Large $z$ Asymptotics for Special Function Solutions of Painlev\'e II in the Complex Plane

TL;DR

This paper derives large complex plane asymptotics for Painlevé II tau functions linked to special solutions, employing Wronskian determinants and steepest descent methods for precise asymptotic analysis.

Contribution

It introduces a novel approach to asymptotic analysis of Painlevé II solutions by expressing tau functions as integrals and applying steepest descent in the complex plane.

Findings

Explicit large $z$ asymptotic expansions obtained.

Representation of tau functions as $n$-fold integrals using Airy functions.

Application of steepest descent method for asymptotic approximation.

Abstract

In this paper we obtain large $z$ asymptotic expansions in the complex plane for the tau function corresponding to special function solutions of the Painlev\'e II differential equation. Using the fact that these tau functions can be written as $n\times n$ Wronskian determinants involving classical Airy functions, we use Heine's formula to rewrite them as $n$-fold integrals, which can be asymptotically approximated using the classical method of steepest descent in the complex plane.