Solitons in field theory and nonlinear analysis:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2001
|
| Series: | Springer monographs in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 525 - 548 |
| Physical Description: | XXIV, 553 S. |
| ISBN: | 038795242X |
Staff View
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| 245 | 1 | 0 | |a Solitons in field theory and nonlinear analysis |c Yisong Yang |
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| 300 | |a XXIV, 553 S. | ||
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface vii
Notation and Convention xxiii
1 Primer of Field Theory 1
1.1 Mechanics and Fields 1
1.1.1 Action principle in classical mechanics 2
1.1.2 Charged particle in electromagnetic field 5
1.1.3 Schrodinger equation via first quantization 6
1.2 Relativistic Dynamics and Electromagnetism 8
1.2.1 Minkowski spacetime and relativistic mechanics ... 8
1.2.2 Klein Gordon fields 12
1.2.3 Maxwell equations 12
1.3 Scalar Fields and Symmetry 15
1.3.1 Variational formalism 15
1.3.2 Noether s theorem and conserved quantities 16
1.3.3 Static solutions and Derrick s theorem 19
1.4 Gauge Field Theory 20
1.4.1 Local symmetry and gauge fields 20
1.4.2 Low temperature and spontaneous symmetry breaking 24
1.4.3 Goldstone particles and Higgs mechanism 25
1.5 Yang Mills Fields 27
1.6 General Relativity and Cosmology 30
1.6.1 Einstein field equations 30
xvi Contents
1.6.2 Cosmological consequences 37
1.7 Remarks 41
2 Sigma Models 43
2.1 Sigma Model and Belavin Polyakov Solution 43
2.1.1 Sigma model for Heisenberg ferromagnet 43
2.1.2 Solution by rational functions 46
2.1.3 Topology 48
2.2 Gauged Sigma Model 50
2.2.1 Field theory and self dual equations 50
2.2.2 Multisolitons: existence theorems 53
2.3 Governing Equations and Characterization 56
2.4 Mathematical Analysis 57
2.4.1 Regularized equation and range of parameter .... 58
2.4.2 Subsolution and variational method 59
2.4.3 Existence of supersolution 67
2.4.4 Existence of bounded solution 68
2.4.5 Asymptotic limit 69
2.4.6 Recovery of original field configurations 71
2.4.7 Magnetic flux and minimum energy value 71
2.4.8 Brouwer degree of map 71
2.4.9 Nonexistence of solution of unit degree 74
2.5 Remarks 76
3 Multiple Instantons and Characteristic Classes 79
3.1 Classical Yang Mills Fields 79
3.1.1 Action principle and self dual equations 80
3.1.2 Energetic and topological characterizations 83
3.1.3 t Hooft instantons 85
3.2 Liouville Equation and Solution 88
3.2.1 Liouville method 88
3.2.2 Backlund transformation method 90
3.3 Witten s Instanton 92
3.3.1 Field configurations and equations 92
3.3.2 Explicit instanton solutions 94
3.4 Instantons and Characteristic Classes 95
3.4.1 Self duality and Witten Tchrakian equations .... 95
3.4.2 Quasilinear elliptic equation 102
3.5 Existence of Weak Solution 103
3.6 Asymptotic Estimates 107
3.7 Topological Charge 116
3.8 Remarks 117
4 Generalized Abelian Higgs Equations 121
4.1 Field Theory Structure 121
Contents xvii
4.1.1 Formulation and main existence theorem 122
4.1.2 Nonlinear elliptic system 125
4.2 General Problems and Solutions 127
4.3 Compact Surface Case 130
4.3.1 Necessary condition 130
4.3.2 Variational principle 130
4.3.3 Existence of solution 133
4.3.4 Uniqueness 134
4.4 Solution on Plane: Existence 135
4.4.1 Variational problem 135
4.4.2 Coercivity 136
4.4.3 Existence and uniqueness of critical point 139
4.5 Solution on Plane: Asymptotic Behavior 140
4.5.1 Pointwise decay near infinity 141
4.5.2 Exponential decay estimates 142
4.5.3 Uniqueness and quantized integrals 143
4.6 Nonexistence Results 144
4.7 Arbitrary Coefficient Matrix Case 151
4.8 Remarks 155
5 Chern—Simons Systems: Abelian Case 157
5.1 Schrodinger Equation 157
5.1.1 Schrodinger fields and Chern Simons dynamics . . . 158
5.1.2 Explicit static solution 160
5.2 Relativistic Chern Simons Model on Plane 164
5.2.1 Field equations and existence results 164
5.2.2 Topological lower energy bound 166
5.3 Construction of Solution 167
5.3.1 Iterative method and control of sequence 168
