# On Airy Solutions of the Second Painleve Equation

Clarkson, Peter (2016) On Airy Solutions of the Second Painleve Equation. Studies in Applied Mathematics, 137 (1). pp. 93-109. ISSN 0022-2526. (doi:10.1111/sapm.12123)

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http://dx.doi.org/10.1111/sapm.12123

## Abstract

In this paper, we discuss Airy solutions of the second Painleve equation and two related equations, the Painleve XXXIV equation and the Jimbo-Miwa-Okamoto sigma form of second Painleve equation, are discussed. It is shown that solutions which depend only on the Airy function Ai(z) have a completely difference structure to those which involve a linear combination of the Airy functions Ai(z) and Bi(z). For all three equations, the special solutions that depend only on inline image are tronquée solutions, i.e., they have no poles in a sector of the complex plane. Further, for both inline image and SII, it is shown that among these tronquée solutions there is a family of solutions that have no poles on the real axis.

Item Type: Article 10.1111/sapm.12123 Q Science > QA Mathematics (inc Computing science) > QA351 Special functionsQ Science > QA Mathematics (inc Computing science) > QA372 Ordinary differential equations Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Applied Mathematics Peter A Clarkson 29 Oct 2015 09:16 UTC 01 Aug 2019 10:39 UTC https://kar.kent.ac.uk/id/eprint/51255 (The current URI for this page, for reference purposes) https://orcid.org/0000-0002-8777-5284