zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
The Painlevé connection problem: An asymptotic approach. I. (English) Zbl 0761.34003

The author clearly and completely develops a rich mathematical structure surrounding the Painlevé equations which is closely related to the well known developed direct methods for solving nonlinear partial differential equations by the implementation of linear integral equations. The integral equation has a nonsingular kernel and the solutions are obtained via a linear integral equation of the form

𝒦(x,y;t)=(x+y;t)+ x 𝒦(x,z;t)𝒩(x;z,y;t)dx,

where satisfies appropriate conditions and then the function μ=d dx𝒦(x,x;t) satisfies the original nonlinear PDE. This paper has the similar back drop in place but adds to this remarkable result. It studies in detail a solution to the Painlevé connection problem by investigating the asymptotic behavior of a solution near a fixed singular point to its behavior near some other (or the same) such point. This additional connection can be solved directly for the first two Painlevé equations through a natural asymptotic method. The construction and significant details are all included in the paper. In particular, a uniformly valid description of the general asymptotic behavior given to leading order by elliptic functions is carefully derived by a generalization of the so called multiple-scales method. The paper carefully develops the results for the first two Painlevé equations P I and P II having the form fg ' 2 u zz +(2f ' g ' +fg '' )u z +f '' u=3 2(f 2 u 2 -x) and fg ' 2 u zz +(2f ' g ' +fg '' )u z +f '' u=2(f 3 u 2 -xfu)+a 0 . Asymptotic behaviors are then one of the main focuses of the paper. It concludes with investigating P 1 near the imaginary z-axis and differential equations satisfied by complete elliptic integrals.

34M55Painlevé and other special equations; classification, hierarchies