Discrete analogues of the Liouville equation

1999/02/23 by V. E. Adler, V. É. Adler, S. Ya. Startsev · 6 citations
Physics and Astronomy · #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems #nlin.SI #solv-int

paper · pdf · doi:10.1007/bf02557219

published as Theoretical and Mathematical Physics 1999, Volume 121, Issue 2, pp 1484-1495 · LaTeX, 15 pages, submitted to Teor. i Mat. Fiz

arxiv created 1999/02/23 · crossref issued 1999/11/01 · crossref published 1999/11/01 · crossref published-print 1999/11/01 · openalex publication_date 1999/11/01 · crossref created 2007/03/22 · arxiv updated 2014/08/27 · openalex created_date 2025/10/10 · crossref deposited 2026/04/06 · crossref indexed 2026/07/30 · openalex updated_date 2026/07/31

Abstract

The notion of Laplace invariants is transferred to the lattices and discrete equations which are difference analogs of hyperbolic PDE's with two independent variables. The sequence of Laplace invariants satisfy the discrete analog of twodimensional Toda lattice. The terminating of this sequence by zeroes is proved to be the necessary condition for existence of the integrals of the equation under consideration. The formulae are presented for the higher symmetries of the equations possessing integrals. The general theory is illustrated by examples of difference analogs of Liouville equation.

Cited by

• On Darboux-integrable semi-discrete chains
• On the classification of Darboux integrable chains
• Darboux transformations for linear operators on two-dimensional regular lattices
• Discrete integrable equations with three independent variables
• Non-Point Invertible Transformations and Integrability of Partial Difference Equations
• Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-difference equations