Stokes Phenomena in Discrete Painlev\'e I

TL;DR

This paper analyzes the asymptotic behavior of solutions to the discrete Painlevé I equation, identifying Stokes phenomena effects and solution regions, revealing a unique and a parametric family of pole-free solutions.

Contribution

It applies exponential asymptotics to discrete Painlevé I, characterizing Stokes phenomena and solution regions, and distinguishes between unique and parametric solution types.

Findings

Identifies two pole-free solution types with different parameterizations.

Determines Stokes lines and their effects on solution validity.

Extends the solution region through parameter choice.

Abstract

In this study, we consider the asymptotic behaviour of the first discrete Painlev\{e} equation in the limit as the independent variable becomes large. Using an asymptotic series expansion, we identify two types of solutions which are pole-free within some sector of the complex plane containing the positive real axis. Using exponential asymptotic techniques, we determine the Stokes Phenomena effects present within these solutions, and hence the regions in which the asymptotic series expression is valid. From a careful analysis of the switching behaviour across Stokes lines, we find that the first type of solution is uniquely defined, while the second type contains two free parameters, and that the region of validity may be extended for appropriate choice of these parameters.