Painlev\'e I and exact WKB: Stokes phenomenon for two-parameter transseries

TL;DR

This paper develops complete two-parameter connection formulae for Painlevé I solutions crossing Stokes lines, combining isomonodromic deformation and transseries methods to analyze the Stokes phenomenon in detail.

Contribution

It introduces new two-parameter connection formulae for Painlevé I solutions, integrating WKB and transseries approaches to advance understanding of Stokes phenomena.

Findings

Derived explicit connection formulae for two-parameter solutions

Recovered and compared Stokes data with recent results

Performed high-precision numerical validation

Abstract

For more than a century, the Painlev\'e I equation has played an important role in both physics and mathematics. Its two-parameter family of solutions was studied in many different ways, yet still leads to new surprises and discoveries. Two popular tools in these studies are the theory of isomonodromic deformation that uses the exact WKB method, and the asymptotic description of transcendents in terms of two-parameter transseries. Combining methods from both schools of thought, and following work by Takei and collaborators, we find complete, two-parameter connection formulae for solutions when they cross arbitrary Stokes lines in the complex plane. These formulae allow us to study Stokes phenomenon for the full two-parameter family of transseries solutions. In particular, we recover the exact expressions for the Stokes data that were recently found by Baldino, Schwick, Schiappa and Vega and compare our connection formulae to theirs. We also explain several ambiguities in relating transseries parameter choices to actual Painlev\'e transcendents, study the monodromy of formal solutions, and provide high-precision numerical tests of our results.