A classification algorithm for integrable two-dimensional lattices via Lie—Rinehart algebras

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 sens...

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Bibliographic Details
Published inTheoretical and mathematical physics Vol. 203; no. 1; pp. 569 - 581
Main Authors Habibullin, I. T., Kuznetsova, M. N.
Format Journal Article
LanguageEnglish
Published Moscow Pleiades Publishing 01.04.2020
Springer Nature B.V
Subjects
Algebra
Algorithms
Applications of Mathematics
Classification
Continuity (mathematics)
Lattices (mathematics)
Mathematical analysis
Mathematical and Computational Physics
Physics
Physics and Astronomy
Theoretical
cutoff condition
integrable reduction
characteristic Lie algebra
open lattice
Darboux-integrable system
two-dimensional integrable lattice
x-integral
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Summary: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.
ISSN:0040-5779
1573-9333
DOI:10.1134/S0040577920040121