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Discrete analogues of the Liouville equation

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Abstract

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.

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Authors and Affiliations

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

    V. E. Adler & S. Ya. Startsev

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  1. V. E. Adler
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  2. S. Ya. Startsev
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 2, pp. 271–284, November, 1999.

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Adler, V.E., Startsev, S.Y. Discrete analogues of the Liouville equation. Theor Math Phys 121, 1484–1495 (1999). https://doi.org/10.1007/BF02557219

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  • DOI: https://doi.org/10.1007/BF02557219

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