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Zbl 0639.58013
Okamoto, Kazuo
Studies on the Painlevé equations. IV: Third Painlevé equation $P\sb{III}$. (English)
[J] Funkc. Ekvacioj, Ser. Int. 30, 305-332 (1987). ISSN 0532-8721

Summary: [For part III see Math. Ann. 275, 221-255 (1986; Zbl 0589.58008).] \par The present article deals with the third Painlevé equation $P\sb{III}$; we consider instead the equation $P\sb{III'}$, equivalent to the former. By defining the Painlevé system ${\cal H}$, we consider the group $G\sb*$ of birational canonical transformations of ${\cal H}$; $G\sb*$ is isomorphic to the affine Weyl group of the root system of the type $B\sb 2$. A sequence of solutions of ${\cal H}$ is obtained from that of $\tau$-functions, satisfying the Toda equation and vice versa. We consider also particular solutions of ${\cal H}$ written in terms of the cylinder function.

MSC 2000:: *37J99 Finite-dimensional Hamiltonian etc. systems
14E05 Birational correspondences

Keywords: third Painlevé equation; birational canonical transformations; affine Weyl group

Citations: Zbl 0601.58032; Zbl 0589.58008

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