Solitons, instantons, and twistors:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford [u.a.]
Oxford Univ. Press
2010
|
| Edition: | 1. publ. |
| Series: | Oxford graduate texts in mathematics
19 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 359 S. graph. Darst. |
| ISBN: | 9780198570622 9780198570639 |
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Record in the Search Index
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| adam_text | Titel: Solitons, instantons and twistors
Autor: Dunajski, Maciej
Jahr: 2010
Contents
List of Figures xii
List of Abbreviations xiii
1 Integrability in classical mechanics 1
1.1 Hamiltonian formalism 1
1.2 Integrability and action-angle variables 4
1.3 Poisson structures 14
2 Soliton equations and the inverse scattering transform 20
2.1 The history of two examples 20
2.1.1 A physical derivation of KdV 21
2.1.2 Backlund transformations for the Sine-Gordon equation 24
2.2 Inverse scattering transform for KdV 25
2.2.1 Direct scattering 28
2.2.2 Properties of the scattering data 29
2.2.3 Inverse scattering 30
2.2.4 Lax formulation 31
2.2.5 Evolution of the scattering data 32
2.3 Reflectionless potentials and solitons 33
2.3.1 One-soliton solution 34
2.3.2 N-soliton solution 35
2.3.3 Two-soliton asymptotics 36
3 Hamiltonian formalism and zero-curvature representation 43
3.1 First integrals 43
3.2 Hamiltonian formalism 46
3.2.1 Bi-Hamiltonian systems 46
Contents
3.3 Zero-curvature representation 48
3.3.1 Riemann-Hilbert problem 50
3.3.2 Dressing method 52
3.3.3 From Lax representation to zero curvature 54
3.4 Hierarchies and finite-gap solutions 56
4 Lie symmetries and reductions 64
4.1 Lie groups and Lie algebras 64
4.2 Vector fields and one-parameter groups of
transformations 67
4.3 Symmetries of differential equations 71
4.3.1 How to find symmetries 74
4.3.2 Prolongation formulae 75
4.4 Painleve equations 78
4.4.1 Painleve test 82
5 Lagrangian formalism and field theory 85
5.1 A variational principle 85
5.1.1 Legendre transform 87
5.1.2 Symplectic structures 88
5.1.3 Solution space 89
5.2 Field theory 90
5.2.1 Solution space and the geodesic
approximation 92
5.3 Scalar kinks 93
5.3.1 Topology and Bogomolny equations 96
5.3.2 Higher dimensions and a scaling argument 98
5.3.3 Homotopy in field theory 99
5.4 Sigma model lumps 100
6 Gauge field theory 105
6.1 Gauge potential and Higgs field 106
6.1.1 Scaling argument 108
6.1.2 Principal bundles 109
6.2 Dirac monopole and flux quantization 110
6.2.1 Hopf fibration 112
6.3 Non-abelian monopoles 114
6.3.1 Topology of monopoles 115
6.3.2 Bogomolny-Prasad-Sommerfeld (BPS) limit 116
Contents
6.4 Yang-Mills equations and instantons 119
6.4.1 Chern and Chern-Simons forms 120
6.4.2 Minimal action solutions and the anti-self-duality
condition 122
6.4.3 Ansatz for ASD fields 123
6.4.4 Gradient flow and classical mechanics 124
7 Integrability of ASDYM and twistor theory 129
7.1 Lax pair 129
7.1.1 Geometric interpretation 132
7.2 Twistor correspondence 133
7.2.1 History and motivation 133
7.2.2 Spinor notation 137
7.2.3 Twistor space 139
7.2.4 Penrose-Ward correspondence 141
8 Symmetry reductions and the integrable chiral model 149
8.1 Reductions to integrable equations 149
8.2 Integrable chiral model 154
8.2.1 Soliton solutions 157
8.2.2 Lagrangian formulation 165
8.2.3 Energy quantization of time-dependent unitons 168
8.2.4 Moduli space dynamics 173
8.2.5 Mini-twistors 181
9 Gravitational instantons 191
9.1 Examples of gravitational instantons 191
9.2 Anti-self-duality in Riemannian geometry 195
9.2.1 Two-component spinors in Riemannian signature 198
9.3 Hyper-Kahler metrics 202
9.4 Multi-centred gravitational instantons 206
9.4.1 Belinskii-Gibbons-Page-Pope class 210
9.5 Other gravitational instantons 212
9.5.1 Compact gravitational instantons and K3 215
9.6 Einstein-Maxwell gravitational instantons 216
9.7 Kaluza-Klein monopoles 221
9.7.1 Kaluza-Klein solitons from Einstein-Maxwell
instantons 222
9.7.2 Solitons in higher dimensions 226
Contents
10 Anti-self-dual conformal structures 229
