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Zbl 1010.34083
Sakai, Hidetaka
Rational surfaces associated with affine root systems and geometry of the Painlevé equations. (English)
[J] Commun. Math. Phys. 220, No.1, 165-229 (2001). ISSN 0010-3616; ISSN 1432-0916/e

Here, the author provides an understanding of (discrete and continuous) Painlevé equations from a purely geometric point of view. The paper is motivated by previous work of Okamoto (and later Takano), but turns this work on its head by beginning with geometric structures and then deriving equations, along with many of their important properties. Thus, in addition to Painlevé equations, this geometric framework is also shown to yield Bäcklund transformations, special integrals and elliptic function limits. The author also gives a generic discrete equation which contains all other discrete equations which appear in the paper.
[A.Pickering (Salamanca)]

MSC 2000:: *34M55 Painlevé and other special equations
37K35 Lie-Bäcklund and other transformations
37K10 Completely integrable systems etc.

Keywords: Painlevé equations; Bäcklund transformations

Cited in: Zbl 1210.37039 Zbl 1119.39018 Zbl 1109.39019 Zbl 1073.34101 Zbl 1076.33015 Zbl 1074.14030 Zbl 1151.34341 Zbl 1044.39014 Zbl 1028.39006 Zbl 1022.34079

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