Related papers: Generalization of Okamoto's equation to arbitrary …
In this paper we construct explicit solutions and calculate the corresponding $\tau$-function to the system of Schlesinger equations describing isomonodromy deformations of $2\times 2$ matrix linear ordinary differential equation whose…
We derive the Christoffel-Geronimus-Uvarov transformations of a system of bi-orthogonal polynomials and associated functions on the unit circle, that is to say the modification of the system corresponding to a rational modification of the…
The sixth Painlev\'e equation (PVI) admits dual isomonodromy representations of type $2$-dimensional Fuchsian and $3$-dimensional Birkhoff. Taking the multiplicative middle convolution of a higher Teichm\"uller coordinatization for the…
In the current paper we study the $q$-analogue introduced by Jimbo and Sakai of the well known Painlev\'e VI differential equation. We explain how it can be deduced from a $q$-analogue of Schlesinger equations and show that for a convenient…
We study a relationship between regular flat structures and generalized Okubo systems. We show that the space of variables of isomonodromic deformations of a regular generalized Okubo system can be equipped with a flat structure. As its…
We study the notion of regular singularities for parameterized complex ordinary linear differential systems, prove an analogue of the Schlesinger theorem for systems with regular singularities and solve both a parameterized version of the…
We represent and analyze the general solution of the sixth Painleve transcendent in the Picard-Hitchin-Okamoto class in the Painleve form as the logarithmic derivative of the ratio of certain $\tau$-functions. These functions are…
We present certain general structures related to the solutions of Painlev\'e equation II and to the solutions of the differential equation satisfied by the corresponding Hamiltonian equations, together with the tau functions. By taking…
Schlesinger transformations are algebraic transformations of a Fuchsian system that preserve its monodromy representation and act on the characteristic indices of the system by integral shifts. One of the important reasons to study such…
Various Schlesinger transformations can be combined with a direct pull-back of a hypergeometric 2x2 system to obtain $RS^2_4$-pullback transformations to isomonodromic 2x2 Fuchsian systems with 4 singularities. The corresponding Painleve VI…
A new method to construct algebro-geometric solutions of rank two Schlesinger systems is presented. For an elliptic curve represented as a ramified double covering of CP^1, a meromorphic differential is constructed with the following…
In this note, we review the notion of Painlev\'e scheme of the sixth Painlev\'e equation from the viewpoint of accessible singular point and its local index in the Hirzebruch surface of degree two ${\Sigma_2}$. The key method is Painlev\'e…
We construct a generalisation of what we call Bureau-Guillot systems, i.e. systems of first order equations with coefficient functions being Painlev\'e transcendents. The same Painlev\'e equation is related to the system and it appears as…
A generalized derivative nonlinear Schr\"odinger equation, \ii q_t + q_{xx} + 2\ii \gamma |q|^2 q_x + 2\ii (\gamma-1)q^2 q^*_x + (\gamma-1)(\gamma-2)|q|^4 q = 0 , is studied by means of Hirota's bilinear formalism. Soliton solutions are…
A local behavior of solutions of the Schlesinger equation is studied. We obtain expansions for this solutions, which converge in some neighborhood of a singular point. As a corollary the similar result for the sixth Painlev\'e equation was…
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 coefficients. This extension is based on the…
In this article, we study a special class of Jimbo-Miwa-Mori-Sato isomonodromy equations, which can be seen as a higher-dimensional generalization of Painlev\'e VI. We first construct its convergent $n\times n$ matrix series solutions…
We study a Hamiltonian system without the Painlev\'e property and show that it admits a kind of regularisation on a bundle of rational surfaces with certain divisors removed, generalising Okamoto's spaces of initial conditions for the…
[Note: important Corrigendum now available at arXiv:1601.04790] 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…
Scaling symmetry of gl_n-type Drinfel'd-Sokolov hierarchy is investigated. Applying similarity reduction to the hierarchy, one can obtain the Schlesinger equation with (n+1) regular singularities. Especially in the case of n=3, the…