On Airy Solutions of the Second Painleve Equation

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.