Archive for Rational Mechanics and Analysis

, Volume 103, Issue 2, pp 97–138

A connection formula for the second Painlevé transcendent

  • Peter A. Clarkson
  • J. Bryce McLeod
Article

DOI: 10.1007/BF00251504

Cite this article as:
Clarkson, P.A. & McLeod, J.B. Arch. Rational Mech. Anal. (1988) 103: 97. doi:10.1007/BF00251504

Abstract

We consider the second Painlevé transcendent
$$\frac{{d^2 y}}{{dx^2 }} = xy + 2y^3 .$$
It is known that if y(x)k Ai (x) as x → + ∞, where −1<k<1 and Ai (x) denotes Airy's function, then
$$y(x) \sim d|x|^{ - \tfrac{1}{4}} sin\{ \tfrac{2}{3}|x|^{\tfrac{3}{2}} - \tfrac{3}{4}d^2 1n|x| - c\} ,$$
where the constants d, c depend on k. This paper shows that
$$d^2 = \pi ^{ - 1} 1n(1 - k^2 )$$
, which confirms a conjecture by Ablowitz & Segur.

Copyright information

© Springer-Verlag GmbH & Co. KG 1988

Authors and Affiliations

  • Peter A. Clarkson
    • 1
    • 2
  • J. Bryce McLeod
    • 1
    • 2
  1. 1.Department of MathematicsExeter UniversityExeterUK
  2. 2.Mathematical InstituteUniversity of OxfordUK