Transition asymptotics for the Painlevé II transcendent

Abstract

We consider real-valued solutions $u=u(x|s),x\in \mathbb{R}$ of the second Painlevé 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 $x\to -\infty$, 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 $\det (I-\gamma K_{\text{Ai}}){|}_{L^2(x,\infty )}$ in a double scaling limit $x\to -\infty ,\gamma \uparrow 1$ as well as asymptotics for the spectrum of $K_{\text{Ai}}$.

71 pages, 26 figures. Version 2 corrects typos

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• Open Problems for Painlevé Equations
• Connection formulas for the Ablowitz-Segur solutions of the inhomogeneous Painlevé II equation
• On the Increasing Tritronquée Solutions of the Painlevé-II Equation
• Fredholm determinant representation of the Painlevé II $\tau$-function
• The $\tau$-function of the Ablowitz-Segur family of solutions to Painlevé II as a Widom constant
• Resurgent Trans-Series for Generalized Hastings-McLeod Solutions
• Asymptotics of the determinant of the modified Bessel functions and the second Painlevé equation
• Global rigidity and exponential moments for soft and hard edge point processes
• The Tracy-Widom distribution at large Dyson index
• Riemann-Hilbert Characterisation of Rational Functions with a General Distribution of Poles on the Extended Real Line Orthogonal with Respect to Varying Exponential Weights: Multi-Point Padé Approximants and Asymptotics