Takasaki, Kanehisa Painlevé-Calogero correspondence revisited. (English) Zbl 1016.34089 J. Math. Phys. 42, No. 3, 1443-1473 (2001). 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. Cited in 1 ReviewCited in 27 Documents MSC: 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies 37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) Keywords:sixth Painlevé equation; Inozemtsev’s extended Calogero systems; Inozemtsev’s classification PDFBibTeX XMLCite \textit{K. Takasaki}, J. Math. Phys. 42, No. 3, 1443--1473 (2001; Zbl 1016.34089) Full Text: DOI arXiv Digital Library of Mathematical Functions: §32.2(iii) Alternative Forms ‣ §32.2 Differential Equations ‣ Properties ‣ Chapter 32 Painlevé Transcendents References: [1] Painlevé, Bull. Soc. Math. Phys. France 28 pp 201– (1900) [2] Painlevé, Acta Math. 21 pp 1– (1902) [3] Gambier, C.R. Acad. Sci. (Paris) 142 pp 266– (1906) [4] Gambier, Acta Math. 33 pp 1– (1910) [5] Fuchs, C. R. Acad. Sci. (Paris) 141 pp 555– (1905) [6] Fuchs, Math. Ann. 63 pp 301– (1907) [7] Painlevé, C. R. Acad. Sci. (Paris) 143 pp 1111– (1906) [8] Okamoto, Ann. Mat. Pura Appl. 146 pp 337– (1987) [9] Manin, Am. Math. Soc. Trans. 186 pp 131– (1998) · doi:10.1090/trans2/186/04 [10] Levin [11] Olshanetsky [12] Olshanetsky, Phys. Rep. 71 pp 313– (1981) [13] Calogero, J. Math. Phys. 12 pp 419– (1971) [14] Calogero, Lett. Nuovo Cimento 13 pp 411– (1975) [15] Inozemtsev, Lett. Math. Phys. 9 pp 13– (1985) [16] Inozemtsev, Lett. Math. Phys. 17 pp 11– (1989) [17] Olshanetsky [18] Okamoto, J. Fac. Sci., Univ. Tokyo, Sect. 1 33 pp 575– (1986) [19] Levi, Phys. Lett. A 103 pp 11– (1984) [20] K. Iwasaki, H. Kimura, S. Shimomura, and M. Yoshida,From Gauss to Painlevé(Vieweg, Braunschweig, 1991). [21] van Diejen, J. Math. Phys. 36 pp 1299– (1995) [22] Malmquist, Ark. Mat., Astron. Fys. 17 pp 1– (1922) [23] Takasaki, J. Math. Phys. 40 pp 5787– (1999) [24] Caseiro This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.