Softcover ISBN:  9780821828038 
Product Code:  CRMP/29 
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eBook ISBN:  9781470439439 
Product Code:  CRMP/29.E 
List Price:  $145.00 
MAA Member Price:  $130.50 
AMS Member Price:  $116.00 
Softcover ISBN:  9780821828038 
eBook: ISBN:  9781470439439 
Product Code:  CRMP/29.B 
List Price:  $290.00 $217.50 
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Softcover ISBN:  9780821828038 
Product Code:  CRMP/29 
List Price:  $145.00 
MAA Member Price:  $130.50 
AMS Member Price:  $116.00 
eBook ISBN:  9781470439439 
Product Code:  CRMP/29.E 
List Price:  $145.00 
MAA Member Price:  $130.50 
AMS Member Price:  $116.00 
Softcover ISBN:  9780821828038 
eBook ISBN:  9781470439439 
Product Code:  CRMP/29.B 
List Price:  $290.00 $217.50 
MAA Member Price:  $261.00 $195.75 
AMS Member Price:  $232.00 $174.00 

Book DetailsCRM Proceedings & Lecture NotesVolume: 29; 2001; 436 ppMSC: Primary 37; 35; 17;
This book is devoted to a classical topic that has undergone rapid and fruitful development over the past 25 years, namely Bäcklund and Darboux transformations and their applications in the theory of integrable systems, also known as soliton theory.
The book consists of two parts. The first is a series of introductory pedagogical lectures presented by leading experts in the field. They are devoted respectively to Bäcklund transformations of Painlevé equations, to the dressing method and Bäcklund and Darboux transformations, and to the classical geometry of Bäcklund transformations and their applications to soliton theory. The second part contains original contributions that represent new developments in the theory and applications of these transformations.
Both the introductory lectures and the original talks were presented at an International Workshop that took place in Halifax, Nova Scotia (Canada). This volume covers virtually all recent developments in the theory and applications of Bäcklund and Darboux transformations.
Titles in this series are copublished with the Centre de Recherches Mathématiques.
ReadershipGraduate students and research mathematicians interested in dynamical systems, ergodic theory, and partial differential equations.

Table of Contents

Introductory Lectures

Bäcklund transformations of the higher order Painlevé equations

Dressing method and Bäcklund and Darboux transformations

The classical geometry of Bäcklund transformations. Introduction to applications in soliton theory

An introduction to integrable difference and differential geometries: Affine spheres, their natural generalization and discretization

Original Contributions

On Bianchi and Bäcklund transformations of two dimensional surfaces in four dimensional Euclidean space

Group invariant solutions without transversality and the principle of symmetric criticality

Multicomponent matrix KP hierarchies as symmetryenhanced scalar KP hierarchies and their DarbouxBäcklund solutions

The DarbouxBäcklund transformation and Clifford algebras

On discrete Painlevé equations as Bäcklund transformations

A Bäcklund transformation for timelike surfaces of constant mean curvature in $\mathbb R^{1,2}$

On Ribaucour transformations

The Ribaucour congruences of spheres within Lie sphere geometry

Bäcklund transformations for the reduced MaxwellBloch equations

Transformations of quasilinear systems originating from the projective theory of congruences

Bäcklund links between different analytic descriptions of constant mean curvature surfaces

On the integrability of Weingarten surfaces

A new immersion formula for surfaces on Lie algebras and integrable equations

Difficulties with the SDiff(2) Toda equation

Superposition formulas based on nonprimitive group action

Symmetries of differential difference equations and Lie algebra contractions

Bäcklund transformations from the bilinear viewpoint

Towards the Lax formulation of SU(2) principal models with nonconstant metric

Transcendental solutions of the sineGordon equation

Three dimensional skyrmions and harmonic maps

Induced surfaces and their integrable deformations

Properties of a class of slowly decaying oscillatory solutions of KdV

Blending two discrete integrability criteria: Singularity confinement and algebraic entropy

Bäcklund transformations of soliton systems from symmetry constraints

Open problems for the super KdV equations

Combinatorial aspects of the Darboux transformation

Bäcklund transformation and nonlinear superposition formula of the KaupKupershmidt and Tzitzéica equations

