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Distributions of Poles to Painlevé Transcendents via Padé Approximations

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Abstract

A version of the Fair–Luke algorithm has been used to find the Padé approximate solutions to the Painlevé I, II, and IV equations. The distributions of poles in the complex plane are studied to check the dynamics of movable poles and the emergence of rational and truncated solutions, as well as various patterns formed by the poles. The high-order approximations allow us to check asymptotic expansions at infinity and estimate the range of asymptotic domains. The Coulomb gas interpretation of the pole ensembles is discussed in view of the patterns arising in Painlevé IV transcendents.

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Authors and Affiliations

  1. Institute of Mathematics, Ufa Scientific Center, Russian Academy of Sciences, Chernyshevsky street 112, 450008, Ufa, Russia

    V. Y. Novokshenov

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  1. V. Y. Novokshenov
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Correspondence to V. Y. Novokshenov.

Additional information

Communicated by Percy Deift and Alexander Its.

This work was supported in part by RFBR Grant # 10-01-00088.

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Novokshenov, V.Y. Distributions of Poles to Painlevé Transcendents via Padé Approximations. Constr Approx 39, 85–99 (2014). https://doi.org/10.1007/s00365-013-9190-6

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  • DOI: https://doi.org/10.1007/s00365-013-9190-6

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