**CRM Proceedings & Lecture Notes**

Volume: 29;
2001;
436 pp;
Softcover

MSC: Primary 37; 35; 17;

**Print ISBN: 978-0-8218-2803-8
Product Code: CRMP/29**

List Price: $138.00

AMS Member Price: $110.40

MAA Member Price: $124.20

**Electronic ISBN: 978-1-4704-3943-9
Product Code: CRMP/29.E**

List Price: $138.00

AMS Member Price: $110.40

MAA Member Price: $124.20

# Bäcklund and Darboux Transformations. The Geometry of Solitons

Share this page *Edited by *
*Alan Coley; Decio Levi; Robert Milson; Colin Rogers; Pavel Winternitz*

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

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 co-published with the Centre de Recherches Mathématiques.

#### Readership

Graduate students and research mathematicians interested in dynamical systems, ergodic theory, and partial differential equations.

# Table of Contents

## Backlund and Darboux Transformations. The Geometry of Solitons

- Cover Cover11
- Title page ii3
- Contents iv5
- Preface viii9
- Albert Victor Bäcklund xii13
- Gaston Darboux xiv15
- List of participants xvi17
- Introductory Lectures 122
- Bäcklund transformations of the higher order Painlevé equations 324
- Dressing method and Bäcklund and Darboux transformations 2950
- The classical geometry of Bäcklund transformations. Introduction to applications in soliton theory 5374
- An introduction to integrable difference and differential geometries: Affine spheres, their natural generalization and discretization 6990
- Original Contributions 89110
- On Bianchi and Bäcklund transformations of two dimensional surfaces in four dimensional Euclidean space 91112
- Group invariant solutions without transversality and the principle of symmetric criticality 95116
- Multi-component matrix KP hierarchies as symmetry-enhanced scalar KP hierarchies and their Darboux-Bäcklund solutions 109130
- The Darboux-Bäcklund transformation and Clifford algebras 121142
- On discrete Painlevé equations as Bäcklund transformations 129150
- A Bäcklund transformation for timelike surfaces of constant mean curvature in ℝ^{1,2} 141162
- On Ribaucour transformations 151172
- The Ribaucour congruences of spheres within Lie sphere geometry 159180
- Bäcklund transformations for the reduced Maxwell-Bloch equations 167188
- Transformations of quasilinear systems originating from the projective theory of congruences 179200
- Bäcklund links between different analytic descriptions of constant mean curvature surfaces 191212
- On the integrability of Weingarten surfaces 199220
- A new immersion formula for surfaces on Lie algebras and integrable equations 207228
- Difficulties with the SDiff(2) Toda equation 217238
- Superposition formulas based on nonprimitive group action 225246
- Symmetries of differential difference equations and Lie algebra contractions 233254
- Bäcklund transformations from the bilinear viewpoint 245266
- Towards the Lax formulation of SU(2) principal models with nonconstant metric 253274
- Transcendental solutions of the sine-Gordon equation 261282
- Three dimensional skyrmions and harmonic maps 273294
- Induced surfaces and their integrable deformations 281302
- Properties of a class of slowly decaying oscillatory solutions of KdV 291312
- Blending two discrete integrability criteria: Singularity confinement and algebraic entropy 299320
- Bäcklund transformations of soliton systems from symmetry constraints 313334
- Open problems for the super KdV equations 325346
- Combinatorial aspects of the Darboux transformation 335356
- Bäcklund transformation and nonlinear superposition formula of the Kaup-Kupershmidt and Tzitzéica equations 345366
- Classification of symmetry-integrable evolution equations 363384
- Integrability of evolution equations and pseudo-spherical surfaces 373394
- Infinitesimal Bäcklund transformations of 𝐾-nets. The 2 + 1-dimensional Sinh-Gordon system 385406
- Isothermic surfaces and the Calapso equation: The full Monty 393414
- Bäcklund transformations induced by symmetries. Application: Discrete mKdV 405426
- Darboux transformation for a spectral problem quadratic in the spectral parameter 411432
- Separation of variables and Darboux transformations 419440
- Bäcklund transformations as nonlinear ordinary differential, or difference equations with superposition formulas 429450
- Back Cover Back Cover1458