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Movable Poles of Painlevé I Transcendents and Singularities of Monodromy Data Manifolds

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Recent Developments in Integrable Systems and Related Topics of Mathematical Physics (MP 2016)

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Abstract

We consider a classification of solutions to the first Painlevé  equation with respect to distribution of their poles at infinity. A connection is found between singularities of two-dimensional monodromy data manifold and analytic properties of solutions parametrized by this manifold. It is proved that solutions of Painlevé  I equation have no poles at infinity at a given critical sector of the complex plane iff the related monodromy data belong to the singular submanifold. Such solutions coincide with the class of “truncated” solutions (intégrales tronquée) by classification of P. Boutroux. We derive further classification based on decomposition of singularities of monodromy data manifold.

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Acknowledgments

This paper was partly supported by the Russian Science Foundation (RSCF grant 17-11-01004).

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Authors and Affiliations

  1. Institute of Mathematics RAS, Chernyshevsky street 112, Ufa, 450008, Russia

    V. Yu. Novokshenov

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  1. V. Yu. Novokshenov
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Correspondence to V. Yu. Novokshenov .

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Editors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

    Victor M. Buchstaber

  2. Centre of Integrable Systems, P.G. Demidov Yaroslavl State University, Yaroslavl, Russia

    Sotiris Konstantinou-Rizos

  3. School of Mathematics, University of Leeds, Leeds, UK

    Alexander V. Mikhailov

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Novokshenov, V.Y. (2018). Movable Poles of Painlevé I Transcendents and Singularities of Monodromy Data Manifolds. In: Buchstaber, V., Konstantinou-Rizos, S., Mikhailov, A. (eds) Recent Developments in Integrable Systems and Related Topics of Mathematical Physics. MP 2016. Springer Proceedings in Mathematics & Statistics, vol 273. Springer, Cham. https://doi.org/10.1007/978-3-030-04807-5_3

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