Nonlinear q-Stokes phenomena for q-Painlevé I

Abstract

We consider the asymptotic behaviour of solutions of the first q-difference Painlevé equation in the limits |q| → 1 and n → ∞. 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.

First-order general differential equation for multi-level asymptotics at higher levels and recurrence relationship of singulants

“…Let us assume the first-level singulant of Q(z) when expanded is u 1 (z) which must satisfy (24). Applying the formula (6) in the pre-factor of (23) gives the first-level singulant by means of…”

“…Therefore, it constitutes "asymptotics beyond all orders". Extracting the exponentially small terms via the growing subject of exponential asymptotics has been studied by many, for example, [5,6,8,9,14,24,25,27,29]. The basis of this work is similar to the methods of hyperasymptotics [6].…”

Capturing the cascade: a transseries approach to delayed bifurcations

“…Transseries methods have been used to study Stokes phenomenon in a wide range of continuous problems. Given that multiple scales methods have been used to study Stokes phenomenon in discrete problems [1,[46][47][48][49][50], it is likely that the transseries approach described here could be used to provide new insight into discrete variants of Stokes phenomenon.…”

“…In addition to the slowly-varying logistic equation, multiple scales-based approaches which describe a system in terms of a fast discrete timescale and a slow continuous timescale have been used to study asymptotic effects in a number of other discrete systems. This includes the study of Stokes phenomena in discrete Painlevé equations [46,48,49], Frenkel-Kontorova models [50], and discrete variants of the Korteweg-de Vries equation [47] and nonlinear Schrödinger equation [1].…”