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

by C. Rogers , W. K. Schief

• ISBN13: 9780521813310
• ISBN10: 052181331X
• Format: Hardcover
• Copyright: 2002-06-24
• Publisher: Cambridge University Press

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, 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.

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