Classification of symmetryintegrable evolution equations

Integrability of evolution equations and pseudospherical surfaces

Infinitesimal Bäcklund transformations of $K$nets. The 2 + 1dimensional SinhGordon system

Isothermic surfaces and the Calapso equation: The full Monty

Bäcklund transformations induced by symmetries. Application: Discrete mKdV

Darboux transformation for a spectral problem quadratic in the spectral parameter

Separation of variables and Darboux transformations

Bäcklund transformations as nonlinear ordinary differential, or difference equations with superposition formulas


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This book is devoted to a classical topic that has undergone rapid and fruitful development over the past 25 years, namely Bäcklund and Darboux transformations and their applications in the theory of integrable systems, also known as soliton theory.
The book consists of two parts. The first is a series of introductory pedagogical lectures presented by leading experts in the field. They are devoted respectively to Bäcklund transformations of Painlevé equations, to the dressing method and Bäcklund and Darboux transformations, and to the classical geometry of Bäcklund transformations and their applications to soliton theory. The second part contains original contributions that represent new developments in the theory and applications of these transformations.
Both the introductory lectures and the original talks were presented at an International Workshop that took place in Halifax, Nova Scotia (Canada). This volume covers virtually all recent developments in the theory and applications of Bäcklund and Darboux transformations.
Titles in this series are copublished with the Centre de Recherches Mathématiques.
Graduate students and research mathematicians interested in dynamical systems, ergodic theory, and partial differential equations.

Introductory Lectures

Bäcklund transformations of the higher order Painlevé equations

Dressing method and Bäcklund and Darboux transformations

The classical geometry of Bäcklund transformations. Introduction to applications in soliton theory

An introduction to integrable difference and differential geometries: Affine spheres, their natural generalization and discretization

Original Contributions

On Bianchi and Bäcklund transformations of two dimensional surfaces in four dimensional Euclidean space

Group invariant solutions without transversality and the principle of symmetric criticality

Multicomponent matrix KP hierarchies as symmetryenhanced scalar KP hierarchies and their DarbouxBäcklund solutions

The DarbouxBäcklund transformation and Clifford algebras

On discrete Painlevé equations as Bäcklund transformations

A Bäcklund transformation for timelike surfaces of constant mean curvature in $\mathbb R^{1,2}$

On Ribaucour transformations

The Ribaucour congruences of spheres within Lie sphere geometry

Bäcklund transformations for the reduced MaxwellBloch equations

Transformations of quasilinear systems originating from the projective theory of congruences

Bäcklund links between different analytic descriptions of constant mean curvature surfaces

On the integrability of Weingarten surfaces

A new immersion formula for surfaces on Lie algebras and integrable equations

Difficulties with the SDiff(2) Toda equation

Superposition formulas based on nonprimitive group action

Symmetries of differential difference equations and Lie algebra contractions

Bäcklund transformations from the bilinear viewpoint

Towards the Lax formulation of SU(2) principal models with nonconstant metric

Transcendental solutions of the sineGordon equation

Three dimensional skyrmions and harmonic maps

Induced surfaces and their integrable deformations

Properties of a class of slowly decaying oscillatory solutions of KdV

Blending two discrete integrability criteria: Singularity confinement and algebraic entropy

Bäcklund transformations of soliton systems from symmetry constraints

Open problems for the super KdV equations

Combinatorial aspects of the Darboux transformation

Bäcklund transformation and nonlinear superposition formula of the KaupKupershmidt and Tzitzéica equations

Classification of symmetryintegrable evolution equations

Integrability of evolution equations and pseudospherical surfaces

Infinitesimal Bäcklund transformations of $K$nets. The 2 + 1dimensional SinhGordon system

Isothermic surfaces and the Calapso equation: The full Monty

Bäcklund transformations induced by symmetries. Application: Discrete mKdV

Darboux transformation for a spectral problem quadratic in the spectral parameter

Separation of variables and Darboux transformations

Bäcklund transformations as nonlinear ordinary differential, or difference equations with superposition formulas