Asymptotic expressions for the second Painlevй functions

The method of isomonodromicJdeformations is used to investigate the asymptotic properties of solutions of the second Painlevé equation (I). The leading terms as $x\to\pm\infty$ of the second Painlevé functions are constructed for the general case. The parameters of the asymptotic behavior are expressed in terms of first inteErals of the Painlevé equation, which are the monodromy data of the associated system (2) of linear ordinary differential equations with rational coefficients. The asymptotic behaviors of the real and imaginary second Painlevé functions are separated.