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Asymptotic expansions of solutions of the sixth Painlevé equation


Authors: A. D. Bruno and I. V. Goryuchkina
Translated by: O. Khleborodova
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 71 (2010).
Journal: Trans. Moscow Math. Soc. 2010, 1-104
MSC (2010): Primary 34E05; Secondary 34M55
DOI: https://doi.org/10.1090/S0077-1554-2010-00186-0
Published electronically: December 28, 2010
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Abstract: We obtain all asymptotic expansions of solutions of the sixth equation near all three singular points $ x=0, x=1$, and $ x=\infty$ for all values of four complex parameters of this equation. The expansions are obtained for solutions of five types: power, power-logarithmic, complicated, semiexotic, and exotic. They form 117 families. These expansions may contain complex powers of the independent variable $ x$. First we use methods of two-dimensional power algebraic geometry to obtain those asymptotic expansions of all five types near the singular point $ x=0$ for which the order of the leading term is less than 1. These expansions are called basic expansions. They form 21 families. All other asymptotic equations near three singular points are obtained from basic ones using symmetries of the equation. The majority of these expansions are new. Also, we present examples and compare our results with previously known ones.


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Additional Information

A. D. Bruno
Affiliation: Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, 4 Miusskaya Ploshchad’, Moscow 125047, Russia
Email: abruno@keldysh.ru

I. V. Goryuchkina
Affiliation: Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, 4 Miusskaya Ploshchad’, Moscow 125047, Russia
Email: chukhareva@yandex.ru

DOI: https://doi.org/10.1090/S0077-1554-2010-00186-0
Published electronically: December 28, 2010
Additional Notes: The work was supported by the Russian Foundation of Fundamental Research (Project 08–01–00082) and the Foundation for the Assistance to Russian Science.
Editorial Note: The following text incorporates changes and corrections submitted by the authors for the English translation.
Article copyright: © Copyright 2010 American Mathematical Society

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