Skip to main content
SpringerLink
Log in

A Laplace Ladder of Discrete Laplace Equations

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

Abstract

We present the notion of a Laplace ladder for a discrete analogue of the Laplace equation. We introduce the adjoint of the discrete Moutard equation and a discrete counterpart of the nonlinear representation for the Goursat equation.

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. H. Jonas, Berlin Sitzungsber, 14, 96 (1915).

    Google Scholar 

  2. L. P. Eisenhart, Transformation of Surfaces, Princeton Univ. Press, Princeton, N. J. (1923).

    Google Scholar 

  3. V. E. Zakharov and S. V. Manakov, Funct. Anal. Appl., 19, 89 (1985).

    Google Scholar 

  4. L. V. Bogdanov and B. G. Konopelchenko, J. Phys. A, 28, L173 (1995); M. Mañas, A. Doliwa, and P. M. Santini, Phys. Lett. A, 232, 99 (1997).

    Google Scholar 

  5. A. Doliwa, P. M. Santini, and M. Mañas, J. Math. Phys., 41, 944 (2000).

    Google Scholar 

  6. J. J. C. Nimmo and W. K. Schief, Proc. Roy. Soc. London A, 453, 255 (1997).

    Google Scholar 

  7. J. Cieslinski, A. Doliwa, and P. M. Santini, Phys. Lett. A, 235, 480 (1997); B. G. Konopelchenko and W. K. Schief, Proc. Roy. Soc. London A, 454, 3075 (1998); A. Doliwa, S. V. Manakov, and P. M. Santini, Comm. Math. Phys., 196, 1 (1998).

    Google Scholar 

  8. A. Doliwa and P. M. Santini, J. Geom. Phys., 36, 60 (2000).

    Google Scholar 

  9. A. Doliwa, J. Geom. Phys., 30, 169 (1999).

    Google Scholar 

  10. C. Athorne, Inverse Problems, 9, 217 (1993).

    Google Scholar 

  11. E. Goursat, Bull. Soc. Math. France, 28, 1 (1900); E. I. Ganzha, Theor. Math. Phys., 122, 50 (2000).

    Google Scholar 

  12. Th.-F. Moutard, J. École Polytec., 45, 1 (1878).

    Google Scholar 

  13. A. Doliwa, Phys. Lett. A, 234, 187 (1997); V. É. Adler and S. Ya. Startsev, Theor. Math. Phys., 121, 271 (1999); I. A. Dynnikov and S. P. Novikov, Russ. Math. Surveys, 52, 1294 (1997).

    Google Scholar 

  14. M. Nieszporski, A. Doliwa, and P. M. Santini, “The integrable discretization of the Bianchi–Ernst system,” nlin.SI/0104065 (2001); A. Doliwa, M. Nieszporski, and P. M. Santini, J. Phys. A, 34, 10423 (2001).

  15. A. Doliwa, “Lattice geometry of the Hirota equation,” in: SIDE III–Symmetries and Integrability of Difference Equations (CRM Proc. Lect. Notes, Vol. 25, D. Levi and O. Ragnisco, eds.), Amer. Math. Soc., Providence, R. I. (2000), p. 93.

    Google Scholar 

  16. A. Doliwa, “Geometric discretization of the Koenigs nets,” nlin.SI/0203011 (2002).

  17. W. K. Schief, “On the unification of classical and novel integrable surfaces: II. Difference geometry,” nlin.SI/ 0104037 (2001).

  18. W. K. Schief, Stud. Appl. Math., 106, 85 (2001).

    Google Scholar 

  19. M. Nieszporski, J. Geom. Phys., 40, 259 (2002).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Theoretical Physics, University of Bialystok, Bialystok, Poland

    M. Nieszposki

Authors
  1. M. Nieszposki
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nieszposki, M. A Laplace Ladder of Discrete Laplace Equations. Theoretical and Mathematical Physics 133, 1576–1584 (2002). https://doi.org/10.1023/A:1021159129804

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1021159129804

Search

Navigation