Research output: Contribution to journal › Article

Thomas Bothner, Peter D. Miller

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

Pages (from-to) | 123-224 |

Number of pages | 102 |

Journal | CONSTRUCTIVE APPROXIMATION |

Volume | 51 |

Issue number | 1 |

Early online date | 29 May 2019 |

DOIs | |

Accepted/In press | 30 Jan 2019 |

E-pub ahead of print | 29 May 2019 |

Published | 1 Feb 2020 |

Additional links |

**Rational Solutions_Accepted 30 Jan 19_GOLD_VoR**Rational_Solutions_Accepted_30_Jan_19_GOLD_VoR.pdf, 3.5 MB, application/pdf

Uploaded date:04 Jun 2019

Version:Final published version

Licence:CC BY

The Painlevé-III equation with parameters Θ = n+ m and Θ
_{∞}= m- n+ 1 has a unique rational solution u(x) = u
_{n}(x; m) with u
_{n}(∞; m) = 1 whenever n∈ Z. Using a Riemann–Hilbert representation proposed in Bothner et al. (Stud Appl Math 141:626–679, 2018), we study the asymptotic behavior of u
_{n}(x; m) in the limit n→ + ∞ with m∈ C held fixed. We isolate an eye-shaped domain E in the y= n
^{- 1}x plane that asymptotically confines the poles and zeros of u
_{n}(x; m) for all values of the second parameter m. We then show that unless m is a half-integer, the interior of E is filled with a locally uniform lattice of poles and zeros, and the density of the poles and zeros is small near the boundary of E but blows up near the origin, which is the only fixed singularity of the Painlevé-III equation. In both the interior and exterior domains we provide accurate asymptotic formulæ for u
_{n}(x; m) that we compare with u
_{n}(x; m) itself for finite values of n to illustrate their accuracy. We also consider the exceptional cases where m is a half-integer, showing that the poles and zeros of u
_{n}(x; m) now accumulate along only one or the other of two “eyebrows,” i.e., exterior boundary arcs of E.

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