Year: 2026
Type: preprint
Abstract: We construct local families of solutions of the $q$-difference second Painlevé equation that vanish at infinity with a common algebraic expansion, parameterised by the amplitude of an exponentially small correction. Considering an appropriate boundary region, a formal expansion that retains two independently excited exponential perturbations is developed, and subsequent transseries summation predicts the location of simple poles. Adopting a $q$-theta function realisation of the exponential factors, we prove an asymptotic pole-location formula for a one-parameter family of true solutions. These poles approach the aforementioned boundaries where exponential behaviour fundamentally changes, providing an analogue of pole arrays near anti-Stokes directions in classical Painlevé asymptotics.
Source: arXiv (Cornell University)
Author: Holroyd, Joshua
Language: English
Topic: Nonlinear Waves and Solitons
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223152652