On Airy Solutions of the Second Painlevé Equation

Abstract

In this paper we discuss Airy solutions of the second Painlevé equation (\mbox{\rm P${}_{\mathrm{II}}$}) and two related equations, the Painlevé XXXIV equation ($\mbox{\rm P}_{34}$) and the Jimbo-Miwa-Okamoto $\sigma$ form of \mbox{\rm P${}_{\mathrm{II}}$}\ (\mbox{\rm S${}_{\mathrm{II}}$}), are discussed. It is shown that solutions which depend only on the Airy function $\mathrm{Ai}(z)$ have a completely difference structure to those which involve a linear combination of the Airy functions $\mathrm{Ai}(z)$ and $\mathrm{Bi}(z)$. For all three equations, the special solutions which depend only on $\mathrm{Ai}(t)$ are \textit{tronquée} solutions, i.e.\ they have no poles in a sector of the complex plane. Further for both $\mbox{\rm P}_{34}$\ and \mbox{\rm S${}_{\mathrm{II}}$}, it is shown that amongst these \textit{tronquée} solutions there is a family of solutions which have no poles on the real axis.

12 pages, 8 figures

References in corpus (2)

• Unique positive solution for an alternative discrete Painlevé I equation
• Correlations of RMT Characteristic Polynomials and Integrability: Hermitean Matrices

Cited by in corpus (3)

• Classical Solutions of the Degenerate Fifth Painlevé Equation
• Generalised Airy Polynomials
• Airy series solution of Painlevé II in electrodiffusion: conjectured convergence