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Backlund and Darboux Transformations : The Geometry of Solitons: AARMS-CRM Workshop, June 4-9, 1999, Halifax, N. S., Canada

By: Alan Coley, Decio Levi, Robert Milson, Colin Rogers, Pavel Winternitz

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Paperback | 1 October 2001

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436 Pages

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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 Backlund 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 Backlund transformations of Painleve equations, to the dressing methodand Backlund and Darboux transformations, and to the classical geometry of Backlund 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 introductorylectures 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 Backlund and Darboux transformations.

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Non-Fiction Mathematics Geometry Differential & Riemannian Geometry Science Physics Classical Mathematics Wave Mechanics including Vibration & Acoustics Applied Mathematics Analytic Geometry

Preface	p. ix
Albert Victor Backlund	p. xiii
Gaston Darboux	p. xv
List of Participants	p. xvii
Introductory Lectures
Backlund Transformations of the Higher Order Painleve Equations	p. 3
Dressing Method and Backlund and Darboux Transformations	p. 29
The Classical Geometry of Backlund Transformations. Introduction to Applications in Soliton Theory	p. 53
An Introduction to Integrable Difference and Differential Geometries: Affine Spheres, Their Natural Generalization and Discretization	p. 69
Original Contributions
On Bianchi and Backlund Transformations of Two Dimensional Surfaces in Four Dimensional Euclidean Space	p. 91
Group Invariant Solutions without Transversality and the Principle of Symmetric Criticality	p. 95
Multi-Component Matrix KP Hierarchies as Symmetry-Enhanced Scalar KP Hierarchies and Their Darboux-Backlund Solutions	p. 109
The Darboux-Backlund Transformation and Clifford Algebras	p. 121
On Discrete Painleve Equations as Backlund Transformations	p. 129
A Backlund Transformation for Timelike Surfaces of Constant Mean Curvature in R[superscript 1,2]	p. 141
On Ribaucour Transformations	p. 151
The Ribaucour Congruences of Spheres Within Lie Sphere Geometry	p. 159
Backlund Transformations for the Reduced Maxwell-Bloch Equations	p. 167
Transformations of Quasilinear Systems Originating from the Projective Theory of Congruences	p. 179
Backlund Links Between Different Analytic Descriptions of Constant Mean Curvature Surfaces	p. 191
On the Integrability of Weingarten Surfaces	p. 199
A New Immersion Formula for Surfaces on Lie Algebras and Integrable Equations	p. 207
Difficulties With the SDiff(2) Toda Equation	p. 217
Superposition Formulas Based on Nonprimitive Group Action	p. 225
Symmetries of Differential Difference Equations and Lie Algebra Contractions	p. 233
Backlund Transformations From the Bilinear Viewpoint	p. 245
Towards the Lax Formulation SU(2) Principal Models with Nonconstant Metric	p. 253
Transcendental Solutions of the Sine-Gordon Equation	p. 261
Three Dimensional Skyrmions and Harmonic Maps	p. 273
Induced Surfaces and Their Integrable Deformations	p. 281
Properties of a Class of Slowly Decaying Oscillatory Solutions of KdV	p. 291
Blending Two Discrete Integrability Criteria: Singularity Confinement and Algebraic Entropy	p. 299
Backlund Transformations of Soliton Systems from Symmetry Constraints	p. 313
Open Problems for the Super KdV Equations	p. 325
Combinatorial Aspects of the Darboux Transformation	p. 335
Backlund Transformation and Nonlinear Superposition Formula of the Kaup-Kupershmidt and Tzitzeica Equations	p. 345
Classification of Symmetry-Integrable Evolution Equations	p. 363
Integrability of Evolution Equations and Pseudo-Spherical Surfaces	p. 373
Infinitesimal Backlund Transformations of K-Nets. The 2 + 1-Dimensional Sinh-Gordon System	p. 385
Isothermic Surfaces and the Calapso Equation: The Full Monty	p. 393
Backlund Transformations Induced by Symmetries Application: Discrete mKdV	p. 405
Darboux Transformation for a Spectral Problem Quadratic in the Spectral Parameter	p. 411
Separation of Variables and Darboux Transformations	p. 419
Backlund Transformations as Nonlinear Ordinary Differential, or Difference Equations with Superposition Formulas	p. 429
Table of Contents provided by Syndetics. All Rights Reserved.

Backlund and Darboux Transformations : The Geometry of Solitons: AARMS-CRM Workshop, June 4-9, 1999, Halifax, N. S., Canada

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