Abstract:
A complete asymptotic description is given for the general real solution of the second Painlevé equation, uxx−xu+2u3=0, including explicit formulas connecting the asymptotics as x→±∞. The approach is based on the asymptotic solution of the direct problem of monodromy theory for a linear system associated with the Painlevé equation in the framework of the method of isomonodromy deformations. There is a brief exposition of the method of isomonodromy deformations itself, which is an analogue in the theory of nonlinear ordinary differential equations of the familiar inverse problem method.
Bibliography: 23 titles.
Citation:
A. R. Its, A. A. Kapaev, “The method of isomonodromy deformations and connection formulas for the second Painlevé transcendent”, Math. USSR-Izv., 31:1 (1988), 193–207
\Bibitem{ItsKap87}
\by A.~R.~Its, A.~A.~Kapaev
\paper The~method of isomonodromy deformations and connection formulas for the second Painlev\'e transcendent
\jour Math. USSR-Izv.
\yr 1988
\vol 31
\issue 1
\pages 193--207
\mathnet{http://mi.mathnet.ru/eng/im1323}
\crossref{https://doi.org/10.1070/IM1988v031n01ABEH001056}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=914864}
\zmath{https://zbmath.org/?q=an:0681.34053}
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https://www.mathnet.ru/eng/im1323
https://doi.org/10.1070/IM1988v031n01ABEH001056
https://www.mathnet.ru/eng/im/v51/i4/p878
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M. Kaan Öztürk, R. A. Wolf, “Bifurcation of drift shells near the dayside magnetopause”, J Geophys Res, 112:a7 (2007), A07207
A R Its, A A Kapaev, “Quasi-linear Stokes phenomenon for the second Painlevé transcendent”, Nonlinearity, 16:1 (2003), 363
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