Journal of Nonlinear Mathematical Physics

Volume 12, Issue Supplement 2, December 2005

Symmetries and Integrability of Difference Equations SIDE VI

Research Article

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

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

2. 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

3. Integrable 1D Toda cellular automata

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

4. 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

5. 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

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

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

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

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

8. Bispectrality for deformed Calogero­Moser­Sutherland systems

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

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

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

12. 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

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

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

14. Finite reductions of the two dimensional Toda chain

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

15. 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

16. Searching for CAC-maps

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

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

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

18. 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

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

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

20. Twisted Volterra equation

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

21. A fuzzy difference equation of a rational form

Pages: 300 - 315
In this paper, we prove some effects concerning a Fuzzy Difference Equation of a rational form.
Research Article

22. 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

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

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...