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Rational Solutions of the Painlevé-III Equation

Research output: Contribution to journalArticle

Thomas Joachim Bothner, Peter D Miller, Yue Sheng

Original languageEnglish
Pages (from-to)626-679
JournalSTUDIES IN APPLIED MATHEMATICS
Volume141
Issue number4
Early online date25 Jun 2018
DOIs
Accepted/In press11 May 2018
E-pub ahead of print25 Jun 2018
PublishedNov 2018

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Abstract

All of the six Painlev´e equations except the first have families of rational
solutions, which are frequently important in applications. The third Painlev´e
equation in generic form depends on two parameters m and n, and it has rational solutions if and only if at least one of the parameters is an integer. We use known algebraic representations of the solutions to study numerically how the distributions of poles and zeros behave as n ∈ Z increases and how the patterns vary with m ∈ C. This study suggests that it is reasonable to consider the rational solutions in the limit of large n ∈ Z with m ∈ C being an auxiliary parameter. To analyze the rational solutions in this limit, algebraic techniques need to be supplemented by analytical ones, and the main new contribution of this paper is to develop a Riemann–Hilbert representation of the rational solutions of Painlevé-III that is amenable to asymptotic analysis. Assuming further that m is a half-integer, we derive from the Riemann–Hilbert representation a finite dimensional Hankel system for the rational solution in which n ∈ Z appears as an explicit parameter.

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