Elliptic asymptotic behaviour of $q$-Painlev\'e transcendents

TL;DR

This paper explores how elliptic functions emerge as the asymptotic limit of $q$-Painlevé transcendents, revealing their behavior as $q$ approaches 1, and connecting discrete and classical Painlevé equations.

Contribution

It provides a detailed asymptotic analysis of the $q$-difference second Painlevé equation, demonstrating elliptic functions as the leading behavior in the limit $|q-1| o 0$, which was previously not established.

Findings

Elliptic functions describe the leading-order asymptotic behavior of $q$-Painlevé transcendents.

Slow modulation of solutions is approximated by complete elliptic integrals.

The analysis bridges discrete $q$-Painlevé equations with classical Painlevé asymptotics.

Abstract

The discrete Painlev\'e equations have mathematical properties closely related to those of the differential Painlev\'e equations. We investigate the appearance of elliptic functions as limiting behaviours of $q$-Painlev\'e transcendents, analogous to the asymptotic theory of classical Painlev\'e transcendents. We focus on the $q$-difference second Painlev\'e equation in the asymptotic regime $|q-1|\ll1$, showing that generic leading-order behaviour is given in terms of elliptic functions and that the slow modulation in this behaviour is approximated in terms of complete elliptic integrals.