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On the transformation group of the second Painlevé equation

  • Hiroshi Umemura (a1)

Abstract

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.

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References

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[MMT] Matano, T., Matsumiya, A. and Takano, K., On some Hamiltonian structures on Painlevé systems II, to appear, J. Math. Soc. Japan.
[Mu] Murata, Y., Rational solutions of the second and the fourth Painlevé equations, Funkcial. Ekvac., 28 (1985), 132.
[O1] Okamoto, K., Sur les feuilletages associées aux équation du second ordre à points critiques fixes de P. Painlevé, Jap. J. Math., 5 (1979), 179.
[O2] Okamoto, K., Studies on the Painlevé equations, Math. Ann., 275 (1986), 221255.
[ST] Shioda, T. and Takano, K., On some Hamiltonian structures of Painlevé systems I, Funkcial. Ekvac., 40 (1997), 271291.
[US] Umemura, H. and Saito, M., Painlevé equations and deformations of rational surfaces with rational double points, preprint.
[UW] Umemura, H., and Watanabe, H., Solutions of the second and fourth Painlevé equations, Nagoya Math. J., 148 (1997), 151198.
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On the transformation group of the second Painlevé equation

  • Hiroshi Umemura (a1)

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