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}
}
  • A. ItsA. Prokhorov
  • 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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References

SHOWING 1-10 OF 27 REFERENCES

Connection Problem for the Sine-Gordon/Painlevé III Tau Function and Irregular Conformal Blocks

The short-distance expansion of the tau function of the radial sine-Gordon/Painlev\'e III equation is given by a convergent series which involves irregular $c=1$ conformal blocks and possesses

Painlevé VI connection problem and monodromy of c = 1 conformal blocks

A bstractGeneric c = 1 four-point conformal blocks on the Riemann sphere can be seen as the coefficients of Fourier expansion of the tau function of Painlevé VI equation with respect to one of its

On the τ-function of the Painlevé equations

THE METHOD OF ISOMONODROMY DEFORMATIONS AND THE ASYMPTOTICS OF SOLUTIONS OF THE “COMPLETE” THIRD PAINLEVÉ EQUATION

A matrix linear ordinary differential equation of the first order is considered whose coefficients depend on an additional parameter having two irregular first order singular points and . The

How instanton combinatorics solves Painlevé VI, V and IIIs

We elaborate on a recently conjectured relation of Painlevé transcendents and 2D conformal field theory. General solutions of Painlevé VI, V and III are expressed in terms of c = 1 conformal blocks

Conformal field theory of Painlevé VI

A bstractGeneric Painlevé VI tau function τ (t) can be interpreted as four-point correlator of primary fields of arbitrary dimensions in 2D CFT with c = 1. Using AGT combinatorial representation of

The Dependence on the Monodromy Data of the Isomonodromic Tau Function

The isomonodromic tau function defined by Jimbo-Miwa-Ueno vanishes on the Malgrange’s divisor of generalized monodromy data for which a vector bundle is nontrivial, or, which is the same, a certain

Painleve Transcendents: The Riemann-hilbert Approach

Introduction. Painleve transcendents as nonlinear special functions Part 1. Riemannian-Hilbert problem, isomonodromy method and special functions: Systems of linear ordinary differential equations