Geodesic
A~classification algorithm for integrable two-dimensional lattices
Teoretičeskaâ i matematičeskaâ fizika, Tome 203 (2020) no. 1, pp. 161-173

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We study the problem of the integrable classification of nonlinear lattices depending on one discrete and two continuous variables. By integrability, we mean the presence of reductions of a chain to a system of hyperbolic equations of an arbitrarily high order that are integrable in the Darboux sense. Darboux integrability admits a remarkable algebraic interpretation: the Lie–Rinehart algebras related to both characteristic directions corresponding to the reduced system of hyperbolic equations must have a finite dimension. We discuss a classification algorithm based on the properties of the characteristic algebra and present some classification results. We find new examples of integrable equations.
Keywords: two-dimensional integrable lattice, $x$-integral, integrable reduction, cutoff condition, open lattice, Darboux-integrable system, characteristic Lie algebra. 
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     author = {I. T. Habibullin and M. N. Kuznetsova},
     title = {A~classification algorithm for integrable two-dimensional lattices},
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I. T. Habibullin; M. N. Kuznetsova. A~classification algorithm for integrable two-dimensional lattices. Teoretičeskaâ i matematičeskaâ fizika, Tome 203 (2020) no. 1, pp. 161-173. http://geodesic.mathdoc.fr/item/TMF_2020_203_1_a11/