On Airy Solutions of the Second Painlev\'e Equation

TL;DR

This paper explores Airy function solutions of the second Painlevé equation and related equations, highlighting differences between solutions involving only Ai(z) and those involving both Ai(z) and Bi(z), including pole-free tronquée solutions.

Contribution

It characterizes Airy solutions of PII, P34, and SII, revealing their structural differences and identifying pole-free tronquée solutions on the real axis.

Findings

Solutions depending solely on Ai(z) are structurally different from those involving Ai(z) and Bi(z).

Existence of pole-free tronquée solutions in specific sectors of the complex plane.

Identification of solutions with no poles on the real axis among tronquée solutions.

Abstract

In this paper we discuss Airy solutions of the second Painlev\'e equation (\mbox{\rm P$_{\rm II}$}) and two related equations, the Painlev\'e XXXIV equation ($\mbox{\rm P}_{34}$) and the Jimbo-Miwa-Okamoto $\sigma$ form of \mbox{\rm P$_{\rm II}$}\ (\mbox{\rm S$_{\rm II}$}), are discussed. It is shown that solutions which depend only on the Airy function $\mathop{\rm Ai}\nolimits(z)$ have a completely difference structure to those which involve a linear combination of the Airy functions $\mathop{\rm Ai}\nolimits(z)$ and $\mathop{\rm Bi}\nolimits(z)$. For all three equations, the special solutions which depend only on $\mathop{\rm Ai}\nolimits(t)$ are \textit{tronqu\'ee} 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$_{\rm II}$}, it is shown that amongst these \textit{tronqu\'ee} solutions there is a family of solutions which have no poles on the real axis.