5.3.2 Global convergence theorems 173
5.4 Symmetric Non topological Solutions 177
5.4.1 Existence theorem 178
5.4.2 Two point boundary value problem 179
5.4.3 Shooting analysis 180
5.5 Solutions on Doubly Periodic Domains 186
5.5.1 Boundary condition modulo gauge symmetry .... 186
5.5.2 Existence versus coupling parameter 188
5.5.3 Construction via sub and supersolutions 189
5.5.4 Alternative variational treatment 194
5.6 Tarantello s Secondary Solution 200
5.6.1 Critical coupling parameter 200
5.6.2 Local minimum 202
5.6.3 Nonminimum via mountain pass lemma 205
5.7 Remarks 208
xviii Contents
6 Chern—Simons Systems: Non Abelian Case 211
6.1 Lie Algebras and Cartan Weyl Bases 211
6.1.1 Simple examples 212
6.1.2 Classification theorem 214
6.1.3 Root vectors and Cartan matrices 219
6.2 Non Abelian Gauged Schrodinger Equations 221
6.2.1 Adjoint representation and elliptic problems 221
6.2.2 Toda systems 226
6.2.3 Explicit non Abelian solutions 231
6.3 Relativistic Chern Simons Systems 232
6.4 Elliptic System and its Variational Principle 236
6.5 Existence of Minimizer 241
6.5.1 Boundary condition 241
6.5.2 Minimization 242
6.5.3 Asymptotic behavior 245
6.5.4 Quantized integrals 248
6.5.5 Original field configuration 248
6.6 Some Examples 249
6.7 Remarks 251
7 Electroweak Vortices 253
7.1 Massive non Abelian Gauge Theory 253
7.1.1 Governing equations 253
7.1.2 Periodic boundary condition 256
7.1.3 First order system and existence theorem 258
7.1.4 Variational proof 260
7.2 Classical Electroweak Theory 263
7.2.1 Unitary gauge framework 263
7.2.2 t Hooft periodic boundary conditions 265
7.2.3 Lower energy bound and its saturation 268
7.3 Multi constrained Variational Approach 269
7.3.1 Elliptic equations 269
7.3.2 Existence via minimization 270
7.3.3 Alternative formulation 274
7.4 Two Higgs Model 277
7.4.1 Physical background 277
7.4.2 Field theory model and equations 277
7.4.3 Periodic multivortices 279
7.4.4 Planar solutions 286
7.5 Remarks 296
8 Dyons 299
8.1 Dirac Monopole 299
8.1.1 Electromagnetic duality 300
8.1.2 Dirac strings and charge quantization 301
Contents xix
8.1.3 Fiber bundle device and removal of strings 303
8.2 Schwinger Theory 305
8.2.1 Rotation symmetry 305
8.2.2 Charge quantization formula for dyons 305
8.3 Julia Zee Dyons 307
8.3.1 Field equations 307
8.3.2 Explicit solutions in BPS limit 309
8.3.3 Existence result in general 311
8.4 Weinberg Salam Electroweak Dyons 322
8.5 Radial Equations and Action Principle 325
8.6 Constrained Variational Method 326
8.6.1 Admissible space 326
8.6.2 Partial coerciveness and minimization 330
8.6.3 Weak solutions of governing equations 338
8.6.4 Full set of boundary conditions 341
8.6.5 Asymptotic estimates 344
8.6.6 Electric and magnetic charges 348
8.7 Remarks 350
9 Ordinary Differential Equations 353
9.1 Existence Results 353
9.2 Dynamical Analysis 355
9.2.1 Local solution via contractive mapping 355
9.2.2 Parameter sets 358
9.2.3 Asymptotic limits 362
9.2.4 Continuous dependence 364
9.2.5 Critical behavior and conclusion of proof 364
9.3 Applications 367
9.4 Remarks 369
10 Strings in Cosmology 371
10.1 Strings, Conical Geometry, and Deficit Angle 371
10.1.1 Localized energy distribution and multiple strings . 372
10.1.2 Harmonic map model 374
10.2 Strings and Abelian Gauge Fields 378
10.2.1 Governing equations over Riemann surfaces 378
10.2.2 Role of defects 380
10.2.3 Obstructions to existence 382
10.2.4 Proof of equivalence and consequences 383
10.3 Existence of Strings: Compact Case 387
10.3.1 Existence for N 3 387
10.3.2 Existence for N = 2 and nonexistence for N = 1 . . 394
10.4 Existence of Strings: Noncompact Case 395
10.4.1 Existence results 395
10.4.2 Construction of solutions 396
xx Contents
10.4.3 Asymptotic decay estimates 403
10.5 Symmetric Solutions 409
10.5.1 Necessary and sufficient condition for existence . . . 409
10.5.2 Equivalence theorem 410
10.5.3 TV strings 411
10.6 Symmetric Solutions on S2 416
10.6.1 Balanced strings at opposite poles 416
10.6.2 Differential equation 417
