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On the transformation group of the second Painlevé equation. (English) Zbl 0972.34073
For the second Painlevé equation \[ y''=2y^3+ty +\alpha, \quad \alpha\in\mathbb C, \] the Bäcklund transformation group is isomorphic to the extended affine Weyl group of type \(\widetilde{A}_1,\) namely the group \(G\) generated by the translations \(t_+(\alpha)=\alpha+1,\) \(t_-(\alpha)=\alpha-1\) and the reflection \(i(\alpha)=-\alpha.\) For the affine spaces \(\mathbb A^4\) and \(\mathbb A^2\) with coordinate systems \((y,y',t,\alpha)\) and \((t,\alpha),\) respectively, consider the vector field \[ \delta(\alpha)=\frac{\partial}{\partial t}+y' \frac{\partial}{\partial y}+(2y^3+ty+\alpha) \frac{\partial}{\partial y'} \] on \(\mathbb A^4\) and a natural fibration \(\pi : \mathbb A^4 \to \mathbb A^2\) by \((y,y',t,\alpha)\mapsto (t,\alpha).\)
Here, the author constructs a projective model of the fibration \(\pi : \mathbb A^4 \to \mathbb A^2\) on which the group \(G\) operates regularly.

34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
Full Text: DOI
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[3] J. Math. Soc. Japan
[4] DOI: 10.1007/BF01458459 · Zbl 0589.58008
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[7] Nagoya Math. J 148 pp 151– (1997) · Zbl 0934.33029
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