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The Painlevé Equations and Discrete Asymptotics

Banff International Research Station for Mathematical Innovation and Discovery

Littlewood reported in his preface to Hardy’s "Divergent Series” that Abel said divergent series were the invention of the devil. But such series arise commonly in the solutions of ODEs in asymptotic limits. The asymptotic description of transcendental solutions of the Painlevé equations has been a longstanding problem, which remains incomplete for many of these equations. We start with a review of these results before describing how such series occur in the solutions of the discrete Painlevé equations. In the latter part of the talk, I will focus on recent studies for additive discrete versions of Painlevé equations and a discrete analogue of the famous tritronquée solutions of the first Painlevé equation for a q-discrete equation. Joshi, N., and C. J. Lustri. "Stokes phenomena in discrete Painlevé I." In Proc. R. Soc. A, vol. 471, no. 2177, p. 20140874. The Royal Society, 2015. Joshi, N., Lustri, C. and Luu, S., 2016. Stokes Phenomena in Discrete Painlev\'e II. arXiv:1607.04494 Joshi N. and Takei, Y., 2016. Toward the exact WKB analysis of discrete Painlev\'e equations, RIMS Kˆokyuˆroku Bessatsu, to appear.

Mathematics; Ordinary differential equations; Dynamical systems and ergodic theory; Dynamical systems

video/mp4

Author affiliation: University of Sydney

2017-04-04

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61096

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/