Scaling limits in the second Painlevé transcendent

By the isomonodromy deformation method, scaling limits in the second Painlevé equation $y_{xx}=2y^3+xy-\alpha$ depending on a complex parameter to and yielding formally equations for an elliptical sine and its degenerations are studied. Results contain the description of discriminant curves on the parameter $t_0$ plane, the proof of the solvability for the system of transcendent equations for an invariant $a_0(t_0)$ for the elliptical asymptotics of the Painlevé transcendent and the description of the main asymptotic terms of the second Painlevé transcendent as $\operatorname{Re}\alpha\to\infty$ for any to with the corresponding connection formulae together in the case of general position. Bibliography: 23 titles.