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 Mat. Sb. (N.S.), 1987, Volume 134(176), Number 3(11), Pages 421–444 (Mi msb2768)

The method of isomonodromy deformations and the asymptotics of solutions of the “complete” third Painlevé equation

A. V. Kitaev

Abstract:$2\times2$ matrix linear ordinary differential equation of the first order is considered whose coefficients depend on an additional parameter $\tau$ having two irregular first order singular points $\lambda=0$ and $\lambda=\infty$. The monodromy data of this equation as $\tau\to0$ and $\tau\to\infty$ are computed. These computations are used to find the asymptotics of the “degenerate” fifth Painlevé equation, which is equivalent to the “complete” third one. This is possible due to the connection of these Painlevé equations with isomonodromy deformations of the coefficients of the matrix linear equation. Bäcklund transformations and their application to asymptotic problems are considered in detail.
Bibliography: 42 titles.

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English version:
Mathematics of the USSR-Sbornik, 1989, 62:2, 421–444

Bibliographic databases:

UDC: 517.9
MSC: Primary 34E05; Secondary 34A20, 34E20, 30D05

Citation: A. V. Kitaev, “The method of isomonodromy deformations and the asymptotics of solutions of the “complete” third Painlevé equation”, Mat. Sb. (N.S.), 134(176):3(11) (1987), 421–444; Math. USSR-Sb., 62:2 (1989), 421–444

Citation in format AMSBIB
\Bibitem{Kit87} \by A.~V.~Kitaev \paper The method of isomonodromy deformations and the asymptotics of solutions of the complete'' third Painlev\'e equation \jour Mat. Sb. (N.S.) \yr 1987 \vol 134(176) \issue 3(11) \pages 421--444 \mathnet{http://mi.mathnet.ru/msb2768} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=922633} \zmath{https://zbmath.org/?q=an:0716.34073|0662.34056} \transl \jour Math. USSR-Sb. \yr 1989 \vol 62 \issue 2 \pages 421--444 \crossref{https://doi.org/10.1070/SM1989v062n02ABEH003247} 

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This publication is cited in the following articles:
1. A V Kitaev, J Phys A Math Gen, 23:15 (1990), 3543
2. D. Levi, P. Winternitz, “Exact solutions of the stimulated-Raman-scattering equations”, Phys Rev A, 44:9 (1991), 6057
3. A V Kitaev, A V Rybin, J Timonen, J Phys A Math Gen, 26:14 (1993), 3583
4. A. L. Kholodenko, A. L. Beyerlein, “Painlevé III and Manning's Counterion Condensation”, Phys Rev Letters, 74:23 (1995), 4679
5. Youmin Lu, Bryce Mcleod, “Asymptotics of the nonnegative solutions of the general fifth Painlevé equation”, Applicable Analysis, 72:3-4 (1999), 501
6. Costin O., “Correlation Between Pole Location and Asymptotic Behavior for Painlevé I Solutions”, Commun. Pure Appl. Math., 52:4 (1999), 461–478
7. Youmin Lu, Zhoude Shao, “Application of uniform asymptotica method to the asymptotics of the solutions of the fifth paonlevé equation when δ=0”, Applicable Analysis, 79:3-4 (2001), 335
8. V.Yu. Novokshenov, “Level spacing functions and connection formulas for Painlevé V transcendent”, Physica D: Nonlinear Phenomena, 152-153 (2001), 225
9. A. V. Kitaev, “Quadratic transformations for the third and fifth Painlevé equations”, J. Math. Sci. (N. Y.), 136:1 (2006), 3586–3595
10. A V Kitaev, A H Vartanian, “Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation: I”, Inverse Problems, 20:4 (2004), 1165
11. J. Math. Sci. (N. Y.), 192:1 (2013), 81–90
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