Google Scholar 

  15. Li, C. Z., Tian, K. L., He, J. S. and Cheng, Y., Recursion operator for a constrained CKP hierarchy, Acta Mathematica Scientia, Ser. B, 31, 2011, 1295–1302.

    Article  MathSciNet  MATH  Google Scholar 

  16. He, J. S., Tu, J. Y., Li, X. D. and Wang, L. H., Explicit flow equations and recursion operator of the ncKP hierarchy, Nonlinearity, 24, 2011, 2875–2890.

    Article  MathSciNet  MATH  Google Scholar 

  17. Hamanaka, M., Noncommutative integrable systems and quasidetermiants, Nonlinear and Modern Mathematical Physics, AIP Conf. Proc., 1212, Amer. Inst. Phys., Melville, NY, 2010, 122–135.

  18. Sooman, G. M., Soliton Solutions of Noncommutative Integrable Systems, University of Glasgow, 2009.

Download references

Acknowledgment

The authors would like to thank Professor Jingsong He for his long term support.

Author information

Authors and Affiliations

  1. Department of Mathematics, Ningbo University, Ningbo, 315211, Zhejiang, China

    Qiufang Liu & Chuanzhong Li

Authors
  1. Qiufang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chuanzhong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chuanzhong Li.

Additional information

This work was supported by the National Natural Science Foundation of China (No. 11571192), the Natural Science Foundation of Ningbo (No. 2015A610157) and K. C. Wong Magna Fund in Ningbo University.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, Q., Li, C. Noncommutative Constrained KP Hierarchy and Multi-component Noncommutative Constrained KP Hierarchy. Chin. Ann. Math. Ser. B 40, 457–468 (2019). https://doi.org/10.1007/s11401-019-0144-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11401-019-0144-6

Keywords

2000 MR Subject Classifications

Search

Navigation