# Journal of Nonlinear Mathematical Physics

Volume 12, Issue Supplement 2, December 2005

### Symmetries and Integrability of Difference Equations SIDE VI

**Research Article**

## 2. Asymptotic behavior of discrete holomorphic maps zc and log(z)

Sergey I. Agafonov

Pages: 1 - 14

It is shown that discrete analogs of zc and log(z), defined via particular "integrable" circle patterns, have the same asymptotic behavior as their smooth counterparts. These discrete maps are described in terms of special solutions of discrete Painlevé-II equations, asymptotics of these solutions providing...

**Research Article**

## 3. A novel approach to the theory of Padé approximants

Christopher Athorne

Pages: 15 - 27

By associating polynomials and power series expansions with sln(C) modules we dscribe the theory of Padé approximants in terms of tensor products of representations and interpret their recurrence relations algebraically. The treatment links with the theory of Hirota derivatives and discrete integrable...

**Research Article**

## 4. Integrable 1D Toda cellular automata

Mariusz Białecki

Pages: 28 - 35

First, we recall the algebro-geometric method of construction of finite field valued solutions of the discrete KP equation, and next we perform a reduction of dKP to the discrete 1D Toda equation.

**Research Article**

## 5. Second Order Dynamic Inclusions

Martin Bohner, Christopher C Tisdell

Pages: 36 - 45

The theory of dynamic inclusions on a time scale is introduced, hence accommodating the special cases of differential inclusions and difference inclusions. Fixed point theory for set-valued upper semicontinuous maps, Green's functions, and upper and lower solutions are used to establish existence results...

**Research Article**

## 6. Bilinear recurrences and addition formulae for hyperelliptic sigma functions

Harry W. Braden, Victor Z. Enolskii, Andrew N.W. Hone

Pages: 46 - 62

The Somos 4 sequences are a family of sequences satisfying a fourth order bilinear recurrence relation. In recent work, one of us has proved that the general term in such sequences can be expressed in terms of the Weierstrass sigma function for an associated elliptic curve. Here we derive the analogous...

**Research Article**

## 7. Orthogonal matrix polynomials satisfying first order differential equations: a collection of instructive examples

Mirta M. Castro, F. Alberto GRUNBAUM

Pages: 63 - 76

We describe a few families of orthogonal matrix polynomials of size N × N satisfying first order differential equations. This problem differs from the recent efforts reported for instance in [7] (Orthogonal matrix polynomials satisfying second order differential equations, Internat. Math. Research Notices,...

**Research Article**

## 8. Integrable flows and Bäcklund transformations on extended Stiefel varieties with application to the Euler top on the Lie group SO(3)

Yuri N. Fedorov

Pages: 77 - 94

We show that the m-dimensional EulerManakov top on so (m) can be represented as a Poisson reduction of an integrable Hamiltonian system on a symplectic extended Stiefel variety ¯V(k, m), and present its Lax representation with a rational parameter. We also describe an integrable two-valued symplectic...

**Research Article**

## 9. Bispectrality for deformed CalogeroMoserSutherland systems

Misha Feigin

Pages: 95 - 136

We prove bispectral duality for the generalized CalogeroMoserSutherland systems related to configurations An,2(m), Cn(l, m). The trigonometric axiomatics of the BakerAkhiezer function is modified, the dual difference operators of rational Madonald type and the BakerAkhiezer functions related to both...

**Research Article**

## 10. Asymptotic symmetries of difference equations on a lattice

Giuseppe Gaeta, Decio Levi, Rosaria Mancinelli

Pages: 137 - 146

It is known that many equations of interest in Mathematical Physics display solutions which are only asymptotically invariant under transformations (e.g. scaling and/or translations) which are not symmetries of the considered equation. In this note we extend the approach to asymptotic symmetries for...

**Research Article**

## 11. Pfaffianization of the q-difference version of the two-dimensional Toda lattice equation

Gegenhasi, Xing-Biao Hu, Hon-Wah Tam

Pages: 147 - 152

**Research Article**

## 12. On Reductions and Real Hamiltonian Forms of Affine Toda Field Theories

Vladimir S. Gerdjikov, Georgi G. Grahovski

Pages: 155 - 168

A family of real Hamiltonian forms (RHF) for the special class of affine 1 + dimensional Toda field theories is constructed. Thus the method, proposed in [1] for systems with finite number of degrees of freedom is generalized to infinite-dimensional Hamiltonian systems. We show that each of these RHF...

