Stokes phenomena in discrete Painlevé II

N. Joshi, C. J. Lustri, S. Luu

    Research output: Contribution to journalArticlepeer-review

    6 Citations (Scopus)


    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 the tronquée and tri-tronquée solutions of the second continuous Painlevé equation.
    Original languageEnglish
    Article number20160539
    Pages (from-to)20160539-1-20160539-20
    Number of pages20
    JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
    Issue number2198
    Publication statusPublished - 28 Feb 2017


    • discrete Painlevé equations
    • nonlinear discrete asymptotics
    • Stokes phenomena
    • exponential asymptotics


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