Related papers: A Riemann--Hilbert approach to Painlev\'e IV
Novel hybrid Ermakov-Painlev\'{e} IV systems are introduced and an associated Ermakov invariant is used in establishing their integrability. B\"{a}cklund transformations are then employed to generate classes of exact solutions via the…
We show that the fourth order nonlinear ODE which controls the pole dynamics in the general solution of equation $P_I^2$ compatible with the KdV equation exhibits two remarkable properties: 1) it governs the isomonodromy deformations of a…
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…
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 study the moduli space of logarithmic connections of rank $2$ on $\mathbb{P}^1 \setminus \{ t_1, \dots, t_5 \}$ with fixed spectral data. The aim of this paper is to compute the cohomology of such space, and this computation will be used…
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…
It is proved that the Painlev\'{e} VI equation $(PVI_{\al,\be,\ga,\de})$ for the special values of constants $(\al=\frac{\nu^2}{4},\be=-\frac{\nu^2}{4}, \ga=\frac{\nu^2}{4},\de=\f1{2}-\frac{\nu^2}{4})$ is a reduced hamiltonian system. Its…
It is well-known that differential Painlev\'e equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique -- there are many very different Hamiltonians that result in the same…
In this paper we describe B\"acklund transformations and hierarchies of exact solutions for the fourth Painlev\'e equation (PIV) $${\d^2 w\over\d z^2}={1\over2w}\left(\d w\over\d z\right)^2 + {{3\over2}}w^3 + 4zw^2 +…
We review the construction of the mixed Painlev\'e P$_{III-V}$ system in terms of a 4-boson integrable model and discuss its symmetries. Such a mixed system consist of an hybrid differential equation that for special limits of its…
We define a variant of the Seiberg-Witten equations using the Rarita-Schwinger operators for closed simply connected spin smooth 4-manifold X. The moduli space of solutions to the system of non-linear differential equations consist of…
The leaves of the Painlev{\'e} foliations appear as the isomonodromic deformations of a rank 2 linear connection on a moduli space of connections. Therefore they are the fibers of the Riemann-Hilbert correspondence that sends each…
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 provide a complete classification and an explicit representation of rational solutions to the fourth Painlev\'e equation PIV and its higher order generalizations known as the $A_{2n}$-Painlev\'e or Noumi-Yamada systems. The construction…
A generalisation of the Stieltjes relations for the Painlev\'e-IV transcendents and their higher analogues determined by the dressing chains is proposed. It is proven that if a rational function from a certain class satisfies these…
In this paper, we describe the moduli space of rank three parabolic logarithmic connections on the projective line with three poles for any local exponents. In particular, we show that the family of moduli spaces of rank three parabolic…
In this survey we present the interpretation of isomondromy preserving equations on Riemann surfaces with marked points as reduced Hamiltonian systems. The upstairs space is the space of smooth connections of GL(N) bundles with simple poles…
We study the singular cohomology of the moduli space of rank 2 parabolic bundles on a Riemann surface where the weights are all 1/4. We give a formula, based on work of Boden, for the Poincar\'e polynomial of this moduli space in general,…
We generalize the surjectivity result of the $p$-adic monodromy for the ordinary locus of a Siegel moduli space by Faltings and Chai (independently by Ekedahl) to that for any $p$-rank stratum. We discuss irreducibility and connectedness of…
We consider the isomonodromy problems for flat $G$-bundles over punctured elliptic curves $\Sigma_\tau$ with regular singularities of connections at marked points. The bundles are classified by their characteristic classes. These classes…