**CRM Proceedings & Lecture Notes**

Volume: 25;
2000;
444 pp;
Softcover

MSC: Primary 22; 33; 42; 39; 53;

**Print ISBN: 978-0-8218-2128-2
Product Code: CRMP/25**

List Price: $117.00

AMS Member Price: $93.60

MAA Member Price: $105.30

**Electronic ISBN: 978-1-4704-3939-2
Product Code: CRMP/25.E**

List Price: $114.00

AMS Member Price: $91.20

MAA Member Price: $102.60

# SIDE III—Symmetries and Integrability of Difference Equations

Share this page *Edited by *
*Decio Levi; Orlando Ragnisco*

A co-publication of the AMS and Centre de Recherches Mathématiques

This volume contains the proceedings of the third meeting on
“Symmetries and Integrability of Difference Equations”
(SIDE III). The collection includes original results not published
elsewhere and articles that give a rigorous but concise overview of
their subject, and provides a complete description of the state of the
art. Research in the field of difference equations—often
referred to more generally as discrete systems—has undergone
impressive development in recent years. In this collection the reader
finds the most important new developments in a number of areas,
including: Lie-type symmetries of differential-difference and
difference-difference equations, integrability of fully discrete
systems such as cellular automata, the connection between
integrability and discrete geometry, the isomonodromy approach to discrete
spectral problems and related discrete Painlevé equations,
difference and q-difference equations and orthogonal polynomials,
difference equations and quantum groups, and integrability and chaos
in discrete-time dynamical systems.

The proceedings will be valuable to mathematicians and theoretical
physicists interested in the mathematical aspects and/or in the physical
applications of discrete nonlinear dynamics, with special emphasis on the
systems that can be integrated by analytic methods or at least admit
special explicit solutions. The research in this volume will also be of
interest to engineers working in discrete dynamics as well as to
theoretical biologists and economists.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

#### Readership

Graduate students in mathematics and theoretical physics; researchers in mathematics, physics and informatics.

#### Reviews & Endorsements

Strongly recommend this volume to all libraries of research institutions and to mathematicians with interest in difference equations.

-- CMS Notes

# Table of Contents

## SIDE III -- Symmetries and Integrability of Difference Equations

- Cover Cover11
- Title page v6
- Contents vii8
- List of participants xi12
- Preface xv16
- Billiard weak solutions of nonlinear PDE’s and Toda flows 118
- Fourier-Gauss transforms of some 𝑞-spectral functions 1330
- Integrable differential-difference systems in (2 + 1)-dimensions and their Hamiltonian structure 2340
- Möbius invariant integrable lattice equations associated with the generalized KP hierarchy 3350
- Wave soliton solutions on a generic background for KPI equation 4764
- Difference and differential equations and convergence acceleration algorithms 5370
- On the dynamics of rational solutions for 1-D Volterra system 6582
- Monodromy data for the discrete first Painlevé hierarchy 7390
- Differential identities, generalized polynomials and applications in physics and mathematics 83100
- Lattice geometry of the Hirota equation 93110
- Integrable discrete geometry: The quadrilateral lattice, its transformations and reductions 101118
- On the Kleinian construction of abelian functions of canonical algebraic curves 121138
- Discrete equations and the singular manifold method 139156
- Discrete versions of some algebraic integrable systems related to generalized Jacobians 147164
- New potential symmetries 161178
- Symmetry classification of systems of differential-difference equations 167184
- Integrable boundary conditions for nonlinear lattices 173190
- Modular invariants and generalized Halphen systems 181198
- Symmetry preserving discretization of the Burgers equation 197214
- Singularity confinement and degree growth 209226
- Time-discretization of soliton equations 217234
- Bäcklund transformations for the Hénon-Heiles and Garnier systems 231248
- Irregular singular behaviour in the first discrete Painlevé equation 237254
- Rational and special solutions of the 𝑃_{𝐼𝐼} hierarchy 245262
- Linearisable systems and the Gambier approach 255272
- Difference equations of Poincaré-Perron type and central two-point connection problems 263280
- Self-similarity in spectral problems and 𝑞-special functions 273290
- Lie point symmetries of discrete and continual SU(∞) Toda field theory 295312
- A note on integrable systems related to discrete time Toda lattice 303320
- The integral formula for the solutions of the quantum Knizhnik-Zamolodchikov equation associated with 𝑈_{𝑞}(̂𝑠𝑙_{𝑛}) for |𝑞|=1 315332
- Symmetries of the discrete Schrödinger equation and its quantum deformations 329346
- Determinant and Pfaffian solutions for discrete soliton equations 339356
- Difference isomonodromy problems and discrete Painlevé equations 347364
- Discrete equations of motion for 𝐴_{𝑘-1} algebra 353370
- Discrete Painlevé equations from nonisospectral soliton equations 363380
- A study of the continuous and discrete Gambier systems 367384
- Discrete surfaces of revolution 381398
- Kac-Moody-Virasoro algebras and integrability of certain higher dimensional nonlinear evolutionary equations 401418
- Different faces of harmonic oscillator 407424
- Nonsymmetric linear difference equations for multiple orthogonal polynomials 415432
- Asymptotics for solution to the Cauchy problem for Volterra lattice with step-like initial values 431448
- Complexity and integrability 439456
- Back Cover Back Cover1462