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Special solutions of the first and second Painlevé equations and singularities of the monodromy data manifold

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Abstract

A classification of solutions of the first and second Painlevé equations corresponding to a special distribution of poles at infinity is considered. The relation between this distribution and singularities of the two-dimensional complex monodromy data manifold used for the parameterization of the solutions is analyzed. It turns out that solutions of the Painlevé equations have no poles in a certain critical sector of the complex plane if and only if their monodromy data lie in the singularity submanifold. Such solutions belong to the so-called class of “truncated” solutions (intégrales tronquée) according to P. Boutroux’s classification. It is shown that all known special solutions of the first and second Painlevé equations belong to this class.

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Authors and Affiliations

  1. Institute of Mathematics with Computing Center, Ufa Research Center of the Russian Academy of Sciences, ul. Chernyshevskogo 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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Dedicated to the 80th birthday of my teacher Arlen Mikhailovich Il’in

Original Russian Text © V.Yu. Novokshenov, 2012, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2012, Vol. 18, No. 2.

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Novokshenov, V.Y. Special solutions of the first and second Painlevé equations and singularities of the monodromy data manifold. Proc. Steklov Inst. Math. 281 (Suppl 1), 105–117 (2013). https://doi.org/10.1134/S0081543813050106

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