We consider real-valued solutions , , of the second Painlevé equation which are parameterized in terms of the monodromy data of the associated Flaschka–Newell system of rational differential equations. Our analysis describes the transition, as , between the oscillatory power-like decay asymptotics for (Ablowitz–Segur) to the power-like growth behavior for (Hastings–McLeod) and from the latter to the singular oscillatory power-like growth for (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 in a double scaling limit , , as well as asymptotics for the spectrum of .
Duke Math. J.
166(2):
205-324
(1 February 2017).
DOI: 10.1215/00127094-3714650
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