Related papers: Geometric Aspects of Painlev\'e Equations
In a 1979 paper, K. Okamoto introduced the space of initial values for the six Painlev\'e equations and their associated Hamiltonian systems, showing that these define regular initial value problems at every point of an augmented phase…
The paper is about a Painlev\'e III equation and its relation to isomonodromic families of vector bundles on P^1 with meromorphic connections. The purpose of the paper is two-fold: it offers a conceptual language for the geometrical objects…
We present a geometric description, based on the affine Weyl group E_{6}^{(1)}, of two discrete analogues of the Painlev\'e VI equation, known as the asymmetric q-P_{V} and asymmetric d-P_{IV}. This approach allows us to describe in a…
A theoretical foundation for a generalization of the elliptic difference Painlev\'e equation to higher dimensions is provided in the framework of birational Weyl group action on the space of point configurations in general position in a…
Over the last decade it has become clear that discrete Painlev\'e equations appear in a wide range of important mathematical and physical problems. Thus, the question of recognizing a given non-autonomous recurrence as a discrete Painlev\'e…
This paper contains a survey of recent developments in investigation of word equations in simple matrix groups and polynomial equations in simple (associative and Lie) matrix algebras along with some new results on the image of word maps on…
We discuss geometric integrability of Hirota's discrete KP equation in the framework of projective geometry over division rings using the recently introduced notion of Desargues maps. We also present the Darboux-type transformations, and we…
We consider several examples of nonautonomous systems of difference equations coming from semi-classical orthogonal polynomials via recurrence coefficients and ladder operators, with respect to various generalisations of Laguerre and…
The symmetries described by Pin groups are the result of combining a finite number of discrete reflections in (hyper)planes. The current work shows how an analysis using geometric algebra provides a picture complementary to that of the…
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…
The theory of orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference…
We find and study a six-parameter family of coupled Painlev\'e III systems in dimension six with affine Weyl group symmetry of type $D_6^{(1)}$. We also find and study its degenerate systems with affine Weyl group symmetry of types…
The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry…
We will describe natural `Lax pairs' for the difference Painleve equations with affine Weyl symmetry groups of types E6, E7 and E8, showing that they do indeed arise as symmetries of certain Fuchsian systems of differential equations.
We show that properties of hypergeometric class equations and functions become transparent if we derive them from appropriate 2nd order differential equations with constant coefficients. More precisely, properties of the hypergeometric and…
This short review is an introduction to a great variety of methods, the collection of which is called the Painlev\'e analysis, intended at producing all kinds of exact (as opposed to perturbative) results on nonlinear equations, whether…
In this paper, we build the Hamiltonian system and the corresponding Lax pairs associated to a twisted connection in $\mathfrak{gl}_2(\mathbb{C})$ admitting an irregular and ramified pole at infinity of arbitrary degree, hence corresponding…
We identify a periodic reduction of the non-autonomous lattice potential Korteweg-de Vries equation with the additive discrete Painlev\'e equation with $E^{(1)}_6$ symmetry. We present a description of a set of symmetries of the reduced…
In this paper, we show that the B\"acklund transformations of Painlev\'e equations come from birational maps of rational surfaces constructed by Okamoto as the spaces of initial conditions. The simultaneous resolutions of rational double…
We consider lattice equations on ${\mathds{Z}}^2$ which are autonomous, affine linear and possess the symmetries of the square. Some basic properties of equations of this type are derived, as well as a sufficient linearization condition and…