×

zbMATH — the first resource for mathematics

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.

MSC:
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Jap. J. Math 5 pp 1– (1979)
[2] Funkcial. Ekvac 28 pp 1– (1985)
[3] J. Math. Soc. Japan
[4] DOI: 10.1007/BF01458459 · Zbl 0589.58008
[6] Funkcial. Ekvac 40 pp 271– (1997)
[7] Nagoya Math. J 148 pp 151– (1997) · Zbl 0934.33029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.