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

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Abstract

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.

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Authors and Affiliations

  1. Institute of Mathematics with Computing Centre - Subdivision of the Ufa Federal Research Centre of the Russian Academy of Science, Ufa, Russia

    I. T. Habibullin & M. N. Kuznetsova

  2. Bashkir State University, Ufa, Russia

    I. T. Habibullin

Authors
  1. I. T. Habibullin
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  2. M. N. Kuznetsova
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Correspondence to I. T. Habibullin or M. N. Kuznetsova.

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Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 203, No. 1, pp. 161–173, April, 2020.

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Habibullin, I.T., Kuznetsova, M.N. A classification algorithm for integrable two-dimensional lattices via Lie—Rinehart algebras. Theor Math Phys 203, 569–581 (2020). https://doi.org/10.1134/S0040577920040121

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  • DOI: https://doi.org/10.1134/S0040577920040121

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