Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 088, 13 pages      arXiv:2102.04207      https://doi.org/10.3842/SIGMA.2021.088

Lax Pair for a Novel Two-Dimensional Lattice

Maria N. Kuznetsova
Institute of Mathematics, Ufa Federal Research Centre, Russian Academy of Sciences,112 Chernyshevsky Street, Ufa 450008, Russia

Received February 09, 2021, in final form September 15, 2021; Published online September 26, 2021

Abstract
In paper by I.T. Habibullin and our joint paper the algorithm for classification of integrable equations with three independent variables was proposed. This method is based on the requirement of the existence of an infinite set of Darboux integrable reductions and on the notion of the characteristic Lie-Rinehart algebras. The method was applied for the classification of integrable cases of different subclasses of equations $u_{n,xy} = f(u_{n+1},u_n,u_{n-1}, u_{n,x},u_{n,y})$ of special forms. Under this approach the novel integrable chain was obtained. In present paper we construct Lax pair for the novel chain. To construct the Lax pair, we use the scheme suggested in papers by E.V. Ferapontov. We also study the periodic reduction of the chain.

Key words: Lax pair; two-dimensional lattice; integrable reduction; characteristic algebra; Lie-Rinehart algebra; Darboux integrable system; higher symmetry; $x$-integral.

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