Algebraic properties of quasilinear two-dimensional lattices connected with integrability

Abstract

In the paper we discuss a classification method for nonlinear integrable equations with three independent variables based on the notion of the integrable reductions. We call an 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 in the framework of this scheme is the class of two dimensional lattices generalizing the well-known Toda lattice. In the present article we study the quasilinear lattices of the form \begin{align*} 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}). \end{align*} 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 describe the class of the lattices integrable in the above sense. There are new examples in the obtained list of chains.