Geodesic
Discrete Toda lattices and the Laplace method
Teoretičeskaâ i matematičeskaâ fizika, Tome 160 (2009) no. 3, pp. 434-443 Cet article a éte moissonné depuis la source Math-Net.Ru

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We apply the Laplace cascade method to systems of discrete equations of the form $u_{i+1,j+1}=f(u_{i+1,j}, u_{i,j+1},u_{i,j}, u_{i,j-1})$, where $u_{ij}$, $i,j\in\mathbb Z$, is an element of a sequence of unknown vectors. We introduce the concept of a generalized Laplace invariant and the related property that the systems is “of the Liouville type”. We prove a series of statements about the correctness of the definition of the generalized invariant and its applicability for seeking solutions and integrals of the system. We give some examples of systems of the Liouville type.
Keywords: nonlinear discrete equation, Laplace method, Darboux integrability. 
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V. L. Vereshchagin. Discrete Toda lattices and the Laplace method. Teoretičeskaâ i matematičeskaâ fizika, Tome 160 (2009) no. 3, pp. 434-443. http://geodesic.mathdoc.fr/item/TMF_2009_160_3_a1/

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