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Integrability, Self-duality, and Twistor Theory [Kõva köide]

(, Mathematical Institute, Oxford), (, Mathematical Institute, Oxford)
  • Formaat: Hardback, 376 pages, kõrgus x laius x paksus: 241x162x25 mm, kaal: 688 g, line figures
  • Sari: London Mathematical Society Monographs 15
  • Ilmumisaeg: 09-May-1996
  • Kirjastus: Oxford University Press
  • ISBN-10: 0198534981
  • ISBN-13: 9780198534983
Teised raamatud teemal:
Integrability, Self-duality, and Twistor TheorySuurem pilt
  • Formaat: Hardback, 376 pages, kõrgus x laius x paksus: 241x162x25 mm, kaal: 688 g, line figures
  • Sari: London Mathematical Society Monographs 15
  • Ilmumisaeg: 09-May-1996
  • Kirjastus: Oxford University Press
  • ISBN-10: 0198534981
  • ISBN-13: 9780198534983
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780198534983.html
Märksõnad:
It has been known for some time that many of the familiar integrable systems of equations are symmetry reductions of self-duality equations on a metric or on a Yang-Mills connection (for example, the Korteweg-de Vries and nonlinear Schrödinger equations are reductions of the self-dual Yang-Mills equation). This book explores in detail the connections between self-duality and integrability, and also the application of twistor techniques to integrable systems. It has two central themes: first, that the symmetries of self-duality equations provide a natural classification scheme for integrable systems; and second that twistor theory provides a uniform geometric framework for the study of B¨ acklund tranformations, the inverse scattering method, and other such general constructions of integrability theory, and that it elucidates the connections between them.

Arvustused

Anybody working in integrable systems or in twistor constructions will want a copy of this book or at least want it in their Library. * Proceedings of the Edinburgh Mathematical Society 1998, 41 *

1 Introduction
1(12)
1.1 Examples of integrability
1(6)
1.2 Outline of the book
7(2)
Notes on
Chapter 1
9(3)
I REDUCTIONS OF THE ASDYM EQUATION 13(124)
2 Mathematical background I
13(19)
2.1 Gauge theories
13(1)
2.2 Space-time
14(2)
2.3 Differential forms
16(3)
2.4 Conformal transformations and compactified space-time
19(4)
2.5 Bundles, connections, and curvature
23(6)
2.6 The Yang-Mills equations
29(1)
Notes on
Chapter 2
30(2)
3 The ASD Yang-Mills equation
32(11)
3.1 ASD electromagnetic fields
32(1)
3.2 Lax pairs
33(1)
3.3 Yang's equation and the K-matrix
34(2)
3.4 Lagrangians for the ASDYM equation
36(2)
3.5 The Hamiltonian formalism
38(4)
Notes on
Chapter 3
42(1)
4 Reduction of the ASDYM equation
43(16)
4.1 Classification of reductions
43(2)
4.2 Reductions of the linear ASD equation
45(1)
4.3 Conformal reduction in the non-Abelian case
46(1)
4.4 Invariant connections and Higgs fields
47(2)
4.5 The space of orbits
49(6)
4.6 Backlund transformations
55(1)
Notes on
Chapter 4
56(3)
5 Reduction to three dimensions
59(8)
5.1 The Bogomolny equation
59(1)
5.2 Hyperbolic monopoles and other generalizations
60(3)
5.3 Reduction by a null translation
63(3)
Notes on
Chapter 5
66(1)
6 Reduction to two dimensions
67(28)
6.1 Two-dimensional groups of conformal motions
67(1)
6.2 Reductions by H++
68(5)
6.3 Reduction by H+0
73(9)
6.4 Reduction by HSD
82(2)
6.5 Reduction by HASD
84(1)
6.6 The Ernst equation
84(5)
6.7 Reduction of Yang's equation
89(2)
6.8 Liouville's equation
91(1)
Notes on
Chapter 6
92(3)
7 Reductions to one dimension
95(16)
7.1 Abelian reduction to one-dimension
95(3)
7.2 Nahm's equations and tops
98(3)
7.3 The motion of an n-dimensional rigid body
101(1)
7.4 The Painleve equations
102(6)
7.5 Non-Abelian reductions
108(1)
Notes on
Chapter 7
109(2)
8 Hierarchies
111(26)
8.1 The KdV flows
111(3)
8.2 The recursion operator for the ASDYM equation
114(1)
8.3 Hamiltonian formalism
115(3)
8.4 ASDYM and Bogomolny hierarchies
118(5)
8.5 Reductions of the ASDYM flows
123(4)
8.6 The generalized ASDYM equation
127(5)
Notes on
Chapter 8
132(5)
II TWISTOR METHODS 137(180)
9 Mathematical background II
137(34)
9.1 Projective spaces and flag manifolds
137(1)
9.2 Twistor space
138(7)
9.3 Birkhoff's factorization theorem
145(4)
9.4 Holomorphic vector bundles: the Cech description
149(4)
9.5 operators
153(2)
9.6 Cohomology
155(2)
9.7 The Grassmannian
157(1)
9.8 Scattering on the real line
158(2)
9.9 Spinors
160(8)
Notes on
Chapter 9
168(3)
10 The twistor correspondence
171(33)
10.1 The concrete from of the Penrose--Ward transform
171(5)
10.2 The abstract form of the transform
176(3)
10.3 The Painleve property
179(1)
10.4 Global solutions in Euclidean signature
180(7)
10.5 Global solution in ultrahyperbolic signature
187(7)
10.6 The GASDYM equation
194(1)
10.7 The truncated GASDYM hierarchy
195(1)
10.8 The linear Penrose transform
196(5)
Notes on
Chapter 10
201(3)
11 Reductions of the Penrose--Ward transform
204(40)
11.1 Symmetries of the twistor correspondence
205(1)
11.2 Symmetries of the twistor bundle
206(5)
11.3 Reduced twistor spaces
211(7)
11.4 The KdV and NLS equations
218(2)
11.5 The initial value problem and inverse scattering
220(11)
11.6 Isomonodromy and the Painleve equations
231(8)
11.7 The Schlesinger equation
239(2)
Notes on
Chapter 11
241(3)
12 Twistor construction of hierarchies
244(40)
12.1 Transformations of the patching matrix
245(5)
12.2 DS operators and the GASDYM hierarchy
250(4)
12.3 The twistor construction of the DS flows
254(10)
12.4 Explicit construction of solutions from twistor data
264(5)
12.5 Hamiltonian formalism
269(5)
12.6 The KP equation and the KP heirarchy
274(8)
Notes on
Chapter 12
282(2)
13 ASD metrics
284(33)
13.1 Self-duality in curved space-time
284(2)
13.2 The Levi-Civita connection
286(3)
13.3 Spinors and the correspondence space
289(5)
13.4 ASD conformal structures
294(6)
13.5 Curved twistor spaces
300(5)
13.6 Reductions
305(2)
13.7 ASDYM fields and the switch map
307(9)
Notes on
Chapter 13
316(1)
A Active and passive gauge transformations 317(2)
B The Drinfeld--Sokolov construction 319(9)
Notes on Appendix B 326(2)
C Poisson and symplectic structures 328(10)
Notes on Appendix C 338(1)
D Reductions of the ASDYM equation 339(4)
References 343(13)
A note on notation 356(1)
Index of notation 357(2)
Index 359