Non-commutative lattice-modified Gel’fand–Dikii systems

Abstract

We introduce integrable multicomponent non-commutative lattice systems, which can be considered as analogues of the modified Gel’fand–Dikii hierarchy. We present the corresponding systems of Lax pairs and show directly the 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 a 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.