The Nonlinear Steepest Descent Approach to the Asymptotics of the Second Painlevé Transcendent in the Complex Domain

Abstract

The asymptotics of the generic second Painleve transcendent in the complex domain is found and justified via the direct asymptotic analysis of the associated Riemann-Hilbert problem based on the Deift-Zhou nonlinear steepest descent method. The asymptotics is proved of the Boutroux type, i.e. it is expressed in terms of the elliptic functions. Explicit connection formulae between the asymptotic phases in the different sectors are obtained as well.