Journal of Non-linear Mathematical Physics

ISSN: 1402-9251
Volume 10, Issue Supplement 2, December 2003

Symmetries and Integrability of Difference Equations (SIDE V)

Foreword
Claude BREZINSKI
Pages: 1 - 12
In this paper, we compare the degrees and the orders of approximation of vector and matrix Padé approximants for series with matrix coefficients. It is shown that, in this respect, vector Padé approximants have better properties. Then, matrix­vector Padé approximants are defined and constructed. Finally,...
Alan K COMMON, Andrew N W HONE, Micheline MUSETTE
Pages: 27 - 40
By considering the Darboux transformation for the third order Lax operator of the Sawada-Kotera hierarchy, we obtain a discrete third order linear equation as well as a discrete analogue of the Gambier 5 equation. As an application of this result, we consider the stationary reduction of the fifth...
Yuri B SURIS, Alexander P VESELOV
Pages: 107 - 118
It is shown that for a certain class of Yang-Baxter maps (or set-theoretical solutions to the quantum Yang-Baxter equation) the Lax representation can be derived straight from the map itself. A similar phenomenon for 3D consistent equations on quagraphs has been recently discovered by A. Bobenko and...
Valery I GROMAK, Galina FILIPUK
Pages: 107 - 118
In this paper we investigate relations between different transformations of the slutions of the sixth Painlevé equation. We obtain nonlinear superposition formulas linking solutions by means of the Bäcklund transformation. Algebraic solutions are also studied with the help of the Bäcklund transformation.
Atsushi NAGAI
Pages: 133 - 142
A fractional q-difference operator is presented and its properties are investigated. Epecially, it is shown that this operator possesses an eigen function, which is regarded as a q-discrete analogue of the Mittag-Leffler function. An integrable nonlinear mapping with fractional q-difference is also...
Yasuhiro OHTA
Pages: 143 - 148
We propose a way of discretization for the soliton equations associated with the toroidal Lie algebra based on the direct method. By the discretization, the symetry of the system is modified so that the discrete time evolutions are no longer compatible with the original continuous ones. The solutions...
A. RAMANI, T. TAMIZHMANI, B. GRAMMATICOS, K. M. TAMIZHMANI
Pages: 149 - 165
We present an extension of a family of second-order integrable mappings to the case where the variables do not commute. In every case we introduce a commutation rule which is consistent with the mapping evolution. Through the proper ordering of variables we ensure the existence of an invariant in...
John A G ROBERTS, Danesh JOGIA, Franco VIVALDI
Pages: 166 - 180
We reduce planar measure-preserving rational maps over finite fields, and study their discrete dynamics. We show that application to the orbit analysis over these fields of the Hasse-Weil bound for the number of points on an algebraic curve gives a strong indication of the existence of an integral...
Cornelia SCHIEBOLD
Pages: 181 - 193
Negatons are a solution class with the following characteristic properties: They consist of solitons which are organized in groups. Solitons belonging to the same group are coupled in the sense that they drift apart from each other only logarithmically. The groups themselves rather behave like particles....
Vincenzo SCIACCA
Pages: 209 - 222
A new approach to discrete KP equation is considered, starting from the GelfanZakhharevich theory for the research of Casimir function for Toda Poisson pencil. The link between the usual approach through the use of discrete Lax operators, is emphsized. We show that these two different formulations of...