Noncommutative discrete equations, symmetries and reductions

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 Painlevé 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-Bäcklund transformation for the Hirota’s KdV equation, and establish their connection with the noncommutative Yang–Baxter map F I I I .