# Connection Problem for the Tau-Function of the Sine-Gordon Reduction of Painlevé-III Equation via the Riemann-Hilbert Approach

@article{Its2015ConnectionPF,
title={Connection Problem for the Tau-Function of the Sine-Gordon Reduction of Painlev{\'e}-III Equation via the Riemann-Hilbert Approach},
author={Alexander Its and Andrei Prokhorov},
journal={International Mathematics Research Notices},
year={2015},
volume={2016},
pages={6856-6883}
}
• Published 24 June 2015
• Mathematics
• International Mathematics Research Notices
We evaluate explicitly, in terms of the Cauchy data, the constant pre-factor in the large $x$ asymptotics of the Painleve III tau-function. Our result proves the conjectural formula for this pre-factor obtained recently by O. Lisovyy, Y. Tykhyy, and the first co-author with the help of the recently discovered connection of the Painleve tau-functions with the Virasoro conformal blocks. Our approach does not use this connection, and it is based on the Riemann-Hilbert method.

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