Transition asymptotics for the Painlevé II transcendent

We consider real-valued solutions u=u(x|s), x∈R, of the second Painlevé equation uxx=xu+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→−∞, between the oscillatory power-like decay asymptotics for |s1|<1 (Ablowitz–Segur) to the power-like growth behavior for |s1|=1 (Hastings–McLeod) and from the latter to the singular oscillatory power-like growth for |s1|>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−γKAi)|L2(x,∞) in a double scaling limit x→−∞, γ↑1, as well as asymptotics for the spectrum of KAi.