Solitons, Instantons, and Twistors

by Dunajski, Maciej

• ISBN13: 9780198570639
• ISBN10: 0198570635
• Format: Paperback
• Copyright: 2010-02-08
• Publisher: Oxford University Press

Summary

Most nonlinear differential equations arising in natural sciences admit chaotic behaviour 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 importantroles 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 ofanti-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, andanti-self-duality equations. The three appendices cover basic differential geometry, complex manifold theory, and the exterior differential system.

Author Biography

Maciej Dunajski read physics in Lodz, Poland and received a PhD in mathematics from Oxford University where he held a Senior Scholarship at Merton College. After spending four years as a lecturer in the Mathematical Institute in Oxford where he was a member of Roger Penrose's research group, he moved to Cambridge, where he holds a Fellowship and lectureship at Clare College and a Newton Trust Lectureship at the Department of Applied Mathematics and Theoretical Physics. Dunajski specialises in twistor theory and differential geometric approaches to integrability and solitons. He is married with two sons.

Table of Contents

List of Figures	p. xii
List of Abbreviations	p. xiii
Integrability in classical mechanics	p. 1
Hamiltonian formalism	p. 1
Integrability and action-angle variables	p. 4
Poisson structures	p. 14
Soliton equations and the inverse scattering transform	p. 20
The history of two examples	p. 20
A physical derivation of KdV	p. 21
Bäcklund transformations for the Sine-Gordon equation	p. 24
Inverse scattering transform for KdV	p. 25
Direct scattering	p. 28
Properties of the scattering data	p. 29
Inverse scattering	p. 30
Lax formulation	p. 31
Evolution of the scattering data	p. 32
Reflectionless potentials and solitons	p. 33
One-soliton solution	p. 34
N-soliton solution	p. 35
Two-soliton asymptotics	p. 36
Hamiltonian formalism and zero-curvature representation	p. 43
First integrals	p. 43
Hamiltonian formalism	p. 46
Bi-Hamiltonian systems	p. 46
Zero-curvature representation	p. 48
Riemann-Hilbert problem	p. 50
Dressing method	p. 52
From Lax representation to zero curvature	p. 54
Hierarchies and finite-gap solutions	p. 56
Lie symmetries and reductions	p. 64
Lie groups and Lie algebras	p. 64
Vector fields and one-parameter groups of transformations	p. 67
Symmetries of differential equations	p. 71
How to find symmetries	p. 74
Prolongation formulae	p. 75
Painlevé equations	p. 78
Painlevé test	p. 82
Lagrangian formalism and field theory	p. 85
A variational principle	p. 85
Legendre transform	p. 87
Symplectic structures	p. 88
Solution space	p. 89
Field theory	p. 90
Solution space and the geodesic approximation	p. 92
Scalar kinks	p. 93
Topology and Bogomolny equations	p. 96
Higher dimensions and a scaling argument	p. 98
Homotopy in field theory	p. 99
Sigma model lumps	p. 100
Gauge field theory	p. 105
Gauge potential and Higgs field	p. 106
Scaling argument	p. 108
Principal bundles	p. 109
Dirac monopole and flux quantization	p. 110
Hopf fibration	p. 112
Non-abelian monopoles	p. 114
Topology of monopoles	p. 115
Bogomolny-Prasad-Sommerfeld (BPS) limit	p. 116
Yang-Mills equations and instantons	p. 119
Chern and Chern-Simons forms	p. 120
Minimal action solutions and the anti-self-duality condition	p. 122
Ansatz for ASD fields	p. 123
Gradient flow and classical mechanics	p. 124
Integrability of ASDYM and twistor theory	p. 129
Lax pair	p. 129
Geometric interpretation	p. 132
Twistor correspondence	p. 133
History and motivation	p. 133
Spinor notation	p. 137
Twistor space	p. 139
Penrose-Ward correspondence	p. 141
Symmetry reductions and the integrable chiral model	p. 149
Reductions to integrable equations	p. 149
Integrable chiral model	p. 154
Soliton solutions	p. 157
Lagrangian formulation	p. 165
Energy quantization of time-dependent unitons	p. 168
Moduli space dynamics	p. 173
Mini-twistors	p. 181
Gravitational instantons	p. 191
Examples of gravitational instantons	p. 191
Anti-self-duality in Riemannian geometry	p. 195
Two-component spinors in Riemannian signature	p. 198
Hyper-Kähler metrics	p. 202
Multi-centred gravitational instantons	p. 206
Belinskii-Gibbons-Page-Pope class	p. 210
Other gravitational instantons	p. 212
Compact gravitational instantons and K3	p. 215
Einstein-Maxwell gravitational instantons	p. 216
Kaluza-Klein monopoles	p. 221
Kaluza-Klein solitons from Einstein-Maxwell instantons	p. 222
Solitons in higher dimensions	p. 226
Anti-self-dual conformal structures	p. 229
¿-surfaces and anti-self-duality	p. 230
Curvature restrictions and their Lax pairs	p. 231
Hyper-Hermitian structures	p. 232
ASD Kähler structures	p. 234
Null-Kähler structures	p. 236
ASD Einstein structures	p. 237
Hyper-Kähler structures and heavenly equations	p. 238
Symmetries	p. 246
Einstein-Weyl geometry	p. 246
Null symmetries and projective structures	p. 253
Dispersionless integrable systems	p. 256
ASD conformal structures in neutral signature	p. 262
Conformal compactification	p. 263
Curved examples	p. 263
Twistor theory	p. 265
Curvature restrictions	p. 270
ASD Ricci-flat metrics	p. 272
Twistor theory and symmetries	p. 283
Manifolds and topology	p. 287
Lie groups	p. 290
Degree of a map and homotopy	p. 294
Homotopy	p. 296
Hermitian projectors	p. 298
Complex analysis	p. 300
Complex manifolds	p. 301
Holomorphic vector bundles and their sections	p. 303
Cech cohomology	p. 307
Deformation theory	p. 308
Overdetermined PDEs	p. 310
Introduction	p. 310
Exterior differential system and Frobenius theorem	p. 314
Involutivity	p. 320
Prolongation	p. 324
Differential invariants	p. 326
Method of characteristics	p. 332
Cartan-Kähler theorem	p. 335
References	p. 344
Index	p. 355
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