Lax Pair for a Novel Two-Dimensional Lattice

TL;DR

This paper constructs a Lax pair for a newly identified integrable two-dimensional lattice chain, expanding the understanding of integrable systems through a novel classification approach and periodic reductions.

Contribution

It introduces a Lax pair for a new integrable chain derived via a classification method based on Darboux reductions and Lie-Rinehart algebras.

Findings

Lax pair constructed for the novel integrable chain

Periodic reduction analyzed

Method extends classification of integrable equations

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.