zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Studies on the Painlevé equations. IV: Third Painlevé equation ${P}_{III}$. (English) Zbl 0639.58013

Summary: [For part III see Math. Ann. 275, 221-255 (1986; Zbl 0589.58008).]

The present article deals with the third Painlevé equation ${P}_{III}$; we consider instead the equation ${P}_{II{I}^{\text{'}}}$, equivalent to the former. By defining the Painlevé system $ℋ$, we consider the group ${G}_{*}$ of birational canonical transformations of $ℋ$; ${G}_{*}$ is isomorphic to the affine Weyl group of the root system of the type ${B}_{2}$. A sequence of solutions of $ℋ$ is obtained from that of $\tau$-functions, satisfying the Toda equation and vice versa. We consider also particular solutions of $ℋ$ written in terms of the cylinder function.

MSC:
 37J99 Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems 14E05 Rational and birational maps