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Symmetries and Integrability of Difference Equations

Edited by: Decio Levi University of Rome III, Rome, Italy
Luc Vinet University of Montreal, Montreal, QC, Canada
Pavel Winternitz University of Montreal, Montreal, QC, Canada
A co-publication of the AMS and Centre de Recherches Mathématiques
Available Formats:
Softcover ISBN: 978-0-8218-0601-2
Product Code: CRMP/9
List Price: $119.00 MAA Member Price:$107.10
AMS Member Price: $95.20 Electronic ISBN: 978-1-4704-3923-1 Product Code: CRMP/9.E List Price:$119.00
MAA Member Price: $107.10 AMS Member Price:$95.20
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List Price: $178.50 MAA Member Price:$160.65
AMS Member Price: $142.80 Click above image for expanded view Symmetries and Integrability of Difference Equations Edited by: Decio Levi University of Rome III, Rome, Italy Luc Vinet University of Montreal, Montreal, QC, Canada Pavel Winternitz University of Montreal, Montreal, QC, Canada A co-publication of the AMS and Centre de Recherches Mathématiques Available Formats:  Softcover ISBN: 978-0-8218-0601-2 Product Code: CRMP/9  List Price:$119.00 MAA Member Price: $107.10 AMS Member Price:$95.20
 Electronic ISBN: 978-1-4704-3923-1 Product Code: CRMP/9.E
 List Price: $119.00 MAA Member Price:$107.10 AMS Member Price: $95.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$178.50 MAA Member Price: $160.65 AMS Member Price:$142.80
• Book Details

CRM Proceedings & Lecture Notes
Volume: 91996; 388 pp
MSC: Primary 39; Secondary 33;

This book is devoted to a topic that has undergone rapid and fruitful development over the last few years: symmetries and integrability of difference equations and $q$-difference equations and the theory of special functions that occur as solutions of such equations. Techniques that have been traditionally applied to solve linear and nonlinear differential equations are now being successfully adapted and applied to discrete equations.

This volume is based on contributions made by leading experts in the field during the workshop on Symmetries and Integrability of Difference Equations held in Estérel, Québec, in May 1994.

Giving an up-to-date review of the current status of the field, the book treats these specific topics: Lie group and quantum group symmetries of difference and $q$-difference equations, integrable and nonintegrable discretizations of continuous integrable systems, integrability of difference equations, discrete Painlevé property and singularity confinement, integrable mappings, applications in statistical mechanics and field theories, Yang-Baxter equations, $q$-special functions and discrete polynomials, and $q$-difference integrable systems.

Graduate students, research mathematicians and physicists working in difference equations, special function theory, applications of Lie groups theory, nonlinear phenomena in general and integrability in particular. Also of interest to pure and applied mathematicians, theoretical and mathematical physicists, and engineers interested in solitons.

• Chapters
• On the numerics of integrable discretizations
• A brief introduction to the world of $q$
• A Ramanujan-type for the Al-Salam and Ismail biorthogonal rational functions
• Knizhnik-Zamolodchikov equations and the algebraic Bethe Ansatz
• Integrable quantum mappings
• Local master symmetries of differential-difference equations
• Bäcklund transformations and hierarchies of exact solutions for the fourth Painlevé equation and their application to discrete equations
• On the diagonalization of difference Calogero-Sutherland systems
• The integrable dynamics of a discrete curve
• Continuous symmetries of finite-difference evolution equations and grids
• Basic Bessel functions and $q$-difference equations
• Difference equations and gauge symmetry
• An operator-valued Riemann-Hilbert problem associated with the XXX model
• Orthogonal polynomials satisfying differential equations: The role of the Darboux transformation
• Quantum isomonodromic deformations and the Knizhnik-Zamolodchikov equations
• Lie symmetries and linearizations of analytic discrete dynamical systems
• $q$-algebra representations of the Euclidean, pseudo-Euclidean, and oscillator algebras, and their tensor products
• $_3F_2$(1) hypergeometric function and quadratic $R$-matrix algebra
• Lie point symmetries of differential difference equations
• The factorization method for the differential-difference relativistic Schrödinger equation and $q$-deformations
• Quantum deformations of $\tau$-functions, bilinear identities, and representation theory
• The factorization method and hierarchies of $q$-oscillator Hamiltonians
• Discrete-time Calogero-Moser model and lattice KP equations
• Bilinear structure and exact solutions of the discrete Painlevé I equation
• Integrable difference equations and numerical analysis algorithms
• An integral representation of the very-well-poised $_{8{\psi }8}$
• Singular analogue of the Fourier transformation for the Askey-Wilson polynomials
• Singularity confinement
• Recurrence relation approach for connection coefficients. Applications to classical discrete orthogonal polynomials
• Linearisation of difference equations using factorisable Lie symmetries
• First integrals of the infinite Toda lattice
• Non semisimple quantum groups and “exponential mappings”
• Discrete Schrödinger equation, Darboux transformations, and orthogonal polynomials
• Integrable cellular automata and AKNS hierarchy
• On some rational Coxeter groups
• Requests

Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
Volume: 91996; 388 pp
MSC: Primary 39; Secondary 33;

This book is devoted to a topic that has undergone rapid and fruitful development over the last few years: symmetries and integrability of difference equations and $q$-difference equations and the theory of special functions that occur as solutions of such equations. Techniques that have been traditionally applied to solve linear and nonlinear differential equations are now being successfully adapted and applied to discrete equations.

This volume is based on contributions made by leading experts in the field during the workshop on Symmetries and Integrability of Difference Equations held in Estérel, Québec, in May 1994.

Giving an up-to-date review of the current status of the field, the book treats these specific topics: Lie group and quantum group symmetries of difference and $q$-difference equations, integrable and nonintegrable discretizations of continuous integrable systems, integrability of difference equations, discrete Painlevé property and singularity confinement, integrable mappings, applications in statistical mechanics and field theories, Yang-Baxter equations, $q$-special functions and discrete polynomials, and $q$-difference integrable systems.

Graduate students, research mathematicians and physicists working in difference equations, special function theory, applications of Lie groups theory, nonlinear phenomena in general and integrability in particular. Also of interest to pure and applied mathematicians, theoretical and mathematical physicists, and engineers interested in solitons.

• Chapters
• On the numerics of integrable discretizations
• A brief introduction to the world of $q$
• A Ramanujan-type for the Al-Salam and Ismail biorthogonal rational functions
• Knizhnik-Zamolodchikov equations and the algebraic Bethe Ansatz
• Integrable quantum mappings
• Local master symmetries of differential-difference equations
• Bäcklund transformations and hierarchies of exact solutions for the fourth Painlevé equation and their application to discrete equations
• On the diagonalization of difference Calogero-Sutherland systems
• The integrable dynamics of a discrete curve
• Continuous symmetries of finite-difference evolution equations and grids
• Basic Bessel functions and $q$-difference equations
• Difference equations and gauge symmetry
• An operator-valued Riemann-Hilbert problem associated with the XXX model
• Orthogonal polynomials satisfying differential equations: The role of the Darboux transformation
• Quantum isomonodromic deformations and the Knizhnik-Zamolodchikov equations
• Lie symmetries and linearizations of analytic discrete dynamical systems
• $q$-algebra representations of the Euclidean, pseudo-Euclidean, and oscillator algebras, and their tensor products
• $_3F_2$(1) hypergeometric function and quadratic $R$-matrix algebra
• Lie point symmetries of differential difference equations
• The factorization method for the differential-difference relativistic Schrödinger equation and $q$-deformations
• Quantum deformations of $\tau$-functions, bilinear identities, and representation theory
• The factorization method and hierarchies of $q$-oscillator Hamiltonians
• Discrete-time Calogero-Moser model and lattice KP equations
• Bilinear structure and exact solutions of the discrete Painlevé I equation
• Integrable difference equations and numerical analysis algorithms
• An integral representation of the very-well-poised $_{8{\psi }8}$
• Singular analogue of the Fourier transformation for the Askey-Wilson polynomials
• Singularity confinement
• Recurrence relation approach for connection coefficients. Applications to classical discrete orthogonal polynomials
• Linearisation of difference equations using factorisable Lie symmetries
• First integrals of the infinite Toda lattice
• Non semisimple quantum groups and “exponential mappings”
• Discrete Schrödinger equation, Darboux transformations, and orthogonal polynomials
• Integrable cellular automata and AKNS hierarchy
• On some rational Coxeter groups
Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.