 | Dunajski M. Solitons, instantons, and twistors. - Oxford: Oxford University Press, 2010. - xi, 359 p.: ill. - Ref.: p.344-354. - Ind.: 355-359. - ISBN 978-0-19-857062-2
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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
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 Painlevé equations ..................................... 78
4.4.1 Painlevé 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
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 A5D 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-Kähler 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 КЗ ........ 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
10 Anti-self-dual conformal structures ........................ 229
10.1 α-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 Kähler structures .......................... 234
10.2.3 Null-Kähler structures ......................... 236
10.2.4 ASD Einstein structures ........................ 237
10.2.5 Hyper-Kähler 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
С.4 Prolongation .......................................... 324
С.4.1 Differential invariants ........................ 326
С.5 Method of characteristics ............................. 332
C.6 Cartan-Kähler theorem ................................. 335
References .................................................... 344
Index ......................................................... 355
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