Related papers: Geometry of the space of monodromy data

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…

This paper studies the space of monodromy data of second order $q$-difference equations through the framework of WKB analysis. We compute the connection matrices associated to the Stokes phenomenon of WKB wavefunctions and develop a general…

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 moduli spaces of meromorphic connections (with arbitrary order poles) over Riemann surfaces together with the corresponding spaces of monodromy data (involving Stokes matrices). Natural symplectic structures are found and described…

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…

Isomonodromy for the fifth Painlev\'e equation ${\rm P}_5$ is studied in detail in the context of certain moduli spaces for connections, monodromy, the Riemann-Hilbert morphism, and Okamoto-Painlev\'e spaces. This involves explicit formulas…

We study the sixth $q$-difference Painlev\'e equation ($q{\textrm{P}_{\textrm{VI}}}$) through its associated Riemann-Hilbert problem (RHP) and show that the RHP is always solvable for irreducible monodromy data. This enables us to identify…

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…

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 This paper has been withdrawn by the…

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…

In this paper, we first give a new interpretation of Jimbo's boundary condition for the generic Painlev\'e VI transcendents, as the shrinking phenomenon in long time behaviour of the Jimbo-Miwa-Mori-Sato equation with rank $n=3$. We then…

In this paper, we give a systematic construction of ten isomonodromic families of connections of rank two on $\mathbb{P}^1$ inducing Painlev\'e equations. The classification of ten families is given by considering the Riemann-Hilbert…

In this short note we give two examples of using the algebro-geometric theory of Painlev\'e equations to solve the Painlev\'e identification problem. The equations that we consider were recently obtained by M. van der Put and J. Top in…

The aim of this paper is to study the effect of isomonodromic deformations of the evolution of scalar fields in Sasaki-Einstein spaces in the context of holography. Here we analyze the monodromy data of the general Heun equation, resulting…

The critical and asymptotic behaviors of solutions of the sixth Painlev\'e equation PVI, obtained in the framework of the monodromy preserving deformation method, and their explicit parametrization in terms of monodromy data, are tabulated.

In this paper we obtain a system of flat coordinates on the monodromy manifold of each of the Painlev\'e equations. This allows us to quantise such manifolds. We produce a quantum confluence procedure between cubics in such a way that…

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 $q\textrm{P}_{\textrm{IV}}$…

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…

Folding transformation of the Painlev\'e equations is an algebraic (of degree greater than 1) transformation between solutions of different equations. In 2005 Tsuda, Okamoto and Sakai classified folding transformations of differential…

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…