Noncommutative discrete equations, symmetries and reductions

TL;DR

This paper develops noncommutative discrete equations, explores their symmetries and reductions, and introduces transformations linking these equations to known integrable systems, expanding the understanding of noncommutative integrability.

Contribution

It constructs new noncommutative differential-difference equations, identifies their symmetries, and connects them to noncommutative Painlevé and Ernst equations, advancing noncommutative integrable systems theory.

Findings

Derived differential-difference equations from noncommutative discrete KdV systems.

Constructed Miura transformations to noncommutative modified Volterra equations.

Established Darboux and Bäcklund transformations linked to noncommutative Yang-Baxter maps.

Abstract

Employing the Lax pairs of the noncommutative discrete potential Korteweg--de Vries (KdV) and Hirota's KdV equations, we derive differential--difference equations that are consistent with these systems and serve as their generalised symmetries. Miura transformations mapping these equations to a noncommutative modified Volterra equation and its master symmetry are constructed. We demonstrate the use of these symmetries to reduce the potential KdV equation, leading to a noncommutative discrete Painlev{\`{e}} equation and to a system of partial differential equations that generalises the Ernst equation and the Neugebauer--Kramer involution. Additionally, we present a Darboux transformation and an auto-B\"acklund transformation for the Hirota KdV equation, and establish their connection with the noncommutative Yang--Baxter map $F_{III}$.