Journal of Nonlinear Mathematical Physics

Volume 20, Issue Supplement 1, November 2013

Research Article

1. Preface to Special Issue on the Geometry of the Painlevé equations

Nalini Joshi, Masatoshi Noumi, Hidetaka Sakai, Claude M. Viallet
Pages: 1 - 2
Research Article

2. Bäcklund transformations for certain rational solutions of Painlevé VI

Johan van de Leur, Henrik Aratyn
Pages: 3 - 16
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...
Research Article

3. A note on minimization of rational surfaces obtained from birational dynamical systems

A. S. Carstea, T. Takenawa
Pages: 17 - 33
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...
Research Article

4. Combinatorics of Matrix Factorizations and Integrable Systems

Anton Dzhamay
Pages: 34 - 47
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...
Research Article

5. On the Recurrence Coefficients for Generalized q-Laguerre Polynomials

Galina Filipuk, Christophe Smet
Pages: 48 - 56
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.
Research Article

6. Higher order Painlevé system of type D2n+2(1) and monodromy preserving deformation

Kenta Fuji, Keisuke Inoue, Keisuke Shinomiya, Takao Suzuki
Pages: 57 - 69
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.
Research Article

7. Prolongability of Ordinary Differential Equations

Yoshishige Haraoka
Pages: 70 - 84
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...
Research Article

8. Properties of the series solution for Painlevé I

A.N.W. Hone, O. Ragnisco, F. Zullo
Pages: 85 - 100
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...
Research Article

9. The space of initial conditions and the property of an almost good reduction in discrete Painlevé II equations over finite fields

Masataka Kanki, Jun Mada, Tetsuji Tokihiro
Pages: 101 - 109
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...
Research Article

10. Point classification of second order ODEs and its application to Painlevé equations

Vera V. Kartak
Pages: 110 - 129
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...
Research Article

11. General Schlesinger Systems and Their Symmetry from the View Point of Twistor theory

Hironobu Kimura, Damiran Tseveenamijil
Pages: 130 - 152
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...
Research Article

12. A new family of discrete Painlevé equations and associated linearisable systems

A. Ramani, B. Grammaticos
Pages: 153 - 164
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...
Research Article

13. A Riemann–Hilbert approach to Painlevé IV

Marius van der Put, Jaap Top
Pages: 165 - 177
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...