Most nonlinear differential equations arising in natural sciences admit chaotic behavior and cannot be solved analytically. Integrable systems lie on the other extreme. They possess regular, stable, and well behaved solutions known as solitons and instantons. These solutions play important roles in pure and applied mathematics as well as in theoretical physics where they describe configurations topologically different from vacuum. While integrable equations in lower space-time dimensions can be solved using the inverse scattering transform, the higher-dimensional examples of anti-self-dual Yang-Mills and Einstein equations require twistor theory. Both techniques rely on an ability to represent nonlinear equations as compatibility conditions for overdetermined systems of linear differential equations. The book provides a self-contained and accessible introduction to the subject. It starts with an introduction to integrability of ordinary and partial differential equations. Subsequent chapters explore symmetry analysis, gauge theory, gravitational instantons, twistor transforms, and anti-self-duality equations. The three appendices cover basic differential geometry, complex manifold theory, and the exterior differential system.
Author(s): Maciej Dunajski
Series: Oxford Graduate Texts in Mathematics
Publisher: Oxford University Press, USA
Year: 2010
Language: English
Pages: 374
Tags: Математика;Дифференциальные уравнения;Дифференциальные уравнения в частных производных;
Contents......Page 8
List of Figures......Page 13
List of Abbreviations......Page 14
1.1 Hamiltonian formalism......Page 16
1.2 Integrability and action–angle variables......Page 19
1.3 Poisson structures......Page 29
2.1 The history of two examples......Page 35
2.1.1 A physical derivation of KdV......Page 36
2.1.2 Bäcklund transformations for the Sine-Gordon equation......Page 39
2.2 Inverse scattering transform for KdV......Page 40
2.2.1 Direct scattering......Page 43
2.2.2 Properties of the scattering data......Page 44
2.2.3 Inverse scattering......Page 45
2.2.4 Lax formulation......Page 46
2.2.5 Evolution of the scattering data......Page 47
2.3 Reflectionless potentials and solitons......Page 48
2.3.1 One-soliton solution......Page 49
2.3.2 N-soliton solution......Page 50
2.3.3 Two-soliton asymptotics......Page 51
3.1 First integrals......Page 58
3.2.1 Bi-Hamiltonian systems......Page 61
3.3 Zero-curvature representation......Page 63
3.3.1 Riemann–Hilbert problem......Page 65
3.3.2 Dressing method......Page 67
3.3.3 From Lax representation to zero curvature......Page 69
3.4 Hierarchies and finite-gap solutions......Page 71
4.1 Lie groups and Lie algebras......Page 79
4.2 Vector fields and one-parameter groups of transformations......Page 82
4.3 Symmetries of differential equations......Page 86
4.3.1 How to find symmetries......Page 89
4.3.2 Prolongation formulae......Page 90
4.4 Painlevé equations......Page 93
4.4.1 Painlevé test......Page 97
5.1 A variational principle......Page 100
5.1.1 Legendre transform......Page 102
5.1.2 Symplectic structures......Page 103
5.1.3 Solution space......Page 104
5.2 Field theory......Page 105
5.2.1 Solution space and the geodesic approximation......Page 107
5.3 Scalar kinks......Page 108
5.3.1 Topology and Bogomolny equations......Page 111
5.3.2 Higher dimensions and a scaling argument......Page 113
5.3.3 Homotopy in field theory......Page 114
5.4 Sigma model lumps......Page 115
6 Gauge field theory......Page 120
6.1 Gauge potential and Higgs field......Page 121
6.1.1 Scaling argument......Page 123
6.1.2 Principal bundles......Page 124
6.2 Dirac monopole and flux quantization......Page 125
6.2.1 Hopf fibration......Page 127
6.3 Non-abelian monopoles......Page 129
6.3.1 Topology of monopoles......Page 130
6.3.2 Bogomolny–Prasad–Sommerfeld (BPS) limit......Page 131
6.4 Yang–Mills equations and instantons......Page 134
6.4.1 Chem and Chem–Simons forms......Page 135
6.4.2 Minimal action solutions and the anti-self-duality condition......Page 137
6.4.3 Ansatz for ASD fields......Page 138
6.4.4 Gradient flow and classical mechanics......Page 139
7.1 Lax pair......Page 144
7.1.1 Geometric interpretation......Page 147
7.2.1 History and motivation......Page 148
7.2.2 Spinor notation......Page 152
7.2.3 Twistor space......Page 154
7.2.4 Penrose–Ward correspondence......Page 156
8.1 Reductions to integrable equations......Page 164
8.2 Integrable chiral model......Page 169
8.2.1 Soliton solutions......Page 172
8.2.2 Lagrangian formulation......Page 180
8.2.3 Energy quantization of time-dependent unitons......Page 183
8.2.4 Moduli space dynamics......Page 188
8.2.5 Mini-twistors......Page 196
9.1 Examples of gravitational instantons......Page 206
9.2 Anti-self-duality in Riemannian geometry......Page 210
9.2.1 Two-component spinors in Riemannian signature......Page 213
9.3 Hyper-Kähler metrics......Page 217
9.4 Multi-centred gravitational instantons......Page 221
9.4.1 Belinskii–Gibbons–Page–Pope class......Page 225
9.5 Other gravitational instantons......Page 227
9.5.1 Compact gravitational instantons and K3......Page 230
9.6 Einstein–Maxwell gravitational instantons......Page 231
9.7 Kaluza–Klein monopoles......Page 236
9.7.1 Kaluza–Klein solitons from Einstein–Maxwell instantons......Page 237
9.7.2 Solitons in higher dimensions......Page 241
10 Anti-self-dual conformal structures......Page 244
10.1 α-surfaces and anti-self-duality......Page 245
10.2 Curvature restrictions and their Lax pairs......Page 246
10.2.1 Hyper-Hermitian structures......Page 247
10.2.2 ASD Kähler structures......Page 249
10.2.3 Null-Kähler structures......Page 251
10.2.4 ASD Einstein structures......Page 252
10.2.5 Hyper-Kähler structures and heavenly equations......Page 253
10.3.1 Einstein–Weyl geometry......Page 261
10.3.2 Null symmetries and projective structures......Page 268
10.3.3 Dispersionless integrable systems......Page 271
10.4 ASD conformal structures in neutral signature......Page 277
10.4.2 Curved examples......Page 278
10.5 Twistor theory......Page 280
10.5.1 Curvature restrictions......Page 285
10.5.2 ASD Ricci-flat metrics......Page 287
10.5.3 Twistor theory and symmetries......Page 298
Appendix A: Manifolds and topology......Page 302
A.1 Lie groups......Page 305
A.2 Degree of a map and homotopy......Page 309
A.2.1 Homotopy......Page 311
A.2.2 Hermitian projectors......Page 313
Appendix B: Complex analysis......Page 315
B.1 Complex manifolds......Page 316
B.2 Holomorphic vector bundles and their sections......Page 318
B.3 Čech cohomology......Page 322
B.3.1 Deformation theory......Page 323
C.1 Introduction......Page 325
C.2 Exterior differential system and Frobenius theorem......Page 329
C.3 Involutivity......Page 335
C.4 Prolongation......Page 339
C.4.1 Differential invariants......Page 341
C.5 Method of characteristics......Page 347
C.6 Cartan–Kähler theorem......Page 350
References......Page 359
C......Page 370
H......Page 371
M......Page 372
S......Page 373
Z......Page 374
