# Noncommutative Painlev\'e equations and systems of Calogero type

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.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-01609147
Contributor : Mattia Cafasso <>
Submitted on : Tuesday, October 3, 2017 - 11:46:30 AM
Last modification on : Wednesday, October 14, 2020 - 4:18:24 AM

### Identifiers

• HAL Id : hal-01609147, version 1
• ARXIV : 1710.00736

### Citation

Mattia Cafasso, Marco Bertola, Vladimir Roubtsov. Noncommutative Painlev\'e equations and systems of Calogero type. 2017. ⟨hal-01609147⟩

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