Research output: Contribution to journal › Article

Thomas Joachim Bothner, Peter D Miller, Yue Sheng

Original language | English |
---|---|

Pages (from-to) | 626-679 |

Journal | STUDIES IN APPLIED MATHEMATICS |

Volume | 141 |

Issue number | 4 |

Early online date | 25 Jun 2018 |

DOIs | |

Accepted/In press | 11 May 2018 |

E-pub ahead of print | 25 Jun 2018 |

Published | Nov 2018 |

**Rational Solutions_BOTHNER_aCCEPTED 11 May 18_GREEN_AAM**Rational_Solutions_BOTHNER_aCCEPTED_11_May_18_GREEN_AAM.pdf, 2 MB, application/pdf

Uploaded date:23 Nov 2018

Version:Accepted author manuscript

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.

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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