Quantum Painlevé Equations: from Continuous to Discrete

Abstract

We examine quantum extensions of the continuous Painlevé equations, expressed as systems of first-order differential equations for non-commuting objects. We focus on the Painlevé equations II, IV and V. From their auto-Bäcklund transformations we derive the contiguity relations which we interpret as the quantum analogues of the discrete Painlevé equations.

Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

References in corpus (1)

• Quantum Painlevé systems of type $A_l^{(1)}$

Cited by in corpus (6)

• On matrix Painlevé-4 equations. Part 2: Isomonodromic Lax pairs
• On matrix Painlevé-4 equations. Part 1: Painlevé--Kovalevskaya test
• A fully noncommutative analog of the Painlevé IV equation and a structure of its solutions
• Quasideterminant solutions of NC Painlevé II equation with the Toda solution at $n=1$ as a seed solution in its Darboux transformation
• Regularity of quantum tau-functions generated by quantum birational Weyl group actions
• Darboux solutions of non-abelian quantum Painlevé II equation in terms of quasideterminants