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On the transformation group of the second Painlevé equation

Published online by Cambridge University Press:  22 January 2016

Hiroshi Umemura
Hiroshi Umemura*
Affiliation:
Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan, umemura@math.nagoya-u.ac.jp
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Abstract

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We show that for the second Painlevé equation y″ = 2y3 + ty + α, the Bäcklund transformation group G, which is isomorphic to the extended affine Weyl group of type Â1, operates regularly on the natural projectification χ(c)/ℂ(c, t) of the space of initial conditions, where c = α - 1/2. χ(c)/ℂ(c, t) has a natural model χ[c]/ℂ(t)[c]. The group G does not operate, however, regularly on χ[c]/ℂ(t)[c]. To have a family of projective surfaces over ℂ(t)[c] on which G operates regularly, we have to blow up the model χ[c] along the projective lines corresponding to the Riccati type solutions.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 157 , 2000 , pp. 15 - 46
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2000

References

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