Series expansions of painlevé transcendents near the point at infinity

Shun Shimomura

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

For the Painlevé equations (I) through (V) near the point at infinity we present several families of two-parameter solutions. Our solutions are expressed by asymptotic series with coefficients polynomial in exponential terms, and also by convergent power series in exponential terms with coefficients expanded into asymptotic series. Both expressions are valid without a restriction on integration constants. We propose a direct method to derive asymptotic solutions, which is also applicable to more general nonlinear equations. As applications of our results, for general solutions of the third and the fifth Painlevé equations, we give estimates for the number of a-points including poles in given sectors.

Original languageEnglish
Pages (from-to)277-319
Number of pages43
JournalFunkcialaj Ekvacioj
Volume58
Issue number2
DOIs
Publication statusPublished - 2015 Aug 13

Fingerprint

Asymptotic series
Series Expansion
Infinity
Asymptotic Solution
Coefficient
Term
Direct Method
Power series
General Solution
Pole
Two Parameters
Nonlinear Equations
Sector
Valid
Restriction
Polynomial
Estimate
Family

Keywords

  • Asymptotic solutions
  • Hamiltonian system
  • Painlevé equations

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Analysis
  • Geometry and Topology

Cite this

Series expansions of painlevé transcendents near the point at infinity. / Shimomura, Shun.

In: Funkcialaj Ekvacioj, Vol. 58, No. 2, 13.08.2015, p. 277-319.

Research output: Contribution to journalArticle

Shimomura, Shun. / Series expansions of painlevé transcendents near the point at infinity. In: Funkcialaj Ekvacioj. 2015 ; Vol. 58, No. 2. pp. 277-319.
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