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Zbl 0816.34039
Its, A.R.; Fokas, A.S.; Kapaev, A.A.
On the asymptotic analysis of the Painlevé equations via the isomonodromy method. (English)
[J] Nonlinearity 7, No.5, 1291-1325 (1994). ISSN 0951-7715; ISSN 1361-6544/e

The authors analyze the global asymptotic properties of Painlevé equations, by means of the isomonodromy method. Their goal is to provide a rigorous justification for the asymptotic analysis of these equations. There are three approaches to be considered; (i) local asymptotic analysis of the Painlevé equation, (ii) the method of Kitaev based on the Brouwer fixed point theorem, and (iii) the method of Deift and Zhou, based on direct asymptotic analysis of the relevant oscillatory Riemann- Hilbert problems. Examples include the connection problem for pure imaginary solutions of the second Painlevé equation $u\sb{xx} - xu - 2u\sp 3 = 0$, $x \in R$, $u \in iR$. There is an interesting and useful analysis of the relation between the Deift-Zhou steepest descent method, and the Kitaev method.
[A.L.Edelson (Davis)]

MSC 2000:: *34E05 Asymptotic expansions (ODE)
34B30 Special ODE

Keywords: global asymptotic properties; Painlevé equations; isomonodromy method; method of Kitaev; method of Deift and Zhou; Riemann-Hilbert problems

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