Classification of a subclass of quasilinear two-dimensional lattices by means of characteristic algebras

Abstract

We consider a classification problem of integrable cases of the Toda type two-dimensional lattices $u_{n,xy} = f(u_{n+1},u_n,u_{n-1}, u_{n,x},u_{n,y})$. The function $f = f(x_1,x_2,\cdots x_5)$ is assumed to be analytic in a domain $D\subset \mathbb{C}^5$. The sought function $u_n = u_n(x,y)$ depends on real $x$, $y$ and integer $n$. Equations with three independent variables are complicated objects for study and especially for classification. It is commonly accepted that for a given equation, the existence of a large class of integrable reductions indicates integrability. Our classification algorithm is based on this observation. We say that a constraint $u_0 = \varphi(x,y)$ defines a degenerate cutting off condition for the lattice if it divides this lattice into two independent semi-infinite lattices defined on the intervals $-\infty