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arXiv · 1510.08326

On Airy Solutions of the Second Painlevé Equation

Peter A Clarkson
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Abstract

In this paper we discuss Airy solutions of the second Painlevé equation (\mbox{\rm P$_{\rm II}$}) and two related equations, the Painlevé XXXIV equation ($\mbox{\rm P}_{34}$) and the Jimbo-Miwa-Okamoto $σ$ 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{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$_{\rm II}$}, it is shown that amongst these \textit{tronquée} solutions there is a family of solutions which have no poles on the real axis.

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Peter A Clarkson. 2016-02-03. On Airy Solutions of the Second Painlevé Equation. https://doi.org/10.1111/sapm.12123

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