The method of isomonodromy deformations and connection formulas for the second Painlevé transcendent

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.
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