# Journal of Nonlinear Mathematical Physics

Volume 10, Issue Supplement 2, December 2003

### Symmetries and Integrability of Difference Equations (SIDE V)

**Research Article**

## 2. Comparisons Between Vector and Matrix Padé Approximants

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,...

**Research Article**

## 3. Hierarchies of Difference Equations and Bäcklund Transformations

Peter A. Clarkson, Andrew N.W. Hone, Nalini Joshi

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.

**Research Article**

## 4. A New Discrete Hénon-Heiles System

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 order...

**Research Article**

## 5. Symmetries, Lagrangian Formalism and Integration of Second Order Ordinary Difference Equations

Vladimir Dorodnitsyn, Roman Kozlov, Pavel Winternitz

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.

**Research Article**

## 6. On the Transformations of the Sixth Painlevé Equation

Valery I. Gromak, Galina Filipuk

Pages: 57 - 68

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.

**Research Article**

## 7. Moyal Deformation of 2D Euler Equation and Discretization

Partha Guha

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.

**Research Article**

## 8. The Discrete Nonlinear Schrödinger Equation and its Lie Symmetry Reductions

R. Hernández Heredero, D. Levi

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 lowest...

**Research Article**

## 9. Burchnall-Chaundy Theory for q-Difference Operators and q-Deformed Heisenberg Algebras

Daniel Larsson, Sergei D. Silvestrov

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.

**Research Article**

## 10. On the Lax pairs of the continuous and discrete sixth Painlevé equations

Robert Conte, Runliang Lin, Micheline Musette

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 Lax...

**Research Article**

## 11. Polynomial Growth for Birational Mappings from Four-State Spin Edge Models

J-M Maillard

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...

**Research Article**

## 12. On a Certain Fractional q-Difference and its Eigen Function

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 presented.

**Research Article**

## 13. Discretization of Toroidal Soliton Equations

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...

**Research Article**

## 14. The Extension of Integrable Mappings to Non-Commuting Variables

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 the...

**Research Article**

## 15. The Hasse-Weil Bound and Integrability Detection in Rational Maps

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 for...

**Research Article**

## 16. On Negatons of the Toda Lattice

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....

**Research Article**

## 17. Lattice Geometry of the Discrete Darboux, KP, BKP and CKP Equations. Menelaus' and Carnot's Theorems

Wolfgang Karl Schief

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...

**Research Article**

## 18. Discrete KP Equation and Momentum Mapping of Toda System

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...

**Research Article**

## 19. Lax Matrices for Yang-Baxter Maps

Yuri B. Suris, Alexander P. Veselov

Pages: 223 - 230

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...

**Research Article**

## 20. Analysis of Non-Linear Recurrence Relations for the Recurrence Coefficients of Generalized Charlier Polynomials

Walter Van Assche, Mama Foupouagnigni

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.

**Research Article**

## 21. An Integrable Coupled Toda Equation And Its Related Equation Via Hirota's Bilinear Approach

Jun-Xiao Zhao, Chun-Xia Li, Xing-Biao Hu

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,...