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Rational Solutions of the Painlevé‐III Equation
Author(s) -
Bothner Thomas,
Miller Peter D.,
Sheng Yue
Publication year - 2018
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12220
Subject(s) - mathematics , algebraic number , rational function , limit (mathematics) , representation (politics) , integer (computer science) , riemann hypothesis , elliptic rational functions , rational number , pure mathematics , mathematical analysis , elliptic curve , politics , political science , computer science , law , quarter period , programming language
All of the six Painlevé equations except the first have families of rational solutions, which are frequently important in applications. The third Painlevé 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.