Noncommutative Painlev\'e equations and systems of Calogero type

TL;DR

This paper extends Painlevé equations to multi-particle Calogero systems, demonstrating their interpretation as isomonodromic equations and providing methods to construct solutions for special parameters.

Contribution

It explicitly constructs isomonodromic Lax pairs for noncommutative Painlevé and Calogero systems, answering Takasaki's open question and enabling solution generation via Schlesinger transforms.

Findings

Established isomonodromic Lax pair formulations for multi-particle systems.

Connected Painlevé equations with Calogero-type interactions in a noncommutative setting.

Demonstrated solution construction for special coupling constants using Schlesinger transforms.

Abstract

All Painlev\'e equations can be written as a time-dependent Hamiltonian system, and as such they admit a natural generalization to the case of several particles with an interaction of Calogero type (rational, trigonometric or elliptic). Recently, these systems of interacting particles have been proved to be relevant in the study of $\beta$-models. An almost two decade old open question by Takasaki asks whether these multi-particle systems can be understood as isomonodromic equations, thus extending the Painlev\'e correspondence. In this paper we answer in the affirmative by displaying explicitly suitable isomonodromic Lax pair formulations. As an application of the isomonodromic representation we provide a construction based on discrete Schlesinger transforms, to produce solutions for these systems for special values of the coupling constants, starting from uncoupled ones; the method is illustrated for the case of the second Painlev\'e equation.