The Coxeter relations and KP map for non-commuting symbols

We give an action of the symmetric group on non-commuting indeterminates in terms of series in the corresponding Mal’cev–Newmann division ring. The action is constructed from the non-Abelian Hirota–Miwa (discrete KP) system. The corresponding companion map, which gives generators of the action, is discussed in the generic case, and the corresponding explicit formulas have been found in the periodic reduction. We discuss also briefly connection of the companion to the KP map with context-free languages.

The squared eigenfunction symmetries of the discrete KP and modified discrete KP hierarchies under the Miura transformations

In this paper, we investigate the squared eigenfunction symmetries under the Miura and reverse-Miura transformations, between the discrete KP and modified discrete KP hierarchies, in unconstrained and constrained cases. What is more, the squared eigenfunction symmetries for the constrained discrete KP and constrained modified discrete KP hierarchies are constructed by imposing the additional conditions on the generating eigenfunctions of the corresponding symmetries.

Gauge transformations of constrained discrete modified KP systems with self-consistent sources

In this paper, we firstly recall some basic facts about the discrete KP(d-KP) and discrete modified KP(d-mKP) hierarchies, and then we find that d-KP hierarchy and d-mKP hierarchy are linked by a gauge transformation. What’s more, we give three gauge transformation operators of the d-mKP hierarchy and give their successive applications. We further construct the ghost symmetry and use this symmetry to give the definition the d-mKP hierarchy with self-consistent sources. We also give gauge transformations of a newly defined constrained d-mKP(cd-mKP) hierarchy and the constrained d-mKP hierarchy with self-consistent sources(cd-mKPHSCSs).