title: Stokes phenomena in discrete Painlevé II.
abstract: We consider the asymptotic behaviour of the second discrete Painlevé equation in the limit as the independent variable becomes large. Using asymptotic power series, we find solutions that are asymptotically pole-free within some region of the complex plane. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. We subsequently apply exponential asymptotic techniques to investigate such phenomena, and obtain mathematical descriptions of the rapid switching behaviour associated with Stokes curves. Through this analysis, we determine the regions of the complex plane in which the asymptotic behaviour is described by a power series expression, and find that the behaviour of these asymptotic solutions shares a number of features with thetronquéeandtri-tronquéesolutions of the second continuous Painlevé equation.
authors.affiliation: School of Mathematics and Statistics, F07 , The University of Sydney , New South Wales 2006, Australia.
authors.forename: N
authors.initials: N
authors.lastname: Joshi
authors.affiliation: Department of Mathematics , Macquarie University , Sydney, New South Wales, Australia.
authors.forename: C J
authors.identifier: 0000-0001-9504-277X
authors.initials: CJ
authors.lastname: Lustri
authors.affiliation: School of Mathematics and Statistics, F07 , The University of Sydney , New South Wales 2006, Australia.
authors.forename: S
authors.identifier: 0000-0002-2249-6591
authors.initials: S
authors.lastname: Luu
keywords: Stokes phenomena; discrete Painlevé equations; exponential asymptotics; nonlinear discrete asymptotics