Darboux-integrable discrete systems
Tóm tắt
We extend Laplace’s cascade method to systems of discrete “hyperbolic” equations of the form ui+1,j+1
= f(ui+1,j, ui,j+1
, ui,j), where uij 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.
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