Backlund and Darboux Transformations

Geometry and Modern Applications in Soliton Theory

By: C. Rogers, W. K. Schief, M. J. Ablowitz (Editor)

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Hardcover | 25 September 2002

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432 Pages

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This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Backlund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Backlund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory. It is with these transformations and the links they afford between the classical differential geometry of surfaces and the nonlinear equations of soliton theory that the present text is concerned. In this geometric context, solitonic equations arise out of the Gauss-Mainardi-Codazzi equations for various types of surfaces that admit invariance under Backlund-Darboux transformations. This text is appropriate for use at a higher undergraduate or graduate level for applied mathematicians or mathematical physics.

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'It is an excellent book for graduate students and young researchers ... very useful for scientists in this field.' Nieuw Archief voor Wiskunde 'The book certainly is a recommendable book for everyone who is interested in these transformations as well as in the related geometry and modern applications.' Bulletin of the Belgian Mathematical Society

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

Preface	p. xv
Acknowledgements	p. xvii
General Introduction and Outline	p. 1
Pseudospherical Surfaces and the Classical Backlund Transformation. The Bianchi System	p. 17
The Gauss-Weingarten Equations for Hyperbolic Surfaces. Pseudospherical Surfaces. The Sine-Gordon Equation	p. 18
The Classical Backlund Transformation for the Sine-Gordon Equation	p. 22
Bianchi's Permutability Theorem. Generation of Multi-Soliton Solutions	p. 28
Bianchi's Permutability Theorem	p. 28
Physical Applications	p. 30
Pseudospherical Soliton Surfaces. Breathers	p. 31
The Pseudosphere	p. 32
A Pseudospherical Helicoid	p. 35
Two-Soliton Surfaces	p. 37
Breathers	p. 38
Stationary Breather Surfaces	p. 39
Parallel Surfaces. Induced Backlund Transformation for a Class of Weingarten Surfaces	p. 41
Surfaces of Constant Mean Curvature. A Theorem of Bonnet	p. 42
An Induced Backlund Transformation	p. 43
The Bianchi System. Its Auto-Backlund Transformation	p. 45
Hyperbolic Surfaces. Spherical Representation	p. 46
A Backlund Transformation for Hyperbolic Surfaces	p. 49
The Bianchi System	p. 53
The Motion of Curves and Surfaces. Soliton Connections	p. 60
Motions of Curves of Constant Torsion or Curvature. The Sine-Gordon Connection	p. 61
A Motion of an Inextensible Curve of Constant Torsion	p. 62
A Motion of an Inextensible Curve of Constant Curvature	p. 63
A 2 x 2 Linear Representation for the Sine-Gordon Equation	p. 64
The Motion of Pseudospherical Surfaces. A Weingarten System and Its Backlund Transformation	p. 68
A Continuum Limit of an Anharmonic Lattice Model	p. 71
A Weingarten System	p. 71
Backlund Transformations	p. 73
The mKdV Equation. Moving Curve and Soliton Surface Representations. A Solitonic Weingarten System	p. 80
The mKdV Equation	p. 80
Motion of a Dini Surface	p. 82
A Triply Orthogonal Weingarten System	p. 85
Tzitzeica Surfaces. Conjugate Nets and the Toda Lattice Scheme	p. 88
Tzitzeica Surfaces. Link to an Integrable Gasdynamics System	p. 89
The Tzitzeica and Affinspharen Equations	p. 89
The Affinspharen Equation in a Gasdynamics Context	p. 95
Construction of Tzitzeica Surfaces. An Induced Backlund Transformation	p. 101
Laplace-Darboux Transformations. The Two-Dimensional Toda Lattice. Conjugate Nets	p. 109
Laplace-Darboux Transformations	p. 110
Iteration of Laplace-Darboux Transformations. The Two-Dimensional Toda Lattice	p. 111
The Two-Dimensional Toda Lattice: Its Linear Representation and Backlund Transformation	p. 113
Conjugate Nets	p. 117
Hasimoto Surfaces and the Nonlinear Schrodinger Equation. Geometry and Associated Soliton Equations	p. 119
Binormal Motion and the Nonlinear Schrodinger Equation. The Heisenberg Spin Equation	p. 120
A Single Soliton NLS Surface	p. 122
Geometric Properties	p. 124
The Heisenberg Spin Equation	p. 128
The Pohlmeyer-Lund-Regge Model. SIT and SRS Connections. Compatibility with the NLS Equation	p. 129
The Pohlmeyer-Lund-Regge Model	p. 130
The SIT Connection	p. 132
The SRS Connection	p. 134
Compatibility of the Maxwell-Bloch System with the NLS Equation	p. 135
Geometry of the NLS Equation. The Auto-Backlund Transformation	p. 137
The Nonlinear Schrodinger Equation	p. 142
The Auto-Backlund Transformation	p. 146
Isothermic Surfaces. The Calapso and Zoomeron Equations	p. 152
The Gauss-Mainardi-Codazzi Equations for Isothermic Surfaces. The Calapso Equation. Dual Isothermic Surfaces	p. 152
The Geometry of Isothermic Surfaces in R[superscript n+2]	p. 156
Conjugate and Orthogonal Coordinates	p. 157
Isothermic Surfaces	p. 159
Specialisations and Generalisations	p. 160
The Vector Calapso System. Its Scalar Lax Pair	p. 162
The Vector Calapso System	p. 162
A Scalar Lax Pair	p. 164
Reductions	p. 166
The Fundamental Transformation	p. 167
