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Zbl 1016.34089
Takasaki, Kanehisa
Painlevé-Calogero correspondence revisited. (English)
[J] J. Math. Phys. 42, No.3, 1443-1473 (2001). ISSN 0022-2488; ISSN 1089-7658/e

Summary: The author extends the work of Fuchs, Painlevé and Martin 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 nonautonomous 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 not included therein. The author further extends the correspondence to the higher-rank models, and obtains a ``multi-component'' version of the Painlevé equations.

MSC 2000:: *34M55 Painlevé and other special equations
37J15 Symmetries, etc.

Keywords: sixth Painlevé equation; Inozemtsev's extended Calogero systems; Inozemtsev's classification

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