Asymptotics of the third Painleve transcendent in the complex plane

Abstract:
An uniform asymptotics in the complex plane for the third Painleve transcendent is constructed. Exact formulas for asymptotic parameters (the module and the phase shift) of the elliptic function in the leading term of the asymptotics in terms of the initial data at zero are given. Using the isomonodromic deformation method, an analog of the Bolibrukh-Its-Kapaev theorem for the approximation of the solution of the corresponding Riemann problem on the complex sphere is proved.