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SIDE III--Symmetries and Integrability of Difference Equations
Edited by: Decio Levi, University of Rome I, Italy, and Orlando Ragnisco, University of Rome III, Italy
A co-publication of the AMS and Centre de Recherches Mathématiques.

CRM Proceedings & Lecture Notes
2000; 444 pp; softcover
Volume: 25
ISBN-10: 0-8218-2128-8
ISBN-13: 978-0-8218-2128-2
List Price: US$114
Member Price: US$91.20
Order Code: CRMP/25
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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.


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


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

-- CMS Notes

Table of Contents

  • M. S. Alber, R. Camassa, and M. Gekhtman -- Billiard weak solutions of nonlinear PDE's and Toda flows
  • N. M. Atakishiyev -- Fourier-Gauss transforms of some \(q\)-spectral functions
  • M. Błaszak and A. Szum -- Integrable differential-difference systems in (2 + 1)-dimensions and their Hamiltonian structure
  • L. V. Bogdanov and B. G. Konopelchenko -- Möbius invariant integrable lattice equations associated with the generalized KP hierarchy
  • M. Boiti, F. Pempinelli, A. K. Pogrebkov, and B. Prinari -- Wave soliton solutions on a generic background for KPI equation
  • C. Brezinski -- Difference and differential equations and convergence acceleration algorithms
  • A. S. Cârstea -- On the dynamics of rational solutions for 1-D Volterra system
  • C. Cresswell -- Monodromy data for the discrete first Painlevé hierarchy
  • G. Dattoli -- Differential identities, generalized polynomials and applications in physics and mathematics
  • A. Doliwa -- Lattice geometry of the Hirota equation
  • A. Doliwa and P. M. Santini -- Integrable discrete geometry: The quadrilateral lattice, its transformations and reductions
  • J. C. Eilbeck, V. Z. Enolskii, and D. V. Leykin -- On the Kleinian construction of abelian functions of canonical algebraic curves
  • P. G. Estévez and P. A. Clarkson -- Discrete equations and the singular manifold method
  • Yu. Fedorov -- Discrete versions of some algebraic integrable systems related to generalized Jacobians
  • M. L. Gandarias -- New potential symmetries
  • D. G. Oteiza, S. Lafortune, and P. Winternitz -- Symmetry classification of systems of differential-difference equations
  • I. T. Habibullin and A. N. Vil'danov -- Integrable boundary conditions for nonlinear lattices
  • J. Harnad and J. McKay -- Modular invariants and generalized Halphen systems
  • R. H. Heredero, D. Levi, and P. Winternitz -- Symmetry preserving discretization of the Burgers equation
  • J. Hietarinta and C. Viallet -- Singularity confinement and degree growth
  • R. Hirota and M. Iwao -- Time-discretization of soliton equations
  • A. N. W. Hone, V. B. Kuznetsov, and O. Ragnisco -- Bäcklund transformations for the Hénon-Heiles and Garnier systems
  • N. Joshi -- Irregular singular behaviour in the first discrete Painlevé equation
  • N. A. Kudryashov and A. Pickering -- Rational and special solutions of the \(P_{\mathrm{II}}\) hierarchy
  • S. Lafortune, B. Grammaticos, and A. Ramani -- Linearisable systems and the Gambier approach
  • W. Lay -- Difference equations of Poincaré-Perron type and central two-point connection problems
  • I. Loutsenko and V. Spiridonov -- Self-similarity in spectral problems and \(q\)-special functions
  • L. Martina -- Lie point symmetries of discrete and continual SU(\(\infty\)) Toda field theory
  • K.-i. Maruno, K. Kajiwara, and M. Oikawa -- A note on integrable systems related to discrete time Toda lattice
  • T. Miwa and Y. Takeyama -- The integral formula for the solutions of the quantum Knizhnik-Zamolodchikov equation associated with \(U_q(\widehat{sl}_n)\) for \(|q| = 1\)
  • L. M. Nieto, J. Negro, F. J. Herranz, and A. Ballesteros -- Symmetries of the discrete Schrödinger equation and its quantum deformations
  • Y. Ohta -- Determinant and Pfaffian solutions for discrete soliton equations
  • V. G. Papageorgiou -- Difference isomonodromy problems and discrete Painlevé equations
  • A. P. Protogenov and V. A. Verbus -- Discrete equations of motion for \(A_{k-1}\) algebra
  • G. R. W. Quispel and D. Levi -- Discrete Painlevé equations from nonisospectral soliton equations
  • A. Ramani, B. Grammaticos, and S. Lafortune -- A study of the continuous and discrete Gambier systems
  • W. K. Schief and B. G. Konopelchenko -- Discrete surfaces of revolution
  • M. Senthilvelan -- Kac-Moody-Virasoro algebras and integrability of certain higher dimensional nonlinear evolutionary equations
  • A. Turbiner -- Different faces of harmonic oscillator
  • W. Van Assche -- Nonsymmetric linear difference equations for multiple orthogonal polynomials
  • V. L. Vereschagin -- Asymptotics for solution to the Cauchy problem for Volterra lattice with step-like initial values
  • C. Viallet -- Complexity and integrability
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