10.6.3 Solution on V 418
10.6.4 Solutions on full S2 422
10.6.5 Nonexistence of unbalanced solutions 422
10.7 Non Abelian Cosmic Strings 425
10.7.1 Massive W boson and strings 426
10.7.2 Einstein Weinberg Salam system 429
10.8 Remarks 436
11 Vortices and Antivortices 439
11.1 Gauge Field Theory and Coexisting Strings 439
11.1.1 Action density 440
11.1.2 Existence theorems 443
11.2 Simplification of Equations 445
11.3 Proof of Existence 450
11.3.1 Vortices and antivortices 450
11.3.2 Strings and antistrings 453
11.3.3 Asymptotic estimates 462
11.4 Quantized Flux, Total Curvature, and Topology 465
11.5 Unique Solutions on Compact Surfaces 469
11.5.1 Formulation on line bundles 470
11.5.2 Number count 472
11.5.3 Solution and fixed point method 473
11.6 Remarks 481
12 Born Infeld Solutions 483
12.1 Nonlinear Electromagnetism 483
12.1.1 Point charge problem 484
12.1.2 Bernstein theorems 488
12.2 Relation of Electrostatic and Magnetostatic Fields 493
12.2.1 Electrostatic fields 493
12.2.2 Magnetostatic fields 494
12.2.3 Generalized Bernstein problem 496
12.2.4 Mixed interaction case 500
12.3 Nonlinear Wave Equations 501
12.3.1 Static solutions 501
12.3.2 In view of Nambu Goto string theory 505
12.4 Abelian Strings 508
Contents xxi
12.4.1 Existence and uniqueness theorems 508
12.4.2 Analysis of compact surface case 516
12.4.3 Solutions on noncompact surfaces 519
12.5 Remarks 522
References 525
Index 549
|
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| id | DE-604.BV013902977 |
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| indexdate | 2024-07-09T18:54:10Z |
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| isbn | 038795242X |
| language | English |
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| spelling | Yang, Yisong 1958- Verfasser (DE-588)123120861 aut Solitons in field theory and nonlinear analysis Yisong Yang New York [u.a.] Springer 2001 XXIV, 553 S. txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Literaturverz. S. 525 - 548 Champs, Théorie des (Physique) Solitons Solitons gtt Field theory (Physics) Soliton (DE-588)4135213-0 gnd rswk-swf Feldtheorie (DE-588)4016698-3 gnd rswk-swf Nichtlineare Analysis (DE-588)4177490-5 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Feldtheorie (DE-588)4016698-3 s Nichtlineare Analysis (DE-588)4177490-5 s Soliton (DE-588)4135213-0 s DE-604 Mathematische Physik (DE-588)4037952-8 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009514411&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Yang, Yisong 1958- Solitons in field theory and nonlinear analysis Champs, Théorie des (Physique) Solitons Solitons gtt Field theory (Physics) Soliton (DE-588)4135213-0 gnd Feldtheorie (DE-588)4016698-3 gnd Nichtlineare Analysis (DE-588)4177490-5 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| subject_GND | (DE-588)4135213-0 (DE-588)4016698-3 (DE-588)4177490-5 (DE-588)4037952-8 |
| title | Solitons in field theory and nonlinear analysis |
| title_auth | Solitons in field theory and nonlinear analysis |
| title_exact_search | Solitons in field theory and nonlinear analysis |
| title_full | Solitons in field theory and nonlinear analysis Yisong Yang |
| title_fullStr | Solitons in field theory and nonlinear analysis Yisong Yang |
| title_full_unstemmed | Solitons in field theory and nonlinear analysis Yisong Yang |
| title_short | Solitons in field theory and nonlinear analysis |
| title_sort | solitons in field theory and nonlinear analysis |
| topic | Champs, Théorie des (Physique) Solitons Solitons gtt Field theory (Physics) Soliton (DE-588)4135213-0 gnd Feldtheorie (DE-588)4016698-3 gnd Nichtlineare Analysis (DE-588)4177490-5 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| topic_facet | Champs, Théorie des (Physique) Solitons Field theory (Physics) Soliton Feldtheorie Nichtlineare Analysis Mathematische Physik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009514411&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT yangyisong solitonsinfieldtheoryandnonlinearanalysis |