Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory

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Cambridge University Press, 24 de juny 2002 - 413 pàgines
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, Bäcklund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Bäcklund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory.
 

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Continguts

Pseudospherical Surfaces and the Classical Backlund Transformation The Bianchi System
17
The GaussWeingarten Equations for Hyperbolic Surfaces Pseudospherical Surfaces The SineGordon Equation
18
The Classical Backlund Transformation for the SineGordon Equation
22
Bianchis Permutability Theorem Generation of MultiSoliton Solutions
28
132 Physical Applications
30
Pseudospherical Soliton Surfaces Breathers
31
141 The Pseudosphere
32
142 A Pseudospherical Helicoid
35
Particular Isothermic Surfaces OneSoliton Surfaces and Cyclides
191
582 A Class of Solutions Generated by the Moutard Transformation
192
583 Dupin Cyclides
198
General Aspects ofSoliton Surfaces Role of Gauge and Reciprocal Transformations
204
The AKNS 2x2 Spectral System 611 The Position Vector of Pseudospherical Surfaces
205
612 The su2 Linear Representation and Its Associated Soliton Surfaces The AKNS Case r q
209
NLS Eigenfunction Hierarchies Geometric Properties The Miura Transformation
216
621 Soliton Surface Position Vectors as Solutions of Eigenfunction Equations
217

