Nobutaka Nakazono

Reduction of lattice equations to the Painlevé equations: PIV and PV

In this paper, we construct a new relation between Adler-Bobenko-Suris equations and Painleve equations. Moreover, using this connection we construct the difference-differential Lax representations of the fourth and fifth Painleve equations.

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

In this paper, we consider the discrete power function associated with the sixth Painlevé equation. This function is a special solution of the so-called cross-ratio equation with a similarity constraint. We show in this paper that this system is embedded in a cubic lattice with W~(3A1(1)) symmetry. By constructing the action of W~(3A1(1)) as a subgroup of W~(D4(1)), i.e. the symmetry group of PVI, we show how to relate W~(D4(1)) to the symmetry group of the lattice. Moreover, by using translations in W~(3A1(1)), we explain the odd–even structure appearing in previously known explicit formulae in terms of the τ function.

Continuous, Discrete and Ultradiscrete Painlevé Equations

We give an introductory lecture on the theory of Painleve equations, which are one of the most important objects in the theory of integrable systems, and their discrete counterparts. The lecture is divided into three parts: the Painleve equations, discrete Painleve equations and ultradiscrete Painleve equations.