Integrability conditions for two-dimensional Toda-like equations

Abstract

In the article some algebraic properties of nonlinear two-dimensional lattices of the form u n,xy = f (u n+1 , u n , u n−1 ) are studied. The problem of exhaustive description of the integrable cases of this kind lattices remains open. By using the approach, developed and tested in our previous works we adopted the method of characteristic Lie-Rinehart algebras to this case. In the article we derived an effective integrability conditions for the lattice and proved that in the integrable case the function f (u n+1 , u n , u n−1 ) is a quasi-polynomial satisfying the following equationwhere C and α are constant parameters and k, m are nonnegative integers.

An algebraic criterion of the Darboux integrability of differential-difference equations and systems

“…The essence of the method is that we put forward the following hypothesis: the condition for the existence of such a reduction of an arbitrarily high order is an integrability test for lattices in 3D. The correctness of this hypothesis was confirmed in the works [2][3][4][5], where the problem of classification of differential-difference equations of the two-dimensional Toda lattice type…”

“…The next step in this direction is to study the problem of adaptation of the classification algorithm worked out in [2][3][4][5] to a class of semi-discrete equations of the form u j n+1,x = F (u j n,x , u j+1 n , u j n+1 , u j n , u j−1 n+1 ),…”

Characteristic algebras and integrable exponential systems

“…In particular, in [2], integrable exponential type systems with Cartan matrices of size 2 were studied, and in [3], systems corresponding to non-degenerate Cartan matrices and, in particular, systems corresponding to Lie algebras of the series , were integrated explicitly. We also mention papers [4]- [6] by the Ufa school , in which reductions of exponential type systems and their connection with higher symmetries were studied, and a series of papers [7]- [17], in which discrete analogs of exponential type systems were studied.…”

Lax Pair for a Novel Two-Dimensional Lattice

“…In a number of recent publications [8,13,14,15,16,18] the problem of integrable classification of two-dimensional lattices u n,xy = f (u n+1 , u n , u n−1 , u n,x , u n,y ), −∞ < n < ∞, (1.1) was studied. Here the sought function u n = u n (x, y) depends on the real variables x, y and the integer variable n. In these papers we proposed the method for seeking and classifying integrable equations with three independent variables based on the requirement of the existence of a set of Darboux integrable reductions and on the notion of the characteristic Lie-Rinehart algebras.…”