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2000 On the transformation group of the second Painlevé equation
Hiroshi Umemura
Nagoya Math. J. 157: 15-46 (2000).

Abstract

We show that for the second Painlevé equation $y'' = 2y^{3}+ty+\alpha$, the Bäcklund transformation group $G$, which is isomorphic to the extended affine Weyl group of type $\hat{A}_{1}$, operates regularly on the natural projectification ${\mathcal X}(c)/\mathbb{C}(c, t)$ of the space of initial conditions, where $c = \alpha-1/2$. ${\cal X}(c)/\mathbb{C}(c, t)$ has a natural model ${\mathcal X}[c]/\mathbb{C}(t)[c]$. The group $G$ does not operate, however, regularly on $\mathcal {X}[c]/\mathbb{C}(t)[c]$. To have a family of projective surfaces over $\mathbb{C}(t)[c]$ on which $G$ operates regularly, we have to blow up the model ${\mathcal X}[c]$ along the projective lines corresponding to the Riccati type solutions.

Citation

Download Citation

Hiroshi Umemura. "On the transformation group of the second Painlevé equation." Nagoya Math. J. 157 15 - 46, 2000.

Information

Published: 2000
First available in Project Euclid: 27 April 2005

zbMATH: 0972.34073
MathSciNet: MR1752473

Subjects:
Primary: 34M55
Secondary: 14E15, 34A26, 34C14, 34M15

Rights: Copyright © 2000 Editorial Board, Nagoya Mathematical Journal

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