**Research Article**

## 13. The Relation between a 2D Lotka-Volterra equation and a 2D Toda Lattice

Claire R. Gilson, Jonathan J.C. Nimmo

Pages: 169 - 179

It is shown that the 2-discrete dimensional Lotka-Volterra lattice, the two dmensional Toda lattice equation and the recent 2-discrete dimensional Toda lattice equation of Santini et al can be obtained from a 2-discrete 2-continuous dimensional Lotka-Volterra lattice.

**Research Article**

## 14. Non-isospectral lattice hierarchies in 2 + 1 dimensions and generalized discrete Painlevé hierarchies

P.R. Gordoa, A. Pickering, Z.N. Zhu

Pages: 180 - 196

In a recent paper we introduced a new 2 + 1-dimensional non-isospectral extension of the Volterra lattice hierarchy, along with its corresponding hierarchy of underlying linear problems. Here we consider reductions of this lattice hierarchy to hierarchies of discrete equations, which we obtain once again...

**Research Article**

## 15. Finite reductions of the two dimensional Toda chain

E.V. Gudkova

Pages: 197 - 205

The problem of the classification of integrable truncations of the Toda chain is dicussed. A new example of the cutting off constraint is found.

**Research Article**

## 16. A factorization for Z × Z-matrices yielding solutions of Toda-type hierarchies

Gerardus Franciscus Helminck

Pages: 206 - 222

In this paper one considers the problem of finding solutions to a number of Todtype hierarchies. All of them are associated with a commutative subalgebra of the k×k-matrices. The first one is formulated in terms of upper triangular Z×Z-matrices, the second one in terms of lower triangular ones and the...

**Research Article**

## 17. Searching for CAC-maps

Jarmo Hietarinta

Pages: 223 - 230

For two-dimensional lattice equations the standard definition of integrability is that it should be possible to extend the map consistently to three dimensions, i.e., that it is "consistent around a cube" (CAC). Recently Adler, Bobenko and Suris conducted a search based on this principle, together with...

**Research Article**

## 18. Darboux transformations for q-discretizations of 2D second order differential equations

P. Malkiewicz, M. Nieszporski

Pages: 231 - 238

We present q-discretizations of a second order differential equation in two independent variables that not only go to the differential counterpart as q goes to 1 but admit Moutard-Darboux transformations as well.

**Research Article**

## 19. Bäcklund transformations for the rational Lagrange chain

Fabio Musso, Matteo Petrera, Orlando Ragnisco, Giovanni Satta

Pages: 240 - 252

We consider a longrange homogeneous chain where the local variables are the geerators of the direct sum of N e(3) interacting Lagrange tops. We call this classical integrable model rational "Lagrange chain" showing how one can obtain it starting from su(2) rational Gaudin models. Moreover we construct...

**Research Article**

## 20. A unitary joint eigenfunction transform for the AOs exp(ia±d/dz) + exp(2z/a)

S.N.M. Ruijsenaars

Pages: 253 - 294

For positive parameters a+ and a- the commuting difference operators exp(ia±d/dz) + exp(2z/a), acting on meromorphic functions f(z), z = x + iy, are formally self-adjoint on the Hilbert space H = L2 (R, dx). Volkov showed that they admit joint eigenfunctions. We prove that the joint eigenfunctions for...

**Research Article**

## 21. Twisted Volterra equation

Sergei D. Silvestrov

Pages: 295 - 299

In this paper an extension of the q-deformed Volterra equation associated with linear rescaling to the general non-linear rescaling is obtained.

**Research Article**

## 22. A fuzzy difference equation of a rational form

G. Stefanidou, G. Papaschinopoulos

Pages: 300 - 315

In this paper, we prove some effects concerning a Fuzzy Difference Equation of a rational form.

**Research Article**

## 23. On a special two-dimensional lattice by Blaszak and Szum: pfaffianization and molecule solutions

Guo-Fu Yu, Chun-Xia Li, Jun-Xiao Zhao

Pages: 316 - 332

In this paper, we first present the Casorati and grammian determinant solutions to a special two-dimensional lattice by Blaszak and Szum. Then, by using the pfaffianiztion procedure of Hirota and Ohta, a new integrable coupled system is generated from the special lattice. Moreover, gram-type pfaffian...

**Research Article**

## 24. Regular algebras of dimension 2, the generalized eigenvalue problem and Padé interpolation

Alexei Zhedanov

Pages: 333 - 356

We consider the generalized eigenvalue problem A = B for two operators A, B. Self-similar closure of this problem under a simplest Darboux transformation gives rise to two possible types of regular algebras of dimension 2 with generators A, B. Realiztion of the operators A, B by tri-diagonal operators...