Geodesic
Darboux-integrable discrete systems
Teoretičeskaâ i matematičeskaâ fizika, Tome 156 (2008) no. 2, pp. 207-219 Cet article a éte moissonné depuis la source Math-Net.Ru

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We extend Laplace's cascade method to systems of discrete “hyperbolic” equations of the form $u_{i+1,j+1}=f(u_{i+1,j},u_{i,j+1},u_{i,j})$, where $u_{ij}$ is a member of a sequence of unknown vectors, $i,j\in\mathbb Z$. 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.
Keywords: Laplace's cascade method, Darboux integrability, nonlinear chain. 
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V. L. Vereshchagin. Darboux-integrable discrete systems. Teoretičeskaâ i matematičeskaâ fizika, Tome 156 (2008) no. 2, pp. 207-219. http://geodesic.mathdoc.fr/item/TMF_2008_156_2_a3/

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