Skip to main content
SpringerLink
Log in

Darboux-integrable discrete systems

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

Abstract

We extend Laplace’s cascade method to systems of discrete “hyperbolic” equations of the form ui+1,j+1 = f(ui+1,j, ui,j+1 , ui,j), where uij is a member of a sequence of unknown vectors, i, j ∊ ℤ. We introduce the notion of a generalized Laplace invariant and the associated property of the system being “Liouville.” We prove several statements on the well-definedness of the generalized invariant and on its use in the search for solutions and integrals of the system. We give examples of discrete Liouville-type systems.

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. G. Darboux, “Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal,” in: Déformation infiniment petite et représentation sphérique, Vol. 4, Gautier-Villars, Paris (1896), pp. 353–548.

    Google Scholar 

  2. I. M. Anderson and N. Kamran, Duke Math. J., 87, 265–319 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  3. A. V. Zhiber, V. V. Sokolov, and S. Ya. Startsev, Dokl. Math., 52, No. 1, 128–130 (1995).

    MATH  Google Scholar 

  4. A. V. Zhiber and V. V. Sokolov, Laplace’s Method of Cascade Integration and Darboux-Integrable Equations [in Russian], RITs BashGU, Ufa (1996).

    Google Scholar 

  5. V. E. Adler and S. Ya. Startsev, Theor. Math. Phys., 121, 1484–1495 (1999).

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  7. D. Levi, L. Pilloni, and P. M. Santini, J. Phys. A, 14, 1567–1575 (1981).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. R. S. Ward, Phys. Lett. A, 199, 45–48 (1995).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. R. Hirota, J. Phys. Soc. Japan, 46, 312–319 (1979).

    Article  ADS  MathSciNet  Google Scholar 

  10. A. V. Zhiber and V. V. Sokolov, Russ. Math. Surveys, 56, 61–101 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  11. A. M. Gur’eva, “Laplace’s method of cascade integration and nonlinear hyperbolic systems of equations,” Candidate’s dissertation phys.-math. sci., UGATU, Ufa (2005) [in Russian].

    Google Scholar 

  12. A. V. Zhiber, V. V. Sokolov, and S. Ya. Startsev, “Nonlinear hyperbolic partial differential systems of the Liouville type,” in: Intl. Conf. “Tikhonov and Contemporary Mathematics” [in Russian] (Moscow, Russia, June 19–25, 2006), Vol. 1 (2006), pp. 1–2.

    Google Scholar 

  13. A. M. Guryeva and A. V. Zhiber, Theor. Math. Phys., 138, 338–355 (2004).

    Article  Google Scholar 

  14. I. T. Habibullin, Theor. Math. Phys., 146, 170–182 (2006).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Mathematics and Computing Center, Urals Science Center, RAS, Ufa, Russia

    V. L. Vereshchagin

Authors
  1. V. L. Vereshchagin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. L. Vereshchagin.

Additional information

__________

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 207–219, August, 2008.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vereshchagin, V.L. Darboux-integrable discrete systems. Theor Math Phys 156, 1142–1153 (2008). https://doi.org/10.1007/s11232-008-0084-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-008-0084-x

Keywords

Search

Navigation