On integrable reductions of two-dimensional Toda-type lattices

TL;DR

This paper investigates integrable reductions of two-dimensional Toda-type lattices, revealing their Darboux integrability through finite-field reductions and providing an algorithm for constructing characteristic integrals.

Contribution

It introduces a method to obtain Darboux integrable finite-field reductions of 2D Toda lattices using boundary conditions, Lax pairs, and Miura transformations.

Findings

Finite-field reductions are Darboux integrable.

An algorithm for constructing characteristic integrals is proposed.

Reductions relate to generalizations of nonlinear Schrödinger equations.

Abstract

The article considers lattices of the two-dimensional Toda type, which can be interpreted as dressing chains for spatially two-dimensional generalizations of equations of the class of nonlinear Schr\"odinger equations. The well-known example of this kind of generalization is the Davey-Stewartson equation. It turns out that the finite-field reductions of these lattices, obtained by imposing cutoff boundary conditions of an appropriate type, are Darboux integrable, i.e., they have complete sets of characteristic integrals. An algorithm for constructing complete sets of characteristic integrals of finite field systems using Lax pairs and Miura-type transformations is discussed.