A connection formula for the second Painlevé transcendent
type: Item
instance of: scholarly article
title: A connection formula for the second Painlevé transcendent
zbMATH Open document ID: 0653.34020
DOI: 10.1007/BF00251504
author: Peter A. Clarkson
author: J. B. McLeod
published in: Archive for Rational Mechanics and Analysis
publication date: 1988-01-01T00:00:00Z
review text: We consider the second Painlevé transcendent \(d^ 2y/dx^ 2=xy+2y^ 3.\) It is known that if y(x)\(\sim k Ai(x)\) as \(x\to +\infty\), where \(- 1<k<1\) and Ai(x) denotes Airy's function, then \[ y(x)\quad \sim \quad d| x|^{-1/4}\sin \{\frac{2}{3}| x|^{3/2}- \frac{3}{2}d^ 2\ell n| x| -c\}, \] as \(x\to -\infty\), where the constants d, c depend on k. This paper shows that \(d^ 2(k)=-\pi^{- 1}\ell n(1-k^ 2),\) which confirms a conjecture by Ablowitz and Segur.
Mathematics Subject Classification ID: 34B30
Mathematics Subject Classification ID: 35Q99
zbMATH DE Number: 4065451
zbMATH Keywords: second Painlevé transcendent
zbMATH Keywords: Airy's function
MaRDI profile type: Publication
cites work: Q5843703
cites work: Q4112233
cites work: A boundary value problem associated with the second Painleve transcendent and the Korteweg-de Vries equation
cites work: Asymptotic Solutions of the Korteweg-deVries Equation
cites work: Asymptotic solutions of nonlinear evolution equations and a Painlevé transcedent
full work available at URL: https://doi.org/10.1007/bf00251504
OpenAlex ID: W4231007983
description: scientific article; zbMATH DE number 4065451