Victor Yu. Novokshenov
(Russia)
Asymptotics in the complex plane of the third Painlev'e transcendent
An uniform asymptotics in the complex plane for the third Painlev\'e transcendent is constructed and proved. The leading term of asymptotics as $|z|\to \infty$ is given by the {\em Boutroux ansatz}, i.e. by an elliptic function with its modulus depending on $\arg z$. A functional equation for the modulus is universal for PIII equation and does not depend on initial conditions. It can be solved as an Abel problem of inversion of elliptic integrals. Another component of the Boutroux ansatz is the phase shift in the elliptic function. It depends on initial data, and we calculate it with the help of Isomonodromic Deformation Method (IDM). By solving a direct monodromy problem for a relevant Lax pair operators, we fit given monodromy data with their approximations, coming from the leading term of asymptotics. This leads to explicit formulas both for the modulus and the phase shift. Since a monodromy data for PIII transcendent can be expressed explicitly through the initial conditions at $z=0$, we come to a connection formulas linking the PIII transcendent asymptotics at infinity and at the origin. Finally, the IDM technique provides the proof of the above constructions, giving an analog of the Bolibrukh-Its-Kapaev theorem proved earlier for the similar asymptotic description of PII transcendent in the complex plane.