9780521813310

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

by
  • ISBN13:

    9780521813310

  • ISBN10:

    052181331X

  • Format: Hardcover
  • Copyright: 2002-06-24
  • Publisher: Cambridge University Press
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Summary

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, Bcklund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Bcklund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory.

Table of Contents

Preface xv
Acknowledgements xvii
General Introduction and Outline 1(370)
Pseudospherical Surfaces and the Classical Backlund Transformation. The Bianchi System
17(43)
The Gauss-Weingarten Equations for Hyperbolic Surfaces. Pseudospherical Surfaces. The Sine-Gordon Equation
18(4)
The Classical Backlund Transformation for the Sine-Gordon Equation
22(6)
Bianchi's Permutability Theorem. Generation of Multi-Soliton Solutions
28(3)
Bianchi's Permutability Theorem
28(2)
Physical Applications
30(1)
Pseudospherical Soliton Surfaces. Breathers
31(10)
The Pseudosphere
32(3)
A Pseudospherical Helicoid
35(2)
Two-Soliton Surfaces
37(1)
Breathers
38(1)
Stationary Breather Surfaces
39(2)
Parallel Surfaces. Induced Backlund Transformation for a Class of Weingarten Surfaces
41(4)
Surfaces of Constant Mean Curvature. A Theorem of Bonnet
42(1)
An Induced Backlund Transformation
43(2)
The Bianchi System. Its Auto-Backlund Transformation
45(15)
Hyperbolic Surfaces. Spherical Representation
46(3)
A Backlund Transformation for Hyperbolic Surfaces
49(4)
The Bianchi System
53(7)
The Motion of Curves and Surfaces. Soliton Connections
60(28)
Motions of Curves of Constant Torsion or Curvature. The Sine-Gordon Connection
61(3)
A Motion of an Inextensible Curve of Constant Torsion
62(1)
A Motion of an Inextensible Curve of Constant Curvature
63(1)
A 2 x 2 Linear Representation for the Sine-Gordon Equation
64(4)
The Motion of Pseudospherical Surfaces. A Weingarten System and Its Backlund Transformation
68(12)
A Continuum Limit of an Anharmonic Lattice Model
71(1)
A Weingarten System
71(2)
Backlund Transformations
73(7)
The mKdV Equation. Moving Curve and Soliton Surface Representations. A Solitonic Weingarten System
80(8)
The mKdV Equation
80(2)
Motion of a Dini Surface
82(3)
A Triply Orthogonal Weingarten System
85(3)
Tzitzeica Surfaces. Conjugate Nets and the Toda Lattice Scheme
88(31)
Tzitzeica Surfaces. Link to an Integrable Gasdynamics System
89(12)
The Tzitzeica and Affinspharen Equations
89(6)
The Affinspharen Equation in a Gasdynamics Context
95(6)
Construction of Tzitzeica Surfaces. An Induced Backlund Transformation
101(8)
Laplace-Darboux Transformations. The Two-Dimensional Toda Lattice. Conjugate Nets
109(10)
Laplace-Darboux Transformations
110(1)
Iteration of Laplace-Darboux Transformations. The Two-Dimensional Toda Lattice
111(2)
The Two-Dimensional Toda Lattice: Its Linear Representation and Backlund Transformation
113(4)
Conjugate Nets
117(2)
Hasimoto Surfaces and the Nonlinear Schrodinger Equation. Geometry and Associated Soliton Equations
119(33)
Binormal Motion and the Nonlinear Schrodinger Equation. The Heisenberg Spin Equation
120(9)
A Single Soliton NLS Surface
122(2)
Geometric Properties
124(4)
The Heisenberg Spin Equation
128(1)
The Pohlmeyer-Lund-Regge Model. SIT and SRS Connections. Compatibility with the NLS Equation
129(8)
The Pohlmeyer-Lund-Regge Model
130(2)
The SIT Connection
132(2)
The SRS Connection
134(1)
Compatibility of the Maxwell-Bloch System with the NLS Equation
135(2)
Geometry of the NLS Equation. The Auto-Backlund Transformation
137(15)
The Nonlinear Schrodinger Equation
142(4)
The Auto-Backlund Transformation
146(6)
Isothermic Surfaces. The Calapso and Zoomeron Equations
152(52)
The Gauss-Mainardi-Codazzi Equations for Isothermic Surfaces. The Calapso Equation. Dual Isothermic Surfaces
152(4)
The Geometry of Isothermic Surfaces in Rn+2
156(6)
Conjugate and Orthogonal Coordinates
157(2)
Isothermic Surfaces
159(1)
Specialisations and Generalisations
160(2)
The Vector Calapso System. Its Scalar Lax Pair
162(5)
The Vector Calapso System
162(2)
A Scalar Lax Pair
164(2)
Reductions
166(1)
The Fundamental Transformation
167(4)
Parallel Nets. The Combescure Transformation
167(1)
The Radial Transformation
168(1)
The Fundamental Transformation
169(2)
A Backlund Transformation for Isothermic Surfaces
171(7)
The Fundamental Transformation for Conjugate Coordinates
171(2)
The Ribaucour Transformation
173(2)
A Backlund Transformation for Isothermic Surfaces
175(3)
Permutability Theorems and Their Geometric Implications
178(9)
A Permutability Theorem for Conjugate Nets. Planarity
