Darboux transformations for linear operators on two dimensional regular lattices

TL;DR

This paper reviews Darboux transformations for linear operators on 2D lattices, focusing on the six point scheme and its reductions, exploring multidimensional aspects and sublattice combinations.

Contribution

It provides a comprehensive review of Darboux transformations on 2D lattices, emphasizing the six point scheme and its multidimensional reductions.

Findings

Analysis of the six point scheme as the master linear problem

Descriptions of various reductions and sublattice combinations

Discussion of multidimensional aspects of the schemes

Abstract

Darboux transformations for linear operators on regular two dimensional lattices are reviewed. The six point scheme is considered as the master linear problem, whose various specifications, reductions, and their sublattice combinations lead to other linear operators together with the corresponding Darboux transformations. The second part of the review deals with multidimensional aspects of (basic reductions of) the four point scheme, as well as the three point scheme.