Painleve-Calogero Correspondence Revisited

Abstract

We extend the work of Fuchs, Painlevé and Manin on a Calogero-like expression of the sixth Painlevé equation (the ``Painlevé-Calogero correspondence'') to the other five Painlevé equations. The Calogero side of the sixth Painlevé equation is known to be a non-autonomous version of the (rank one) elliptic model of Inozemtsev's extended Calogero systems. The fifth and fourth Painlevé equations correspond to the hyperbolic and rational models in Inozemtsev's classification. Those corresponding to the third, second and first are seemingly new. We further extend the correspondence to the higher rank models, and obtain a ``multi-component'' version of the Painlevé equations.

latex2e using amsmath and amssymb packages, 40 pages, no figure; (v2) bibliographic comments on degenerate cases are added; (v3) typos corrected

References in corpus (3)

• Elliptic Calogero-Moser Systems and Isomonodromic Deformations
• Algebraic Linearization of Dynamics of Calogero Type for any Coxeter Group
• Painlevé type equations and Hitchin systems

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