Skip to main content
SpringerLink
Log in

Hirota difference equation: Inverse scattering transform, darboux transformation, and solitons

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the direct and inverse problems for the Hirota difference equation. We introduce the Jost solutions and scattering data and describe their properties. In a special case, we show that the Darboux transformation allows finding the evolution in discrete time and obtaining a recursive procedure for sequentially constructing the Jost solution at an arbitrary time for a given initial value. We consider some properties of the soliton solutions.

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. R. Hirota, J. Phys. Soc. Japan, 43, 2074–2078 (1977).

    Article  ADS  MathSciNet  Google Scholar 

  2. R. Hirota, J. Phys. Soc. Japan, 50, 3785–3791 (1981).

    Article  ADS  MathSciNet  Google Scholar 

  3. T. Miwa, Proc. Japan. Acad. Ser. A, 58, 9–12 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  4. L. V. Bogdanov and B. G. Konopelchenko, Phys. D, 152–153, 85–96 (2001).

    Article  MathSciNet  Google Scholar 

  5. A. V. Zabrodin, Theor. Math. Phys., 113, 1347–1392 (1997).

    Article  MathSciNet  Google Scholar 

  6. A. V. Zabrodin, Theor. Math. Phys., 155, 567–584 (2008); arXiv:0705.4006v1 [hep-th] (2007).

    Article  MATH  MathSciNet  Google Scholar 

  7. S. Saito, J. Nonlinear Math. Phys., 19, 539–550 (2012).

    ADS  Google Scholar 

  8. I. Krichever, P. Wiegmann, and A. Zabrodin, Commun. Math. Phys., 193, 373–396 (1998).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. A. K. Pogrebkov, St. Petersburg Math. J., 22, 473–483 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Boiti, F. Pempinelli, A. K. Pogrebkov, and M. K. Polivanov, Theor. Math. Phys., 93, 1200–1224 (1992).

    Article  MathSciNet  Google Scholar 

  11. M. Boiti, F. Pempinelli, A. K. Pogrebkov, and M. C. Polivanov, Inverse Problems, 8, 331–364 (1992).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. M. Boiti, F. Pempinelli, and A. K. Pogrebkov, J. Math. Phys., 35, 4683–4718 (1994).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. M. Boiti, F. Pempinelli, A. K. Pogrebkov, and B. Prinari, Inverse Problems, 17, 937–957 (2001).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. M. Boiti, F. Pempinelli, A. K. Pogrebkov, and B. Prinari, J. Math. Phys., 44, 3309–3340 (2003).

    Article  ADS  MATH  MathSciNet  Google Scholar 

  15. A. K. Pogrebkov, Theor. Math. Phys., 154, 405–417 (2008); arXiv:nlin/0703018v1 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  16. A. K. Pogrebkov, “2D Toda chain and associated commutator identity,” in: Geometry, Topology, and Mathematical Physics: S. P. Novikov’s Seminar: 2006–2007 (Amer. Math. Soc. Transl. Ser. 2, Vol. 224, V. M. Buchstaber and I. M. Krichever, eds.), Amer. Math. Soc., Providence, R. I. (2008), pp. 261–270.

    Google Scholar 

  17. M. Boiti, F. Pempinelli, and A. K. Pogrebkov, Theor. Math. Phys., 172, 1037–1051 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  18. B. Grammaticos, A. Ramani, V. Papageorgiou, J. Satsuma, and R. Willox, J. Phys. A, 40, 12619–12627 (2007).

    Article  ADS  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute, RAS Moscow, Russia

    A. K. Pogrebkov

  2. National Research University Higher School of Economics, Moscow, Russia

    A. K. Pogrebkov

Authors
  1. A. K. Pogrebkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. K. Pogrebkov.

Additional information

Dedicated to the 75th birthday of Andrei Alekseevich Slavnov

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pogrebkov, A.K. Hirota difference equation: Inverse scattering transform, darboux transformation, and solitons. Theor Math Phys 181, 1585–1598 (2014). https://doi.org/10.1007/s11232-014-0237-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-014-0237-z

Keywords

Search

Navigation