Parametric Stokes phenomenon for the second Painlev\'e equation with a large parameter

TL;DR

This paper investigates the parametric Stokes phenomenon in the second Painlevé equation with a large parameter using exact WKB analysis, deriving connection formulas confirmed by Voros coefficients and isomonodromic deformation theory.

Contribution

It formulates and confirms the connection formulas for the parametric Stokes phenomenon in ($P_{II}$) through two distinct methods, unifying different analytical approaches.

Findings

Connection formulas for the parametric Stokes phenomenon are derived.

The formulas from WKB analysis and isomonodromic deformation theory coincide.

The study advances understanding of Stokes phenomena in Painlevé equations.

Abstract

The second Painlev\'e equation with a large parameter ($P_{II}$) is analyzed by using the exact WKB analysis. The purpose of this study is to investigate the problem of the degeneration of $P$-Stokes geometry of ($P_{II}$), which relates to a kind of Stokes phenomena for asymptotic (formal) solutions of ($P_{II}$). We call this Stokes phenomenon a "parametric Stokes phenomenon". We formulate the connection formula for this Stokes phenomenon, and confirm it in two ways: the first one is by computing the "Voros coefficient" of ($P_{II}$), and the second one is by using the isomonodromic deformation theory. Our main claim is that the connection formulas derived by these two completely different methods coincide.