Softcover ISBN:  9780821806012 
Product Code:  CRMP/9 
List Price:  $119.00 
MAA Member Price:  $107.10 
AMS Member Price:  $95.20 
Electronic ISBN:  9781470439231 
Product Code:  CRMP/9.E 
List Price:  $119.00 
MAA Member Price:  $107.10 
AMS Member Price:  $95.20 

Book DetailsCRM Proceedings & Lecture NotesVolume: 9; 1996; 388 ppMSC: 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 uptodate 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, YangBaxter equations, \(q\)special functions and discrete polynomials, and \(q\)difference integrable systems.ReadershipGraduate 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.

Table of Contents

Chapters

On the numerics of integrable discretizations

A brief introduction to the world of $q$

A Ramanujantype for the AlSalam and Ismail biorthogonal rational functions

KnizhnikZamolodchikov equations and the algebraic Bethe Ansatz

Integrable quantum mappings

Local master symmetries of differentialdifference equations

Bäcklund transformations and hierarchies of exact solutions for the fourth Painlevé equation and their application to discrete equations

On the diagonalization of difference CalogeroSutherland systems

The integrable dynamics of a discrete curve

Continuous symmetries of finitedifference evolution equations and grids

Basic Bessel functions and $q$difference equations

Difference equations and gauge symmetry

An operatorvalued RiemannHilbert problem associated with the XXX model

Orthogonal polynomials satisfying differential equations: The role of the Darboux transformation

Quantum isomonodromic deformations and the KnizhnikZamolodchikov equations

Lie symmetries and linearizations of analytic discrete dynamical systems

$q$algebra representations of the Euclidean, pseudoEuclidean, 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 differentialdifference 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

Discretetime CalogeroMoser 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 verywellpoised $_{8{\psi }8}$

Singular analogue of the Fourier transformation for the AskeyWilson 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


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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 uptodate 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, YangBaxter 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 Ramanujantype for the AlSalam and Ismail biorthogonal rational functions

KnizhnikZamolodchikov equations and the algebraic Bethe Ansatz

Integrable quantum mappings

Local master symmetries of differentialdifference equations

Bäcklund transformations and hierarchies of exact solutions for the fourth Painlevé equation and their application to discrete equations

On the diagonalization of difference CalogeroSutherland systems

The integrable dynamics of a discrete curve

Continuous symmetries of finitedifference evolution equations and grids

Basic Bessel functions and $q$difference equations

Difference equations and gauge symmetry

An operatorvalued RiemannHilbert problem associated with the XXX model

Orthogonal polynomials satisfying differential equations: The role of the Darboux transformation

Quantum isomonodromic deformations and the KnizhnikZamolodchikov equations

Lie symmetries and linearizations of analytic discrete dynamical systems

$q$algebra representations of the Euclidean, pseudoEuclidean, 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 differentialdifference 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

Discretetime CalogeroMoser 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 verywellpoised $_{8{\psi }8}$

Singular analogue of the Fourier transformation for the AskeyWilson 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