# Geometric aspects of Painlevé equations

@article{Kajiwara2015GeometricAO, title={Geometric aspects of Painlev{\'e} equations}, author={Kenji Kajiwara and Masatoshi Noumi and Yasuhiko Yamada}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2015}, volume={50} }

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…

## 114 Citations

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

- Mathematics, Computer ScienceProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2017

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

- Mathematics
- 2020

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

- Mathematics
- 2016

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

- MathematicsJournal 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

- Mathematics
- 2017

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

- MathematicsNonlinear 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

- Physics
- 2021

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

- Mathematics
- 2017

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

- MathematicsMathematical Physics, Analysis and Geometry
- 2022

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

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2018

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…

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