Desargues maps and their reductions

TL;DR

This paper explores the geometric and algebraic structures of Desargues maps and their reductions, establishing multidimensional consistency and connections to integrable systems like the KP hierarchy and Painleve equations.

Contribution

It introduces new results on the multidimensional consistency of non-commutative KP maps and links Desargues maps to various integrable systems and their reductions.

Findings

Proved three-dimensional consistency of non-commutative KP map.

Connected Desargues maps to quadrilateral lattice maps.

Derived non-isospectral modified lattice Gel'fand-Dikii and q-Painleve equations.

Abstract

We present recent developments on geometric theory of the Hirota system and of the non-commutative discrete Kadomtsev-Petviashvili (KP) hierarchy adding also some new results which make the picture more complete. We pay special attention to multidimensional consistency of the Desargues maps and of the resulting non-linear non-commutative systems. In particular, we show three-dimensional consistency of the non-commutative KP map in its edge formulation. We discuss also relation of Desargues maps and quadrilateral lattice maps. We study from that point of view reductions of the Hirota system to discrete B-KP and C-KP systems presenting also a novel constraint which leads to the Miwa equations. By imposing periodicity reduction of the discrete KP hierarchy we obtain non-isospectral versions of the modified lattice Gel'fand-Dikii equations. To close the picture from below, we apply additional self-similarity constraint on the non-isospectral non-autonomous modified lattice Korteweg-de Vries system to recover known q-Painleve equation of type A_2+A_1.