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.
Keywords:
Painlevé equations, isomonodromic deformations, distribution of poles, special solutions, Padé approximations.
Citation:
V. Yu. Novokshenov, “Special solutions of the first and second Painlevé equations and singularities of the monodromy data manifold”, Trudy Inst. Mat. i Mekh. UrO RAN, 18, no. 2, 2012, 179–190; Proc. Steklov Inst. Math. (Suppl.), 281, suppl. 1 (2013), 105–117
\Bibitem{Nov12}
\by V.~Yu.~Novokshenov
\paper Special solutions of the first and second Painlev\'e equations and singularities of the monodromy data manifold
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2012
\vol 18
\issue 2
\pages 179--190
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\elib{https://elibrary.ru/item.asp?id=17736196}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2013
\vol 281
\issue , suppl. 1
\pages 105--117
\crossref{https://doi.org/10.1134/S0081543813050106}
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Linking options:
https://www.mathnet.ru/eng/timm818
https://www.mathnet.ru/eng/timm/v18/i2/p179
This publication is cited in the following 1 articles:
B. I. Suleimanov, “Isomonodromic quantization of the second Painlevé equation by means of conservative Hamiltonian systems with two degrees of freedom”, St. Petersburg Math. J., 33:6 (2022), 995–1009
Citation in format AMSBIB
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