Related papers: A tau-function solution to the sixth Painleve tran…
The paper is an essentially extended version of the work math.CA/0601371, supplemented with an application. We present new results in the theory of classical $\theta$-functions of Jacobi and $\sigma$-functions of Weierstrass: ordinary…
In this paper we obtain explicit expressions for tau-functions related to Picard type solutions of the Painlev\'e VI equation in terms of theta functions and their derivatives.
We exhibit a remarkable connection between sixth equation of Painleve list and infinite families of explicitly uniformizable algebraic curves. Fuchsian equations, congruences for group transformations, differential calculus of functions and…
We prove that certain polynomials previously introduced by the author can be identified with tau functions of Painlev\'e VI, obtained from one of Picard's algebraic solutions by acting with a four-dimensional lattice of B\"acklund…
We consider connection between the Painleve-6 equation and explicitly uniformizable orbifolds
We study some properties of tau-functions of an isomonodromic deformation leading to the fifth Painlev\'e equation. In particular, here is given an elementary proof of Miwa's formula for the logarithmic differential of a tau-function.
Painleve's transcendental differential equation P_{VI} may be expressed as the consistency condition for a pair of linear differential equations with 2 by 2 matrix coefficients with rational entries. By a construction due to Tracy and…
Using the description of Paileve' VI family of differential equations in terms of a universal elliptic curve, going back to R. Fuchs (cf. [Ma96]), we translate it into the realm of Arithmetic Differential Equations (cf. [Bu05]), where the…
Explicit determinant formulas are presented for the $\tau$ functions of the generalized Painlev\'e equations of type $A$. This result allows an interpretation of the $\tau$-functions as the Pl\"ucker coordinates of the universal Grassmann…
We investigate the structure of $\tau$-functions for the elliptic difference Painlev\'e equation of type $E_8$. Introducing the notion of ORG $\tau$-functions for the $E_8$ lattice, we construct some particular solutions which are expressed…
We present some observations on the tau-function for the fourth Painlev\'e equation. By considering a Hirota bilinear equation of order four for this tau-function, we describe the general form of the Taylor expansion around an arbitrary…
Iorgov, Lisovyy, and Teschner established a connection between isomonodromic deformation of linear differential equations and Liouville conformal field theory at $c=1$. In this paper we present a $q$ analog of their construction. We show…
Dirac hamiltonian on the Poincare disk in the presence of an Aharonov-Bohm flux and a uniform magnetic field admits a one-parameter family of self-adjoint extensions. We determine the spectrum and calculate the resolvent for each element of…
We present some observations on the asymptotic behaviour of the coefficients in the Laurent series expansion of solutions of the first Painleve equation. For the general solution, explicit recursive formulae for the Taylor expansion of the…
For the Painlev\'e 6 transcendents, we provide a unitary description of the critical behaviours, the connection formulae, their complete tabulation, and the asymptotic distribution of the poles close to a critical point.
An algebro-geometric setting for the study of the Painlev\'e VI equation is introduced. Hamiltonian form of the equation is realized on a twisted relative cotangent bundle to the universal elliptic curve with labelled points of order two.…
Building upon the recent works of Bertola; Fasondini, Olver and Xu, we define a class of orthogonal polynomials on elliptic curves and establish a corresponding Riemann-Hilbert framework. We then focus on the special case, defined by a…
A determinant formula for a class of algebraic solutions to Painlev\'e VI equation (P$_{\rm VI}$) is presented. This expression is regarded as a special case of the universal characters. The entries of the determinant are given by the…
We study the underlying relationship between Painleve equations and infinite-dimensional integrable systems, such as the KP and UC hierarchies. We show that a certain reduction of these hierarchies by requiring homogeneity and periodicity…
For a pair of coupled Painlev\'e equations obtained as a similarity reduction of the Hirota-Satsuma systems we describe special parameter-families of solutions given in terms of mixtures of rational and Airy functions, and in terms of a…