10.1 o(-surfaces and anti-self-duality 230
10.2 Curvature restrictions and their Lax pairs 231
10.2.1 Hyper-Hermitian structures 232
10.2.2 ASD Kahler structures 234
10.2.3 Null-Kahler structures 236
10.2.4 ASD Einstein structures 237
10.2.5 Hyper-Kahler structures and heavenly equations 238
10.3 Symmetries 246
10.3.1 Einstein-Weyl geometry 246
10.3.2 Null symmetries and projective structures 253
10.3.3 Dispersionless integrable systems 256
10.4 ASD conformal structures in neutral signature 262
10.4.1 Conformal compactification 263
10.4.2 Curved examples 263
10.5 Twistor theory 265
10.5.1 Curvature restrictions 270
10.5.2 ASD Ricci-flat metrics 272
10.5.3 Twistor theory and symmetries 283
Appendix A: Manifolds and topology 287
A.1 Lie groups 290
A.2 Degree of a map and homotopy 294
A.2.1 Homotopy 296
A.2.2 Hermitian projectors 298
Appendix B: Complex analysis 300
B.1 Complex manifolds 301
B.2 Holomorphic vector bundles and their sections 303
B.3 Cech cohomology 307
B.3.1 Deformation theory 308
Appendix C: Overdetermined PDEs 310
C.1 Introduction 310
C.2 Exterior differential system and Frobenius theorem 314
C.3 Involutivity 320
Contents
C.4 Prolongation 324
C.4.1 Differential invariants 326
C.5 Method of characteristics 332
C.6 Cartan-Kahler theorem 335
References 344
Index 355
|
| any_adam_object | 1 |
| author | Dunajski, Maciej |
| author_GND | (DE-588)140454047 |
| author_facet | Dunajski, Maciej |
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| dewey-search | 530.12/4 |
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| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| edition | 1. publ. |
| format | Book |
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| spelling | Dunajski, Maciej Verfasser (DE-588)140454047 aut Solitons, instantons, and twistors Maciej Dunajski 1. publ. Oxford [u.a.] Oxford Univ. Press 2010 XI, 359 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Oxford graduate texts in mathematics 19 Mathematik Solitons Mathematics Geometry, Differential Wave-motion, Theory of Twistor theory Konforme Struktur (DE-588)4500911-9 gnd rswk-swf Eichtheorie (DE-588)4122125-4 gnd rswk-swf Soliton (DE-588)4135213-0 gnd rswk-swf Twistor (DE-588)4186504-2 gnd rswk-swf Instanton (DE-588)4161874-9 gnd rswk-swf Soliton (DE-588)4135213-0 s Instanton (DE-588)4161874-9 s Twistor (DE-588)4186504-2 s Eichtheorie (DE-588)4122125-4 s Konforme Struktur (DE-588)4500911-9 s DE-604 Oxford graduate texts in mathematics 19 (DE-604)BV011416591 19 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018908797&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dunajski, Maciej Solitons, instantons, and twistors Oxford graduate texts in mathematics Mathematik Solitons Mathematics Geometry, Differential Wave-motion, Theory of Twistor theory Konforme Struktur (DE-588)4500911-9 gnd Eichtheorie (DE-588)4122125-4 gnd Soliton (DE-588)4135213-0 gnd Twistor (DE-588)4186504-2 gnd Instanton (DE-588)4161874-9 gnd |
| subject_GND | (DE-588)4500911-9 (DE-588)4122125-4 (DE-588)4135213-0 (DE-588)4186504-2 (DE-588)4161874-9 |
| title | Solitons, instantons, and twistors |
| title_auth | Solitons, instantons, and twistors |
| title_exact_search | Solitons, instantons, and twistors |
| title_full | Solitons, instantons, and twistors Maciej Dunajski |
| title_fullStr | Solitons, instantons, and twistors Maciej Dunajski |
| title_full_unstemmed | Solitons, instantons, and twistors Maciej Dunajski |
| title_short | Solitons, instantons, and twistors |
| title_sort | solitons instantons and twistors |
| topic | Mathematik Solitons Mathematics Geometry, Differential Wave-motion, Theory of Twistor theory Konforme Struktur (DE-588)4500911-9 gnd Eichtheorie (DE-588)4122125-4 gnd Soliton (DE-588)4135213-0 gnd Twistor (DE-588)4186504-2 gnd Instanton (DE-588)4161874-9 gnd |
| topic_facet | Mathematik Solitons Mathematics Geometry, Differential Wave-motion, Theory of Twistor theory Konforme Struktur Eichtheorie Soliton Twistor Instanton |
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