Noncommutative discrete equations, symmetries and reductions

2025/07/06 by Xenitidis, Pavlos · 1 citation
#Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2507.04472

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 Painlevè 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 KdV equation, and establish their connection with the noncommutative Yang--Baxter map F_III.

Citations

• Integrability of Nonabelian Differential-Difference Equations: the Symmetry Approach
• Systems of difference equations, symmetries and integrability conditions

Cited by

• Integrable discretisations of the noncommutative NLS equation

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