Symmetries and Integrability of Difference Equations

by Edited by Peter A. Clarkson , Frank W. Nijhoff

• ISBN13: 9780521596992
• ISBN10: 0521596998
• Format: Paperback
• Copyright: 1999-02-28
• Publisher: Cambridge University Press

Summary

There has in recent years been a remarkable growth of interest in the area of discrete integrable systems. Much progress has been made by applying symmetry groups to the study of differential equations, and connections have been made to other topics such as numerical methods, cellular automata and mathematical physics. This volume comprises state of the art articles from almost all the leading workers in this important and rapidly developing area, making it a necessary resource for all researchers interested in discrete integrable systems or related subjects.

Table of Contents

Preface	ix	(6)
List of Participants	xv
1 Partial Difference Equations	3	(40)
Discrete linearisable Gambier equations	3	(12)
A.K. Common
E. Hesameddini
M. Musette
Generalized Backlund transformation and new explicit solutions of the two-dimensional Toda equation	15	(8)
X.-B. Hu
P.A. Clarkson
Different aspects of relativistic Toda chain	23	(20)
S. Kharchev
A. Mironov
A. Zhedanov
2 Integrable Mappings	43	(54)
Integrable symplectic maps	43	(13)
A.P. Fordy
An iterative process on quartics and integrable symplectic maps	56	(8)
J.P. Francoise
O. Ragnisco
Integrable mappings of KdV type and hyperelliptic addition formulae	64	(15)
F.W. Nijhoff
V.Z. Enolskii
R-matrix hierarchies, integrable lattice systems and their integrable discretizations	79	(18)
Yu.B. Suris
3 Discrete Geometry	97	(54)
Discrete conformal maps and surfaces	97	(12)
A.I. Bobenko
The Backlund transformation for discrete isothermic surfaces	109	(13)
J. Cieslinski
Integrable discrete geometry with ruler and compass	122	(15)
A. Doliwa
Self-dual Einstein spaces and a discrete Tzitzeica equation, a permutability theorem link	137	(14)
W. Schief
4 Asymptotic Analysis	151	(46)
New solutions of nonstationary Schrodinger and Kadomtsev-Petviashvili equations	151	(14)
M.J. Ablowitz
J. Villaroel
On asymptotic analysis of orthogonal polynomials via the Riemann-Hilbert method	165	(13)
P. Bleher
A. Its
A new spectral transform for solving the continuous and spatially discrete heat equations on simple trees	178	(19)
P. Bressloff
A.S. Fokas
5 Discrete Painleve Equations	197	(50)
The discrete Painleve I hierarchy	197	(9)
C. Cresswell
N. Joshi
Rational solutions to dPIV	206	(11)
J. Hietarinta
K. Kajiwara
The discrete Painleve II equation and the classical special functions	217	(11)
K. Kajiwara
Freud's equations for orthogonal polynomials as discrete Painleve equations	228	(19)
A. Magnus
6 Symmetries of Difference Equations	247	(40)
An approach to master symmetries of lattice equations	247	(14)
B. Fuchssteiner
W.-X. Ma
Symmetries and generalized symmetries for discrete dynamical systems	261	(14)
D. Levi
Nonlinear difference equations with superposition formulae	275	(12)
P. Winternitz
7 Numerical Methods and Miscellaneous	287	(26)
Generalised Painleve truncation: expansion in Riccati pseudopotentials	287	(12)
A.P. Fordy
A. Pickering
Symplectic Runge-Kutta schemes	299	(14)
W. Oevel
8 Cellular Automata	313	(32)
Soliton Cellular Automata	313	(12)
C. Gilson
Painleve equations and cellular automata	325	(9)
B. Grammaticos
A. Ramani
Soliton cellular automaton	334	(11)
S. Moriwaki
A. Nagai
J. Satsuma
T. Tokihiro
M. Torii
D. Takahashi
J. Matsukidaira
9 q-Special Functions and q-Difference Equations	345	(42)
Fourier-Gauss transform of some q-special functions	345	(8)
N.M. Atakishiyev
The Wilson bispectral involution: some elementary examples	353	(17)
F.A. Grunbaum
L. Haine
Factorisation of Macdonald polynomials	370	(17)
V.B. Kuznetsov
E.K. Sklyanin
10 Quantum Aspects and Yang-Baxter Equations	387
Schrodinger equation on quantum homogeneous spaces	387	(15)
F. Bonechi
R. Giachetti
E. Sorace
M. Tarlini
Local Yang-Baxter relations associated with Hirota's discrete equation	402	(8)
R.M. Kashaev
Some algebraic solutions of discrete equations from anticommuting variables	410	(10)
C. Viallet
q-combinatorics and quantum integrability	420
A. Yu. Volkov