Communications in Mathematical Physics

, Volume 363, Issue 2, pp 503–530 | Cite as

Noncommutative Painlevé Equations and Systems of Calogero Type

  • M. Bertola
  • M. Cafasso
  • V. Rubtsov


All Painlevé 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 Painlevé 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 Painlevé equation.


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We thank P. Boalch for pointing out the relations between this work and his article [5], and G. Rembado for some interesting discussions on the deformation quantization of simply laced isomonodromy systems. The three authors acknowledge the support of the project IPaDEGAN (H2020-MSCA-RISE-2017), Grant Number 778010. The research of M.B. was supported in part by the Natural Sciences and Engineering Research Council of Canada Grant RGPIN-2016-06660 and by the FQRNT grant “Applications des systèmes intégrables aux surfaces de Riemann et aux espaces de modules”. The research of M.C. and V.R. was partially supported by a project “Nouvelle équipe” funded by the region Pays de la Loire. V. R. acknowledges the support of the Russian Foundation for Basic Research under the Grants RFBR 18-01-00461 and 16-51-53034-GFEN. M.C. and V.R. thank the Centre “Henri Lebesgue” ANR-11-LABX-0020-01 for creating an attractive mathematical environment. They also thank the International School of Advanced Studies (SISSA) in Trieste for the hospitality during part of the preparation of this work.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsConcordia UniversityMontrealCanada
  2. 2.SISSA/ISASTriesteItaly
  3. 3.Centre de recherches mathématiquesUniversité de MontréalMontrealCanada
  4. 4.LAREMAUniversité d’AngersAngersFrance
  5. 5.ITEP, Theory DivisionMoscowRussia

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