Algebraic Properties of Quasilinear Two-Dimensional Lattices connected with integrability

TL;DR

This paper classifies a class of quasilinear two-dimensional lattices related to integrability, identifying new examples by analyzing their algebraic properties and reductions to Darboux integrable systems.

Contribution

It introduces a classification method for quasilinear lattices connected with integrability, providing new examples and characterizing their algebraic properties.

Findings

Identified conditions for integrability of quasilinear lattices.

Derived new examples of integrable lattice chains.

Connected lattice integrability with Darboux reduction techniques.

Abstract

In the article a classification method for nonlinear integrable equations with three independent variables is discussed based on the notion of the integrable reductions. We call the equation integrable if it admits a large class of reductions being Darboux integrable systems of hyperbolic type equations with two independent variables. The most natural and convenient object to be studied within the frame of this scheme is the class of two dimensional lattices generalizing the well-known Toda lattice. In the present article we deal with the quasilinear lattices of the form $u_{n,xy}=\alpha(u_{n+1} ,u_n,u_{n-1} )u_{n,x}u_{n,y} + \beta(u_{n+1},u_n,u_{n-1})u_{n,x}+\gamma(u_{n+1} ,u_n,u_{n-1} )u_{n,y}+\delta(u_{n+1} ,u_n,u_{n-1})$. We specify the coefficients of the lattice assuming that there exist cutting off conditions which reduce the lattice to a Darboux integrable hyperbolic type system of the arbitrarily high order. Under some extra assumption of nondegeneracy we described the class of the lattices integrable in the sense indicated above. There are new examples in the obtained list of chains.