Full Asymptotic Expansion of Monodromy Data for the First Painlev\'{e} Transcendent: Applications to Connection Problems

TL;DR

This paper develops a refined complex WKB method to derive the full asymptotic expansion of monodromy data for the first Painlevé transcendent, providing rigorous proofs and detailed asymptotics for eigenvalues and pole parameters.

Contribution

It introduces a higher-accuracy complex WKB approach to asymptotically analyze monodromy data and pole parameters of the first Painlevé transcendent, with rigorous proofs and applications.

Findings

Rigorous proof of asymptotic expansion of nonlinear eigenvalues.

Full asymptotic expansion for pole parameters of the real tritronquée.

Refined complex WKB method with higher-order accuracy.

Abstract

We study the full asymptotic expansion of the monodromy data ({\it i.e.}, Stokes multipliers) for the first Painlev\'{e} transcendent (PI) with large initial data or large pole parameters. Our primary approach involves refining the complex WKB method, also known as the method of uniform asymptotics, to approximate the second-order ODEs derived from PI's Lax pair with higher-order accuracy. As an application, we provide a rigorous proof of the full asymptotic expansion of the nonlinear eigenvalues proposed numerically by Bender, Komijani, and Wang. Additionally, we present the full asymptotic expansion for the pole parameters $(p_{n}, H_{n})$ corresponding to the $n$-th pole of the real tritronqu\'{e}e solution of the PI equation as $n \to +\infty$.