Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0589.58008
Okamoto, Kazuo
Studies on the Painlevé equations. III: Second and fourth Painlevé equations, $P\sb{II}$ and $P\sb{IV}$. (English)
[J] Math. Ann. 275, 221-255 (1986). ISSN 0025-5831; ISSN 1432-1807/e

[Part II is to appear in Jap. J. Math., Part I in Ann. Mat. Pura Appl., IV. Ser.] \par In this paper, which is the third part of the series of papers "Studies on the Painlevé equations", we study the second and the fourth Painlevé equations by means of the method of birational canonical transformations. We associate with each equation the nonautonomous Hamiltonian system (H), called the Painlevé system. The group of birational canonical transformations of (H) is investigated by the use of the notion of the affine Weyl group; we attach the root system (R) to each (H). We consider also the families of particular solutions of the Painlevé systems, written in terms of the Airy functions or the Hermite functions. In particular, the rational solutions of (H) are studied in detail. The $\tau$-functions related to (H) is the other main object of this article; it is shown that the sequence $\{\tau\sb n$; $n\in {\bbfZ}\}$ of $\tau$-functions of (H) satisfies the Toda equation: $\delta\sp 2\log \tau\sb n=\tau\sb{n-1}\tau\sb{n+1}/\tau\sp 2\sb n$.

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

Keywords: Painlevé equations; birational canonical transformations; Hamiltonian system

Cited in: Zbl 1042.82019 Zbl 1141.34357 Zbl 0881.34052 Zbl 0947.34077 Zbl 0639.58013

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]