Transition asymptotics for the Painlev\'e II transcendent

TL;DR

This paper analyzes the asymptotic behavior of solutions to the Painlevé II equation, revealing transition phenomena between different asymptotic regimes and expressing these transitions using elliptic functions, with applications to Airy kernel determinants.

Contribution

It provides a detailed description of transition asymptotics for Painlevé II solutions in terms of elliptic functions, connecting different known asymptotic regimes and applying results to kernel determinants.

Findings

Transition asymptotics are of Boutroux type and involve Jacobi elliptic functions.

Describes the transition from oscillatory decay to growth in solutions as parameters vary.

Provides asymptotics for Airy kernel determinants and their spectra in a double scaling limit.

Abstract

We consider real-valued solutions $u=u(x|s),x\in\mathbb{R}$ of the second Painlev\'e equation $u_{xx}=xu+2u^3$ which are parametrized in terms of the monodromy data $s\equiv(s_1,s_2,s_3)\subset\mathbb{C}^3$ of the associated Flaschka-Newell system of rational differential equations. Our analysis describes the transition, as $x\rightarrow-\infty$, between the oscillatory power-like decay asymptotics for $|s_1|<1$ (Ablowitz-Segur) to the power-like growth behavior for $|s_1|=1$ (Hastings-McLeod) and from the latter to the singular oscillatory power-like growth for $|s_1|>1$ (Kapaev). It is shown that the transition asymptotics are of Boutroux type, i.e. they are expressed in terms of Jacobi elliptic functions. As applications of our results we obtain asymptotics for the Airy kernel determinant $\det(I-\gamma K_{\textnormal{Ai}})|_{L^2(x,\infty)}$ in a double scaling limit $x\rightarrow-\infty,\gamma\uparrow 1$ as well as asymptotics for the spectrum of $K_{\textnormal{Ai}}$.