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

  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},
  • 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. 

Figures from this paper

On the connection problem for Painlevé I

We study the dependence of the tau function of the Painlevé I equation on the generalized monodromy of the associated linear problem. In particular, we compute the connection constants relating the

Symplectic aspects of the tt*-Toda equations

  • Ryosuke Odoi
  • Mathematics
    Journal of Physics A: Mathematical and Theoretical
  • 2022
We evaluate explicitly, in terms of the asymptotic data, the ratio of the constant pre-factors in the large and small x asymptotics of the tau functions for global solutions of the tt*-Toda

On the tau function of the hypergeometric equation

On Some Hamiltonian Properties of the Isomonodromic Tau Functions

We discuss some new aspects of the theory of the Jimbo–Miwa–Ueno tau function which have come to light within the recent developments in the global asymptotic analysis of the tau functions related to

Irregular conformal blocks and connection formulae for Painlevé V functions

We prove a Fredholm determinant and short-distance series representation of the Painleve V tau function τt associated with generic monodromy data. Using a relation of τt to two different types of

Monodromy dependence and connection formulae for isomonodromic tau functions

We discuss an extension of the Jimbo–Miwa–Ueno differential 1-form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational

Generating function of monodromy symplectomorphism for $2\times 2$ Fuchsian systems and its WKB expansion

. We study the WKB expansion of 2 × 2 system of linear differential equations with four fuchsian singularities. The main focus is on the generating function of the monodromy symplectomorphism which,

Pure 𝑆𝑈(2) gauge theory partition function and generalized Bessel kernel

We show that the dual partition function of the pure $\mathcal N=2$ $SU(2)$ gauge theory in the self-dual $\Omega$-background (a) is given by Fredholm determinant of a generalized Bessel kernel and

Three-point Liouville correlation function as generating function of hypergeometric monodromy map

The monodromy map for a rank-two system of differential equations with three Fuchsian singularities is classically solved by the Kummer formulæ for Gauss’ hypergeometric functions. We define the

Tau-Functions and Monodromy Symplectomorphisms

We derive a new Hamiltonian formulation of Schlesinger equations in terms of the dynamical r-matrix structure. The corresponding symplectic form is shown to be the pullback, under the monodromy map,



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


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