Theoretical and Mathematical Physics

, Volume 121, Issue 2, pp 1484–1495 | Cite as

Discrete analogues of the Liouville equation

  • V. E. Adler
  • S. Ya. Startsev


The notion of Laplace invariants is generalized to lattices and discrete equations that are difference analogues of hyperbolic partial differential equations with two independent variables. The sequence of Laplace invariants satisfies the discrete analogue of the two-dimensional Toda lattice. We prove that terminating this sequence by zeros is a necessary condition for the existence of integrals of the equation under consideration. We present formulas for the higher symmetries of equations possessing such integrals. We give examples of difference analogues of the Liouville equation.


Dynamic Variable Liouville Equation Continuous Case Discrete Case Discrete Analogue 
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  1. 1.
    J. Liouville,J. Math. Pure Appl.,18, 71 (1853).Google Scholar
  2. 2.
    G. Darboux,Leçons sur la théorie générale des surfaces et les applications geometriques du calcul infinitesimal, Gauthier-Villars, Paris (1896).zbMATHGoogle Scholar
  3. 3.
    E. Goursat,Leçons sur l'intégration des équations aux dérivées partielles du second ordre à deux variables independantes, Hermann, Paris (1896).Google Scholar
  4. 4.
    E. Vessiot,J. Math. Pure Appl.,18, 1 (1939);21, 1 (1942).zbMATHMathSciNetGoogle Scholar
  5. 5.
    A. R. Forsyth,Theory of Differential Equations, Dover, New York (1959).zbMATHGoogle Scholar
  6. 6.
    A. V. Zhiber and A. B. Shabat,Dokl. Akad. Nauk SSSR,247, 1103 (1979).MathSciNetGoogle Scholar
  7. 7.
    A. V. Zhiber, N. Kh. Ibragimov, and A. B. Shabat,Sov. Math. Dokl. 20. 1183 (1979).zbMATHGoogle Scholar
  8. 8.
    A. B. Shabat and R. I. Yamilov, Exponential systems of type I and Cartan matrices [in Russian], Bashkir Branch Acad. Sci. USSR, Ufa (1981).Google Scholar
  9. 9.
    A. V. Zhiber and A. B. Shabat,Sov. Math. Dokl. 30, 23 (1984).zbMATHGoogle Scholar
  10. 10.
    A. V. Zhiber,Russ. Acad. Sci. Izv. Math. 45, No. 1, 33 (1995).CrossRefMathSciNetGoogle Scholar
  11. 11.
    I. M. Anderson and N. Kamran,Duke Math. J.,87, 265 (1997).zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    A. V. Zhiber, V. V. Sokolov, and S. Ya. Startsev,Dokl. Math.,52, No. 1, 128 (1995).zbMATHGoogle Scholar
  13. 13.
    V. V. Sokolov and A. V. Zhiber,Phys. Lett. A. 208, 303 (1995).zbMATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    I. M. Anderson and M. Juras,Duke Math. J.,89, 351 (1997).zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    A. P. Veselov and A. B. Shabat,Funct. Anal. Appl. 27, No. 2, 81 (1993).zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    V. G. Papageorgiou, F. W. Nijhoff, and H. W. Capel,Phys. Lett. A,147, 106 (1990).CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    S. P. Novikov and I. A. Dynnikov,Russ. Math. Surv.,52, 1057 (1997).zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    R. Hirota,J. Phys. Soc. Japan,50, 3785 (1981).CrossRefMathSciNetGoogle Scholar
  19. 19.
    T. Miwa,Proc. Japan Acad.,58, 9 (1982).zbMATHMathSciNetGoogle Scholar
  20. 20.
    D. Levi, L. Pilloni, and P. M. Santini,J. Phys. A,14, 1567 (1981).CrossRefADSMathSciNetGoogle Scholar
  21. 21.
    A. Ramani, B. Grammaticos, and J. Satsuma,Phys. Lett. A,169, 323 (1992).CrossRefADSMathSciNetGoogle Scholar
  22. 22.
    I. G. Korepanov, “Integrable systems in discrete space-time and inhomogeneous models in two-dimensional statistical physics”, Doctoral dissertation, POMI, St. Petersburg (1995); Preprint solv-int/9506003 (1995).Google Scholar
  23. 23.
    A. B. Shabat,Phys. Lett. A,200, 121 (1995).zbMATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 1999

Authors and Affiliations

  • V. E. Adler
    • 1
  • S. Ya. Startsev
    • 1
  1. 1.Institute for Mathematics and Computation Center, Ufa Science CenterRASUfaRussia

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