Journal of Nonlinear Mathematical Physics

We introduce certain Bäcklund transformations for rational solutions of the Painlevé VI equation. These transformations act on a family of Painlevé VI tau functions. They are obtained from reducing the Hirota bilinear equations that describe the relation between certain points in the 3 component polynomial...

In many cases rational surfaces obtained by desingularization of birational dynamical systems are not relatively minimal. We propose a method to obtain coordinates of relatively minimal rational surfaces by using blowing down structure. We apply this method to the study of various integrable or linearizable...

We study relations between the eigenvectors of rational matrix functions on the Riemann sphere. Our main result is that for a subclass of functions that are products of two elementary blocks it is possible to represent these relations in a combinatorial–geometric way using a diagram of a cube. In this...

In this paper we consider a semi-classical variation of the weight related to the q-Laguerre polynomials and study their recurrence coefficients. In particular, we obtain a second degree second order discrete equation which in particular cases can be reduced to either the qPV or the qPIII equation.

The higher order Painlevé system of type D2n+2(1) was proposed by Y. Sasano as an extension of PVI for the affine Weyl group symmetry with the aid of algebraic geometry for Okamoto initial value space. In this article, we give it as the monodromy preserving deformation of a Fuchsian system.

We extend the notion of deformation to inverse operations of restrictions of completely integrable systems to regular or singular locus, and call the extended notion prolongation. We show that a prolongability determines uniquely a Fuchsian ordinary differential equation of rank three with three regular...

We present some observations on the asymptotic behaviour of the coefficients in the Laurent series expansion of solutions of the first Painlevé equation. For the general solution, explicit recursive formulae for the Taylor expansion of the tau-function around a zero are given, which are natural extensions...

We investigate the discrete Painlevé equations (dPII and qPII) over finite fields. We first show that they are well defined by extending the domain according to the theory of the space of initial conditions. Then we treat them over local fields and observe that they have a property that is similar to...

The first part of this work is a review of the point classification of second order ODEs done by Ruslan Sharipov. His works were published in 1997–1998 in the Electronic Archive at LANL. The second part is an application of this classification to Painlevé equations. In particular, it allows us to solve...

Isomonodromic deformation of linear differential equations on ℙ1 with regular and irregular singular points is considered from the view point of twistor theory. We give explicit form of isomonodromic deformation using the maximal abelian subgroup H of G = GLN+1(ℂ) which appeared in the theory of general...

We derive discrete systems which result from a second, not studied up to now, form of the q-PVI equation. The derivation is based on two different procedures: “limits” and “degeneracies”. We obtain several new discrete Painlevé equations along with some linearisable systems. The parallel between the...

The methods of [vdP-Sa, vdP1, vdP2] are applied to the fourth Painlevéequation. One obtains a Riemann–Hilbert correspondence between moduli spaces of rank two connections on ℙ1 and moduli spaces for the monodromy data. The moduli spaces for these connections are identified with Okamoto–Painlevé varieties...