Parallel Nets. The Combescure Transformation	p. 167
The Radial Transformation	p. 168
The Fundamental Transformation	p. 169
A Backlund Transformation for Isothermic Surfaces	p. 171
The Fundamental Transformation for Conjugate Coordinates	p. 171
The Ribaucour Transformation	p. 173
A Backlund Transformation for Isothermic Surfaces	p. 175
Permutability Theorems and Their Geometric Implications	p. 178
A Permutability Theorem for Conjugate Nets. Planarity	p. 178
A Permutability Theorem for Orthogonal Conjugate Nets. Cyclicity	p. 181
A Permutability Theorem for Isothermic Surfaces. Constant Cross-Ratio	p. 184
An Explicit Permutability Theorem for the Vector Calapso System	p. 187
The Ribaucour-Moutard Connection	p. 187
A Permutability Theorem	p. 189
Particular Isothermic Surfaces. One-Soliton Surfaces and Cyclides	p. 191
One-Soliton Isothermic Surfaces	p. 191
A Class of Solutions Generated by the Moutard Transformation	p. 192
Dupin Cyclides	p. 198
General Aspects of Soliton Surfaces. Role of Gauge and Reciprocal Transformations	p. 204
The AKNS 2 x 2 Spectral System	p. 205
The Position Vector of Pseudospherical Surfaces	p. 205
The su(2) Linear Representation and Its Associated Soliton Surfaces. The AKNS Case r = -q	p. 209
NLS Eigenfunction Hierarchies. Geometric Properties. The Miura Transformation	p. 216
Soliton Surface Position Vectors as Solutions of Eigenfunction Equations	p. 217
The Serret-Frenet Equations and the NLS Hierarchy	p. 220
Reciprocal Transformations. Loop Solitons	p. 222
Reciprocal Transformations and the Loop Soliton Equation	p. 222
Loop Solitons	p. 225
The Dym, mKdV, and KdV Hierarchies. Connections	p. 229
Invariance under Reciprocal Transformations. A Class of Planar Curve Motions	p. 230
The Dym, mKdV, and KdV Hierarchies	p. 233
A Permutability Theorem	p. 235
A Geometric Derivation of the mKdV Hierarchy	p. 237
The Binormal Motion of Curves of Constant Curvature. Extended Dym Surfaces	p. 240
Curves of Constant Curvature	p. 242
Extended Dym Surfaces. The su(2) Linear Representation	p. 246
A CC-Ideal Formulation	p. 249
A Matrix Darboux Transformation. A Backlund Transformation for the Extended Dym and m[superscript 2]KdV Equations	p. 252
Solition Surfaces	p. 255
The Binormal Motion of Curves of Constant Torsion. The Extended Sine-Gordon System	p. 258
The Extended Sine-Gordon System	p. 259
Fundamental Forms. An su(2) Linear Representation	p. 260
A Backlund Transformation	p. 262
An Analogue of the Bianchi Transformation. Dual Surfaces	p. 263
Backlund Transformation and Darboux Matrix Connections	p. 266
The Connection for Pseudospherical and Nonlinear Schrodinger Surfaces	p. 267
Pseudospherical Surfaces	p. 267
NLS Surfaces	p. 271
Darboux Matrix and Induced Backlund Transformations for the AKNS System. The Constant Length Property	p. 277
An Elementary Matrix Darboux Transformation	p. 277
Invariance of a su(2) Constraint	p. 280
The AKNS Class r = -q and Its Elementary Backlund Transformation	p. 282
The Constant Length Property	p. 285
Iteration of Matrix Darboux Transformations. Generic Permutability Theorems	p. 287
Iteration of Matrix Darboux Transformations	p. 288
Generic Permutability Theorems	p. 292
Bianchi and Ernst Systems. Backlund Transformations and Permutability Theorems	p. 297
Bianchi Surfaces. Application of the Sym-Tafel Formula	p. 298
Matrix Darboux Transformations for Non-Isospectral Linear Representations	p. 300
Invariance of the su(2) Constraint. A Distance Property	p. 302
The Ernst Equation of General Relativity	p. 303
Linear Representations	p. 305
The Dual 'Ernst Equation'	p. 306
The Ehlers and Matzner-Misner Transformations	p. 309
The Neugebauer and Harrison Backlund Transformations	p. 311
A Matrix Darboux Transformation for the Ernst Equation	p. 319
A Permutability Theorem for the Ernst Equation and Its Dual. A Classical Bianchi Connection	p. 324
Projective-Minimal and Isothermal-Asymptotic Surfaces	p. 329
Analogues of the Gauss-Mainardi-Codazzi Equations in Projective Differential Geometry	p. 330
Projective-Minimal, Godeaux-Rozet, and Demoulin Surfaces	p. 333
Linear Representations	p. 335
The Wilczynski Tetrahedral and a 4 x 4 Linear Representation	p. 336
The Plucker Correspondence and a 6 x 6 Linear Representation	p. 337
The Demoulin System as a Periodic Toda Lattice	p. 341
A Backlund Transformation for Projective-Minimal Surfaces	p. 343
Invariance of the so(3, 3) Linear Representation	p. 345
Invariance of the sl(4) Linear Representation	p. 350
One-Soliton Demoulin Surfaces	p. 353
Isothermal-Asymptotic Surfaces. The Stationary mNVN Equation	p. 357
The Stationary mNVN Equation	p. 358
The Stationary NVN Equation	p. 360
A Backlund Transformation for Isothermal-Asymptotic Surfaces	p. 365
An Invariance of the mNVN Equation	p. 365
An Invariance of the NVN Equation and a Backlund Transformation for Isothermal-Asymptotic Surfaces	p. 367
The su(2)-so(3) Isomorphism	p. 371
CC-Ideals	p. 374
Biographies	p. 380
Bibliography and Author Index	p. 383
Subject Index	p. 403
Table of Contents provided by Syndetics. All Rights Reserved.

Backlund and Darboux Transformations

Geometry and Modern Applications in Soliton Theory

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