Joshua Holroyd | Searcharxiv

arXiv2025

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

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.

math.CA↗

arXiv2022

On the Perturbed Second Painlevé Equation

We consider a perturbed version of the second Painlevé equation ($\textrm{P}_{\textrm{II}}$), which arises in applications, and show that it possesses solutions analogous to the celebrated Hastings-McLeod and tritronquée solutions of $\textrm{P}_{\textrm{II}}$. The Hastings-McLeod-type solution of the perturbed equation is holomorphic, real-valued and positive on the whole real-line, while the tritronquée-type solution is holomorphic in a large sector of the complex plane. These properties also characterise the corresponding solutions of $\textrm{P}_{\textrm{II}}$ and are surprising because the perturbed equation does not possess additional distinctive properties that characterise $\textrm{P}_{\textrm{II}}$, particularly the Painlevé property.

math-ph↗