A non-commutative discrete first Painlevé hierarchy: the Lax pair approach

2026/05/28 by Irina Bobrova · 1 voice
Physics and Astronomy · Mathematics · #nlin.SI #math-ph #math.RA

paper · pdf

Abstract

Using a non-commutative analogue of the isomonodromic problem associated with the discrete first Painlevé hierarchy, we construct a non-commutative version of this hierarchy, denoted by d-PI_m^nc. We show that both hierarchies, d-PI_m and d-PI_m^nc, can be expressed in terms of the polynomials S_s^k(n), which we call the Svinin polynomials. We also derive a reduction of the non-commutative Volterra lattice hierarchy to the d-PI_m^nc hierarchy and present explicit continuous limits for the first three members of the d-PI_m^nc, thereby recovering non-commutative analogues of the first three members of the differential first Painlevé hierarchy.

Citations

• On a non-commutative sixth q-Painlevé system: from discrete system to surface theory
• Affine Weyl groups and non-Abelian discrete systems: an application to the d-Painlevé equations
• On integrals for some class of ordinary difference equations admitting a Lax pair representation
• Singularity confinement for matrix discrete Painleve Equations
• Non-commutative q-Painleve VI equation
• On some classes of discrete polynomials and ordinary difference equations
• On some class of homogeneous polynomials and explicit form of integrable hierarchies of differential-difference equations
• On some class of reductions for Itoh-Narita-Bogoyavlenskii lattice
• The Hamiltonian Structure of the Second Painleve Hierarchy

Discussions

• Irina Bobrova: A non-commutative discrete first Painlev\'e hierarchy: the Lax pair approach https://arxiv.org/abs/2605.29722 https://arxiv.org/pdf/2605.29722 https://arxiv.org/html/2605.29722 [bsky, 0 points, 0 comments]

Related

• The Painlevé I hierarchy: Correspondence between the isomonodromic approach and the minimal models of the KP hierarchy
• Integrable Mappings Related to the Extended Discrete KP Hierarchy
• Invariant Submanifolds of Darboux-KP Chain and Extension of the Discete KP Hierarchy
• Noncommutative nonisospectral Toda and Lotka-Volterra lattices, and matrix discrete Painlevé equations
• On a non-commutative sixth q-Painlevé system: from discrete system to surface theory