Related papers: On the monodromy manifold of $q$-Painlev\'e VI and…

A Riemann-Hilbert problem for a $q$-difference Painlev\'e equation, known as $q\textrm{P}_{\textrm{IV}}$, is shown to be solvable. This yields a bijective correspondence between the transcendental solutions of $q\textrm{P}_{\textrm{IV}}$…

Explicit solutions to the Riemann-Hilbert problem will be found realising some irreducible non-rigid local systems. The relation to isomonodromy and the sixth Painleve equation will be described. Keywords: Riemann-Hilbert problem, Painleve…

We present the discrete, q-, form of the Painlev\'e VI equation written as a three-point mapping and analyse the structure of its singularities. This discrete equation goes over to P_{VI} at the continuous limit and degenerates towards the…

The Painlev\'e equations possess transcendental solutions $y(t)$ with special initial values that are symmetric under rotation or reflection in the complex $t$-plane. They correspond to monodromy problems that are explicitly solvable in…

We investigate a Riemann-Hilbert problem (RHP), whose solution corresponds to a group of $q$-orthogonal polynomials studied earlier by Ismail et al. Using RHP theory we determine new asymptotic results in the limit as the degree of the…

We describe the close connection between the linear system for the sixth Painlev\'e equation and the general Heun equation, formulate the Riemann-Hilbert problem for the Heun functions and show how, in the case of reducible monodromy, the…

For a generic Painlev\'e 5 equation we characterise all the asymptotics in a right half plane near the point at infinity, that is, we find classified explicit solutions that are, by the Riemann-Hilbert correspondence, labelled with…

We develop a dynamical study of the sixth Painleve equation for all parameters generalizing an earlier work for generic parameters. Here the main focus of this paper is on non-generic parameters, for which the corresponding character…

We formulate a geometric Riemann-Hilbert correspondence that applies to the derivation by Jimbo and Sakai of equation $q$-PVI from ``isomonodromy'' conditions. This is a step within work in progress towards the application of…

A starting point of this paper is a classification of quadratic polynomial transformations of the monodromy manifold for the 2x2 isomonodromic Fuchsian systems associated to the Painleve VI equation. Up to birational automorphisms of the…

In the context of $q$-Painlev\'e VI with generic parameter values, the Riemann-Hilbert correspondence induces a one-to-one mapping between solutions of the nonlinear equation and points on an affine Segre surface. Upon fixing a generic…

In this paper, we will give a complete geometric background for the geometry of Painlev\'e $VI$ and Garnier equations. By geometric invariant theory, we will construct a smooth coarse moduli space $M_n^{\balpha}(\bt, \blambda, L) $ of…

The Riemann-Hilbert approach for the equations ${\rm PIII(D_6)}$ and ${\rm PIII(D_7)}$ is studied in detail, involving moduli spaces for connections and monodromy data, Okamoto-Painlev\'e varieties, the Painlev\'e property, special…

We study the monodromy of Painlev\'e VI equation from a dynamical point of view. This is applied to the description of bounded orbits, and to a proof of the irreducibility of Painlev\'e VI equation in the sens of Casale and Malgrange. On…

This is a sequel to papers by the last two authors making the Riemann-Hilbert correspondence and isomonodromy explicit. For the degenerate fifth Painlev\'e equation, the moduli spaces for connections and for monodromy are explicitly…

This paper applies methods of Van der Put and Van derPut-Saito to the fourth Painlev\'e equation. One obtains a Riemann--Hilbert correspondence between moduli spaces of rank two connections on $\mathbb{P}^1$ and moduli spaces for the…

The critical and asymptotic behaviors of solutions of the sixth Painlev\'e equation, an their parametrization in terms of monodromy data, are synthetically reviewed. The explicit formulas are given. This paper has been withdrawn by the…

We will study two types of special solutions of the sixth Painleve equation, which are invariant under the symmetries obtained from the Backlund transformations. In most cases, the fixed points of the Backlund transformations are classical…

This paper first discusses irreducibility of a Painlev\'e equation $P$. We explain how the Painlev\'e property is helpful for the computation of special classical and algebraic solutions. As in a paper of Morales-Ruiz we associate an…

We study the solutions of the sixth Painlev\'e equation with a logarithmic asymptotic behavior at a critical point. We compute the monodromy group associated to the solutions by the method of monodromy preserving deformations and we…