Volume 10, Issue Supplement 2, December 2003

Foreword

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,...

Pages: 13 - 26

In this paper we present a method for deriving infinite sequences of difference equations
containing well known discrete Painlevé equations by using the Bäcklund transformtions for the equations in the second Painlevé equation hierarchy.

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...

Pages: 41 - 56

An integration technique for difference schemes possessing Lie point symmetries is
proposed. The method consists of determining an invariant Lagrangian and using
a discrete version of Noether's theorem to obtain first integrals. This lowers the
order of the invariant difference scheme.

Pages: 69 - 76

In this paper we discuss the Moyal deformed 2D Euler flows and its Lax pairs. This
in turn yields the semi-discrete version of 2D Euler equation.

Pages: 77 - 94

The Lie algebra L(h) of symmetries of a discrete analogue of the non-linear Schrödinger
equation (NLS) is studied. A five-dimensional subspace of L(h), generated by both
point and generalized symmetries, transforms into the five-dimensional point symmtry algebra L(0) of the NLS equation. We use the...

Pages: 95 - 106

This paper is devoted to an extension of Burchnall-Chaundy theory on the inteplay between algebraic geometry and commuting differential operators to the case of
q-difference operators.

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...

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.

Pages: 107 - 118

Among the recently found discretizations of the sixth Painlevé equation P6, only the
one of Jimbo and Sakai admits a discrete Lax pair, which does establish its integrabiity. However, a subtle restriction in this Lax pair prevents the possibility to generalize
it in order to find the other missing...

Pages: 119 - 132

We classify all four-state spin edge models according to their behavior under a specific
group of birational symmetry transformations generated from the so-called inversion
relations. This analysis uses the measure of complexity of the action of birational symetries of these lattice models, and aims...

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...

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...

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...

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...

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....

Pages: 194 - 208

Möbius invariant versions of the discrete Darboux, KP, BKP and CKP equations are
derived by imposing elementary geometric constraints on an (irregular) lattice in a
three-dimensional Euclidean space. Each case is represented by a fundamental theorem
of plane geometry. In particular, classical theorems...

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...

Pages: 231 - 237

The recurrence coefficients of generalized Charlier polynomials satisfy a system of nolinear recurrence relations. We simplify the recurrence relations, show that they are
related to certain discrete Painlevé equations, and analyze the asymptotic behaviour.

Pages: 238 - 245

A coupled Toda equation and its related equation are derived from 3-coupled bilinear
equations. The corresponding Bäcklund transformation and nonlinear superposition
formula are presented for the 3-coupled bilinear equations. As an application of the
results, solition solutions are derived. Besides,...