Geometric aspects of Painlevé equations

  title={Geometric aspects of Painlev{\'e} equations},
  author={Kenji Kajiwara and Masatoshi Noumi and Yasuhiko Yamada},
  journal={Journal of Physics A: Mathematical and Theoretical},
In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlevé equations, with a particular emphasis on the discrete Painlevé equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on P 1 × P 1 and classified according to the degeneration of points. We give a systematic description of the equations and their various… 

Geometric description of a discrete power function associated with the sixth Painlevé equation

It is shown that the discrete power function associated with the sixth Painlevé equation is embedded in a cubic lattice with W~(3A1(1)) symmetry, and the odd–even structure appearing in previously known explicit formulae in terms of the τ function is explained.

Theory and Applications of the Elliptic Painlevé Equation

This note is intended to provide an introduction to the theory of discrete Painleve equations focusing mainly on the elliptic difference case. The elliptic Painleve equation is the master case of the

Painleve equations and orthogonal polynomials

In this thesis we classify all of the special function solutions to Painleve equations and all their associated equations produced using their Hamiltonian structures. We then use these special

Full-parameter discrete Painlevé systems from non-translational Cremona isometries

  • Alexander Stokes
  • Mathematics
    Journal of Physics A: Mathematical and Theoretical
  • 2018
Since the classification of discrete Painlevé equations in terms of rational surfaces, there has been much interest in the range of integrable equations arising from each of the 22 surface types in

Cluster integrable systems, q-Painlevé equations and their quantization

A bstractWe discuss the relation between the cluster integrable systems and q-difference Painlevé equations. The Newton polygons corresponding to these integrable systems are all 16 convex polygons

A review of elliptic difference Painlevé equations

  • N. JoshiN. Nakazono
  • Mathematics
    Nonlinear Systems and Their Remarkable Mathematical Structures
  • 2019
Discrete Painlev\'e equations are nonlinear, nonautonomous difference equations of second-order. They have coefficients that are explicit functions of the independent variable $n$ and there are three

Transformations of Hamiltonian systems connected with the fifth Painlevé equation

The talk will be about the Painlevé equations, especially about the fifth one PV . I am going to present three different Hamiltonians and Hamiltonian systems connected with PV (KNY Hamiltonian,

Graphic Enumerations and Discrete Painlev\'e Equations via Random Matrix Models

We revisit the enumeration problems of random discrete surfaces (RDS) based on solutions of the discrete equations derived from the matrix models. For RDS made of squares, the recursive coefficients

Regularising Transformations for Complex Differential Equations with Movable Algebraic Singularities

In a 1979 paper, Okamoto introduced the space of initial values for the six Painlevé equations and their associated Hamiltonian systems, showing that these define regular initial value problems at

On Some Applications of Sakai's Geometric Theory of Discrete Painlevé Equations

Although the theory of discrete Painleve (dP) equations is rather young, more and more examples of such equations appear in interesting and important applications. Thus, it is essential to be able to



Studies on the Painlevé equations

SummaryIn this series of papers, we study birational canonical transformations of the Painlevé system ℋ, that is, the Hamiltonian system associated with the Painlevé differential equations. We

Birational canonical transformations and classical solutions of the sixth Painlevé equation

Two topics on the sixth Painleve equation are treated in this paper. In Section 1, a simple construction of a group of birational canonical transformations of the sixth equation isomorphic to the

Rational Surfaces Associated with Affine Root Systems¶and Geometry of the Painlevé Equations

Abstract: We present a geometric approach to the theory of Painlevé equations based on rational surfaces. Our starting point is a compact smooth rational surface X which has a unique anti-canonical

An affine Weyl group approach to the eight-parameter discrete Painlevé equation

We present a geometrical construction of the eight-parameter discrete Painleve equations. Our starting point is the E(1)8 affine Weyl group. We assume that the multi-dimensional τ-function lives on

Riccati solutions of discrete Painlevé equations with Weyl group symmetry of type E8(1)

We present special solutions of the discrete Painleve equations associated with A0(1), A0(1)* and A0(1)**-surfaces. These solutions can be expressed by solutions of linear difference equations. Here


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

Symmetries of Quantum Lax Equations for the Painlevé Equations

Based on the fact that the Painlevé equations can be written as Hamiltonian systems with affine Weyl group symmetries, a canonical quantization of the Painlevé equations preserving such symmetries

Padé Method to Painlevé Equations

A class of special solutions of Painleve/Garnier systems arising as the Backlund or Schlesinger transformations of the Riccati solutions is known. In the past several years, the corresponding

Symmetries in the fourth Painlevé equation and Okamoto polynomials

Abstract The fourth Painlevé equation PIV is known to have symmetry of the affine Weyl group of type with respect to the Bäcklund transformations. We introduce a new representation of PIV , called