Full Asymptotic Expansion of Monodromy Data for the First Painlevé Transcendent: Applications to Connection Problems
Wen-Gao Long, Yun-Jiang Jiang, Yu-Tian Li
TL;DR
The paper addresses the connection problem for the first Painlevé transcendent by deriving a full asymptotic expansion of the monodromy data as initial data or pole parameters become large. It advances the complex WKB (uniform asymptotics) approach by incorporating higher-order corrections to the Lax-pair ODEs, enabling precise expansion terms for the Stokes multipliers. These expansions are then applied to rigorously prove full asymptotics for nonlinear eigenvalues $a_n$, $b_n$ and pole parameters $(p_n,H_n)$, including the real tritronquée solution, thereby refining existing numerical conjectures. The results yield refined connection formulas and deepen understanding of the isomonodromic structure underlying Painlevé I, with explicit asymptotics expressed through Airy-type functions and higher-order correction terms.
Abstract
We study the full asymptotic expansion of the monodromy data ({\it i.e.}, Stokes multipliers) for the first Painlevé 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 tritronquée solution of the PI equation as $n \to +\infty$.