Non-commutative lattice modified Gel'fand-Dikii systems

TL;DR

This paper introduces integrable non-commutative lattice systems as analogs of the Gel'fand-Dikii hierarchy, demonstrating their multidimensional consistency, reductions, and geometric interpretations, including non-isospectral generalizations.

Contribution

It presents the first integrable multicomponent non-commutative lattice systems related to the Gel'fand-Dikii hierarchy with geometric and reduction frameworks.

Findings

Demonstrated multidimensional consistency of the systems

Derived systems as reductions of the non-commutative KP hierarchy

Established geometric interpretation via Desargues maps

Abstract

We introduce integrable multicomponent non-commutative lattice systems, which can be considered as analogs of the modified Gel'fand-Dikii hierarchy. We present the corresponding systems of Lax pairs and we show directly multidimensional consistency of these Gel'fand-Dikii type equations. We demonstrate how the systems can be obtained as periodic reductions of the non-commutative lattice Kadomtsev-Petviashvilii hierarchy. The geometric description of the hierarchy in terms of Desargues maps helps to derive non-isospectral generalization of the non-commutative lattice modified Gel'fand-Dikii systems. We show also how arbitrary functions of single arguments appear naturally in our approach when making commutative reductions, which we illustrate on the non-isospectral non-autonomous versions of the lattice modified Korteweg-de Vries and Boussinesq systems.