Darboux-integrable discrete systems

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 ∊ 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. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 207–219, August, 2008.