相关论文: The fifty-two icosahedral solutions to Painleve VI
We classify all functions satisfying non-trivial families of PVI equations. It turns out that, up to an Okamoto equivalence, there are exactly four families parameterized by affine planes or lines. Each affine space is generated by points…
We will describe a method for constructing explicit algebraic solutions to the sixth Painleve equation, generalising that of Dubrovin-Mazzocco. There are basically two steps: First we explain how to construct finite braid group orbits of…
It is well known that the sixth Painlev\'e equation $\PVI$ admits a group of B\"acklund transformations which is isomorphic to the affine Weyl group of type $\mathrm{D}_4^{(1)}$. Although various aspects of this unexpectedly large symmetry…
After recalling some of the geometry of the sixth Painleve equation, we will describe how the Okamoto symmetries arise naturally from symmetries of Schlesinger's equations and summarise the classification of the Platonic Painleve six…
We will explain how some new algebraic solutions of the sixth Painleve equation arise from complex reflection groups, thereby extending some results of Hitchin and Dubrovin-Mazzocco for real reflection groups. The problem of finding…
The Painleve-IV equation has two families of rational solutions generated respectively by the generalized Hermite polynomials and the generalized Okamoto polynomials. We apply the isomonodromy method to represent all of these rational…
In families of Painlev\'e VI differential equations having common algebraic solutions we classify all the members which come from geometry, i.e. the corresponding linear differential equations which are Picard-Fuchs associated to families…
We will revise Garnier-Okamoto's coalescent diagram of isomonodromic deformations and give a complete coalescent diagram. In our viewpoint, we have ten types of isomonodromic deformations and two of them give the same type of the Painlev\'e…
We find four kinds of six-parameter family of coupled Painlev\'e VI systems in dimension four with affine Weyl group symmetry of types $B_6^{(1)}$, $D_6^{(1)}$ and $D_7^{(2)}$. Each system is the first example which gave higher-order…
We study the global analytic properties of the solutions of a particular family of Painleve' VI equations with the parameters $\beta=\gamma=0$, $\delta={1\over2}$ and $\alpha$ arbitrary. We introduce a class of solutions having critical…
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…
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…
Every finite branch solutions to the sixth Painleve equation around a fixed singular point is an algebraic branch solution. In particular a global solution is an algebraic solution if and only if it is finitely many-valued globally. The…
We present a general scheme to derive higher-order members of the Painleve VI (PVI) hierarchy of ODE's as well as their difference analogues. The derivation is based on a discrete structure that sits on the background of the PVI equation…
The paper discusses P$_{III-V}$ equation for special values of its parameters for which this equation reduces to P$_{III}$, I$_{12}$, as well as, to some special cases of I$_{38}$ and I$_{49}$ equations from the Ince's list of $50$ second…
Starting with a rational solution to Painleve' VI, coming from a Riccati equation, using Okamoto's theory a four-parametric rational solution is obtained.
We study critical behaviour and connection problem for a Painleve' 6 equation. We construct solutions of WDVV eqs. using the isomonodromic deformation method and the Painleve' equations. We find algebraic solutions of WDVV and Gromov-Witten…
We construct a family of second-order linear difference equations parametrized by the hypergeometric solution of the elliptic Painlev\'e equation (or higher-order analogues), and admitting a large family of monodromy-preserving…
We announce some results which might bring a new insight into the classification of algebraic solutions to the sixth Painleve equation. The main results consist of the rationality of parameters, trigonometric Diophantine conditions, and…
This is basically the text of a survey talk (entitled 'Painleve, Klein and the icosahedron') given at Hitchin's 60th birthday conference. It discusses the search for and construction of algebraic solutions of the sixth Painleve differential…