143 TwoSoliton Surfaces
37
144 Breathers
38
145 Stationary Breather Surfaces
39
Parallel Surfaces Induced Bãcklund Transformation for a Class of Weingarten Surfaces
41
151 Surfaces of Constant Mean Curvature A Theorem of Bonnet
42
152 An Induced Bdcklund Transformation
43
The Bianchi System Its AutoBãcklund Transformation
45
161 Hyperbolic Surfaces Spherical Representation
46
162 A Bãcklund Transformation for Hyperbolic Surfaces
49
163 The Bianchi System
53
The Motion of Curves and Surfaces Soliton Connections
60
Motions of Curves of Constant Torsion or Curvature The SineGordon Connection
61
211 A Motion of an Inextensible Curve of Constant Torsion
62
212 A Motion of an Inextensible Curve of Constant Curvature
63
A 2 x 2 Linear Representation for the SineGordon Equation
64
The Motion of Pseudospherical Surfaces A Weingarten System and Its Bãcklund Transformation
68
231 A Continuum Limit of an Anharmonic Lattice Model
71
233 Bãcklund Transformations
73
24 The mKdV Equation Moving Curve and Soliton Surface Representations A Solitonic Weingarten System
80
242 Motion of a Dini Surface
82
243 A Triply Orthogonal Weingarten System
85
Tzitzeica Surfaces Conjugate Nets and the Toda Lattice Scheme
88
31 Tzitzeica Surfaces Link to an Integrable Gasdynamics System
89
312 The Affinspharen Equation in a Gasdynamics Context
95
Construction ofTzitzeica Surfaces An Induced Bãcklund Transformation
101
LaplaceDarboux Transformations The TwoDimensional Toda Lattice Conjugate Nets
109
331 LaplaceDarboux Transformations
110
332 Iteration of LaplaceDarboux Transformations The TwoDimensional Toda Lattice
111
Its Linear Representation and Backlund Transformation
113
334 Conjugate Nets
117
Hasimoto Surfaces and the Nonlinear Schrodinger Equation Geometry and Associated Soliton Equations
119
Binormal Motion and the Nonlinear Schrodinger Equation The Heisenberg Spin Equation
120
411 A Single Soliton NLS Surface
122
412 Geometric Properties
124
413 The Heisenberg Spin Equation
128
The PohlmeyerLundRegge Model SIT and SRS Connections Compatibility with the NLS Equation
129
421 The PohlmeyerLundRegge Model
130
422 The SIT Connection
132
423 The SRS Connection
134
424 Compatibility of the MaxwellBloch System with the NLS Equation
135
Geometry of the NLS Equation The AutoBãcklund Transformation
137
431 The Nonlinear Schrodinger Equation
142
432 The AutoBacklund Transformation
146
Isothermic Surfaces The Calapso and Zoomeron Equations
152
The Geometry of Isothermic Surfaces in Rn+2
156
527 Conjugate and Orthogonal Coordinates
157
522 Isothermic Surfaces
159
523 Specialisations and Generalisations
160
The Vector Calapso System Its Scalar Lax Pair
162
532 A Scalar Lax Pair
164
533 Reductions
166
The Fundamental Transformation
167
542 The Radial Transformation
168
543 The Fundamental Transformation
169
A Backlund Transformation for Isothermic Surfaces
171
552 The Ribaucour Transformation
173
553 A Backlund Transformation for Isothermic Surfaces
175
Permutability Theorems and Their Geometric Implications
178
562 A Permutability Theorem for Orthogonal Conjugate Nets Cyclicity
181
563 A Permutability Theorem for Isothermic Surfaces Constant CrossRatio
184
An Explicit Permutability Theorem for the Vector Calapso System
187
572 A Permutability Theorem
189
622 The SerretFrenet Equations and the NLS Hierarchy
220
Reciprocal Transformations Loop Solitons
222
632 Loop Solitons
225
The Dym mKdV and KdV Hierarchies Connections
229
Curve Motions
230
642 The Dym mKdV and KdV Hierarchies
233
643 A Permutability Theorem
235
644 A Geometric Derivation of the mKdV Hierarchy
237
The Binormal Motion of Curves of Constant Curvature Extended Dym Surfaces
240
651 Curves of Constant Curvature
242
652 Extended Dym Surfaces The su2 Linear Representation
246
653 A CCIdeal Formulation
249
654 A Matrix Darboux Transformation A Backlund Transformation for the Extended Dym and m2KdV Equations
252
655 Soliton Surfaces
255
The Binormal Motion of Curves of Constant Torsion The Extended SineGordon System
258
661 The Extended SineGordon System
259
662 Fundamental Forms An su2 Linear Representation
260
663 A Backlund Transformation
262
664 An Analogue of the Bianchi Transformation Dual Surfaces
263
Bãcklund Transformation and Darboux Matrix Connections
266
The Connection for Pseudospherical and Nonlinear Schrodinger Surfaces
267
712 NLS Surfaces
271
Darboux Matrix and Induced Backlund Transformations for the AKNS System The Constant Length Property
277
722 Invariance of a su2 Constraint
280
723 The AKNS Class r q and Its Elementary Backlund Transformation
282
724 The Constant Length Property
285
Iteration of Matrix Darboux Transformations Generic Permutability Theorems
287
731 Iteration of Matrix Darboux Transformations
288
732 Generic Permutability Theorems
292
Bianchi and Ernst Systems Bãcklund Transformations and Permutability Theorems
297
Bianchi Surfaces Application of the SymTafei Formula
298
Matrix Darboux Transformations for NonIsospectral Linear Representations
300
Invariance of the su2 Constraint A Distance Property
302
The Ernst Equation of General Relativity
303
841 Linear Representations
305
842 The Dual Ernst Equation
306
The Ehlers and MatznerMisner Transformations
309
The Neugebauer and Harrison Backlund Transformations
311
A Matrix Darboux Transformation for the Ernst Equation
319
A Permutability Theorem for the Ernst Equation and Its Dual A Classical Bianchi Connection
324
ProjectiveMinimal and IsothermalAsymptotic Surfaces
329
Analogues of the GaussMainardiCodazzi Equations in Projective Differential Geometry
330
ProjectiveMinimal GodeauxRozet and Demoulin Surfaces
333
Linear Representations
335
931 The Wilczynski Tetrahedral and a 4 x 4 Linear Representation
336
932 The Pliicker Correspondence and a 6 X 6 Linear Representation
337
The Demoulin System as a Periodic Toda Lattice
341
A Bãcklund Transformation for ProjectiveMinimal Surfaces
343
951 Invariance of the so3 3 Linear Representation
345
952 Invariance of the sl4 Linear Representation
350
OneSoliton Demoulin Surfaces
353
IsothermalAsymptotic Surfaces The Stationary mNVN Equation
357
977 The Stationary mNVN Equation
358
972 The Stationary NVN Equation
360
A Bãcklund Transformation for IsothermalAsymptotic Surfaces
365
982 An Invariance of the NVNEquation and a Bãcklund Transformation for IsothermalAsymptotic Surfaces
367
The su2so3 Isomorphism
371
CCIdeals
374
Biographies
380
Bibliography and Author Index
383
Subject Index
403
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