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Zbl 0869.34047
Deift, P.A.; Zhou, X.
Asymptotics for the Painlevé II equation. (English)
[J] Commun. Pure Appl. Math. 48, No.3, 277-337 (1995). ISSN 0010-3640

The paper is concerned with the further development of a new and general nonlinear steepest descent-type method for analysing the asymptotics of oscillatory Riemann-Hilbert (RH) problems proposed earlier by the authors [Ann. Math., II Ser. 137, No. 2, 295-368 (1993; Zbl 0771.35042)]. The main purpose of the paper is to give a rigorous justification of certain well-known asymptotic results for the second Painlevé equation and to derive directly error bounds by using the steepest descent method. The method proceeds by deforming contours, and in the simplest cases the RH problem localizes near the points of stationary phase and the localized RH problems can be solved explicitly in terms of classical special functions, though in more complicated cases the RH problem localizes on a line segment rather than at the stationary phase points. The solution of the RH problem localized on a segment occupies the major part of the paper.
[Y.V.Rogovchenko (Firenze)]

MSC 2000:: *34E05 Asymptotic expansions (ODE)
34M55 Painlevé and other special equations
34M50 Inverse problems in theory of DE in the complex domain
34A34 Nonlinear ODE and systems, general
34A55 Inverse problems of ODE

Keywords: steepest descent-type method; asymptotics; oscillatory Riemann-Hilbert (RH) problems; second Painlevé equation

Citations: Zbl 0771.35042

Cited in: Zbl 1149.42021 Zbl 1149.42020

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