Nonlinear $q$-Stokes phenomena for $q$-Painlev\'{e} I

TL;DR

This paper investigates the complex asymptotic behaviour and Stokes phenomena of solutions to the $q$-Painlevé I equation as parameters approach critical limits, revealing hidden solution structures and Stokes curves.

Contribution

It introduces exponential asymptotic techniques to analyze nonlinear $q$-difference equations and describes the geometry of Stokes curves as $q$-spirals.

Findings

Identification of four families of asymptotic solutions with free parameters

Description of Stokes phenomena and rapid switching behaviour

Determination of regions where power series describe the solutions

Abstract

We consider the asymptotic behaviour of solutions of the first $q$-difference Painlev\'{e} equation in the limits $|q|\rightarrow 1$ and $n\rightarrow\infty$. Using asymptotic power series, we describe four families of solutions that contain free parameters hidden beyond-all-orders. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. In order to investigate such phenomena we apply exponential asymptotic techniques to obtain mathematical descriptions of the rapid switching behaviour associated with Stokes curves. Through this analysis, we also determine the regions of the complex plane in which the asymptotic behaviour is described by a power series expression, and find that the Stokes curves are described by curves known as $q$-spirals.