Transition asymptotics for the Painlevé II transcendent

We consider real-valued solutions u D u.xjs/, x 2 R, of the second Painlevé equation uxx D xu C 2u3 which are parameterized in terms of the monodromy data s .s1; s2; s3/ C3 of the associated Flaschka–Newell system of rational differential equations. Our analysis describes the transition, as x ! 1, between the oscillatory power-like decay asymptotics for js1j < 1 (Ablowitz–Segur) to the power-like growth behavior for js1j D 1 (Hastings–McLeod) and from the latter to the singular oscillatory power-like growth for js1j > 1 (Kapaev). It is shown that the transition asymptotics are of Boutroux type; that is, they are expressed in terms of Jacobi elliptic functions. As applications of our results we obtain asymptotics for the Airy kernel determinant det.I KAi/jL2.x;1/ in a double scaling limit x ! 1, " 1, as well as asymptotics for the spectrum of KAi.