The sixth painlevé transcendents and the associated schlesinger equation

Shun Shimqmura

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5 Citations (Scopus)

Abstract

For the Schlesinger equation associated with the sixth Painlevé equation (PVI) near the critical point, we present families of solutions expanded into convergent series with matrix coefficients. These families yield four basic solutions of (PVI) in Guzzetti's table describing the critical behaviours of the sixth Painleve transcendents; two of the basic solutions are of complex power type, and two are of logarithmic type. Consequently, the convergence of the logarithmic solutions is verified. Furthermore we obtain more information on these basic solutions as well as on inverse logarithmic solutions. For complex power solutions, examining related inverse oscillatory ones, we discuss sequences of zeros and poles, non-decaying exponential oscillation and the analytic continuation around the critical point, and show the spiral distribution of poles conjectured by Guzzetti.

Original languageEnglish
Pages (from-to)417-463
Number of pages47
JournalPublications of the Research Institute for Mathematical Sciences
Volume51
Issue number3
DOIs
Publication statusPublished - 2015 Aug 28

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Keywords

  • Distribution of poles
  • Isomonodromy deformation
  • Painlevé transcendents
  • Schlesinger equation

ASJC Scopus subject areas

  • Mathematics(all)

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