Geodesic
The~method of isomonodromy deformations and connection formulas for the second Painlev\'e transcendent
Izvestiya. Mathematics , Tome 31 (1988) no. 1, pp. 193-207

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A complete asymptotic description is given for the general real solution of the second Painlevé equation, $u_{xx}-xu+2u^3=0$, including explicit formulas connecting the asymptotics as $x\to\pm\infty$. 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. 
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     author = {A. R. Its and A. A. Kapaev},
     title = {The~method of isomonodromy deformations and connection formulas for the second {Painlev\'e} transcendent},
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A. R. Its; A. A. Kapaev. The~method of isomonodromy deformations and connection formulas for the second Painlev\'e transcendent. Izvestiya. Mathematics , Tome 31 (1988) no. 1, pp. 193-207. http://geodesic.mathdoc.fr/item/IM2_1988_31_1_a7/