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Large z Asymptotics for Special Function Solutions of Painlevé II in the Complex Plane

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URI: https://hdl.handle.net/10016/32004
ISSN: 1815-0659
DOI: https://doi.org/10.3842/SIGMA.2018.107
UXXI: AR/0000026483
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large-z_SIGMA_2018.pdf (2.8 MB)
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2018-10-03
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Kiïv: Department of Applied Research Institute of Mathematics of National Academy of Science of Ukraine
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Abstract
In this paper we obtain large z asymptotic expansions in the complex plane forthe tau function corresponding to special function solutions of the Painlevé II differentialequation. Using the fact that these tau functions can be written as n × n Wronskiandeterminants involving classical Airy functions, we use Heine's formula to rewrite them asn-fold integrals, which can be asymptotically approximated using the classical method ofsteepest descent in the complex plane.
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This paper is a contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev. The full collection is available at https://www.emis.de/journals/SIGMA/Kapaev.html
Keywords
Painlevé equations, Asymptotic expansions, Airy functions
Bibliographic citation
SIGMA, (2018), v.14, 107, [19] p.
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