178(3)
A Permutability Theorem for Orthogonal Conjugate Nets. Cyclicity
181(3)
A Permutability Theorem for Isothermic Surfaces. Constant Cross-Ratio
184(3)
An Explicit Permutability Theorem for the Vector Calapso System
187(4)
The Ribaucour-Moutard Connection
187(2)
A Permutability Theorem
189(2)
Particular Isothermic Surfaces. One-Soliton Surfaces and Cyclides
191(13)
One-Soliton Isothermic Surfaces
191(1)
A Class of Solutions Generated by the Moutard Transformation
192(6)
Dupin Cyclides
198(6)
General Aspects of Soliton Surfaces. Role of Gauge and Reciprocal Transformations
204(62)
The AKNS 2 x 2 Spectral System
205(11)
The Position Vector of Pseudospherical Surfaces
205(4)
The su(2) Linear Representation and Its Associated Soliton Surfaces. The AKNS Case r = -q
209(7)
NLS Eigenfunction Hierarchies. Geometric Properties. The Miura Transformation
216(6)
Soliton Surface Position Vectors as Solutions of Eigenfunction Equations
217(3)
The Serret-Frenet Equations and the NLS Hierarchy
220(2)
Reciprocal Transformations. Loop Solitons
222(7)
Reciprocal Transformations and the Loop Soliton Equation
222(3)
Loop Solitons
225(4)
The Dym, mKdV, and KdV Hierarchies. Connections
229(11)
Invariance under Reciprocal Transformations. A Class of Planar Curve Motions
230(3)
The Dym, mKdV and KdV Hierarchies
233(2)
A Permutability Theorem
235(2)
A Geometric Derivation of the mKdV Hierarchy
237(3)
The Binormal Motion of Curves of Constant Curvature. Extended Dym Surfaces
240(18)
Curves of Constant Curvature
242(4)
Extended Dym Surfaces. The su(2) Linear Representation
246(3)
A CC-Ideal Formulation
249(3)
A Matrix Darboux Transformation. A Backlund Transformation for the Extended Dym and m2KdV Equations
252(3)
Soliton Surfaces
255(3)
The Binormal Motion of Curves of Constant Torsion. The Extended Sine-Gordon System
258(8)
The Extended Sine-Gordon System
259(1)
Fundamental Forms. An su(2) Linear Representation
260(2)
A Backlund Transformation
262(1)
An Analogue of the Bianchi Transformation. Dual Surfaces
263(3)
Backlund Transformation and Darboux Matrix Connections
266(31)
The Connection for Pseudospherical and Nonlinear Schrodinger Surfaces
267(10)
Pseudospherical Surfaces
267(4)
NLS Surfaces
271(6)
Darboux Matrix and Induced Backlund Transformations for the AKNS System. The Constant Length Property
277(10)
An Elementary Matrix Darboux Transformation
277(3)
Invariance of a su(2) Constraint
280(2)
The AKNS Class r = -q and Its Elementary Backlund Transformation
282(3)
The Constant Length Property
285(2)
Iteration of Matrix Darboux Transformations. Generic Permutability Theorems
287(10)
Iteration of Matrix Darboux Transformations
288(4)
Generic Permutability Theorems
292(5)
Bianchi and Ernst Systems. Backlund Transformations and Permutability Theorems
297(32)
Bianchi Surfaces. Application of the Sym-Tafel Formula
298(2)
Matrix Darboux Transformations for Non-Isospectral Linear Representations
300(2)
Invariance of the su(2) Constraint. A Distance Property
302(1)
The Ernst Equation of General Relativity
303(6)
Linear Representations
305(1)
The Dual `Ernst Equation'
306(3)
The Ehlers and Matzner-Misner Transformations
309(2)
The Neugebauer and Harrison Backlund Transformations
311(8)
A Matrix Darboux Transformation for the Ernst Equation
319(5)
A Permutability Theorem for the Ernst Equation and Its Dual. A Classical Bianchi Connection
324(5)
Projective-Minimal and Isothermal-Asymptotic Surfaces
329(42)
Analogues of the Gauss-Mainardi-Codazzi Equations in Projective Differential Geometry
330(3)
Projective-Minimal, Godeaux-Rozet, and Demoulin Surfaces
333(2)
Linear Representations
335(6)
The Wilczynski Tetrahedral and a 4 x 4 Linear Representation
336(1)
The Plucker Correspondence and a 6 x 6 Linear Representation
337(4)
The Demoulin System as a Periodic Toda Lattice
341(2)
A Backlund Transformation for Projective-Minimal Surfaces
343(10)
Invariance of the so(3, 3) Linear Representation
345(5)
Invariance of the sl(4) Linear Representation
350(3)
One-Soliton Demoulin Surfaces
353(4)
Isothermal-Asymptotic Surfaces. The Stationary mNVN Equation
357(8)
The Stationary mNVN Equation
358(2)
The Stationary NVN Equation
360(5)
A Backlund Transformation for Isothermal-Asymptotic Surfaces
365(6)
An Invariance of the mNVN Equation
365(2)
An Invariance of the NVN Equation and a Backlund Transformation for Isothermal-Asymptotic Surfaces
367(4)
Appendix A The su(2)-so(3) Isomorphism 371(3)
Appendix B CC-Ideals 374(6)
Appendix C Biographies 380(3)
Bibliography and Author Index 383(20)
Subject Index 403

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