On Airy Solutions of the Second Painleve Equation

In this paper, we discuss Airy solutions of the second Painleve equation (P-II) and two related equations, the Painleve XXXIV equation (P34) and the Jimbo-Miwa-Okamoto sigma form of P-II (S-II), are discussed. It is shown that solutions that depend only on the Airy function Ai (z) have a completely different structure to those that involve a linear combination of the Airy functions Ai (z) and Bi (z). For all three equations, the special solutions that depend only on Ai (t) are tronquee solutions, i.e., they have no poles in a sector of the complex plane. Further, for both P34 and S-II, it is shown that among these tronquee solutions there is a family of solutions that have no poles on the real axis.