Nonlinear waves and solitons on contours and closed surfaces:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2012
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Springer Series in Synergetics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | 520 S. |
| ISBN: | 9783642228940 9783642228957 3642228941 |
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| 245 | 1 | 0 | |a Nonlinear waves and solitons on contours and closed surfaces |c Andrei Ludu |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2012 | |
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Datensatz im Suchindex
| _version_ | 1805146629998641152 |
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| adam_text |
IMAGE 1
CONTENTS
PART I MATHEMATICAL PREREQUISITES 1 INTRODUCTION 3
1.1 INTRODUCTION TO SOLITON THEORY 3
1.2 ALGEBRAIC AND GEOMETRIC APPROACHES 4
1.3 A LIST OF USEFUL DERIVATIVES IN FINITE DIMENSIONAL SPACES 6
2 MATHEMATICAL PREREQUISITES 9
2.1 ELEMENTS OF TOPOLOGY 9
2.1.1 SEPARATION AXIOMS 11
2.1.2 COMPACTNESS 13
2.1.3 WEIERSTRASS-S TONE THEOREM 15
2.1.4 CONNECTEDNESS, CONNECTIVITY, AND HOMOTOPY 17
2.1.5 SEPARABILITY AND BASIS 18
2.1.6 METRIC AND NORMED SPACES 19
2.2 ELEMENTS OF HOMOLOGY 20
2.3 GROUP ACTION 21
3 THE IMPORTANCE O F THE BOUNDARY 23
3.1 THE POWER O F COMPACT BOUNDARIES: REPRESENTATION FORMULAS 23
3.1.1 REPRESENTATION FORMULA FOR N = 1: TAYLOR SERIES 24
3.1.2 REPRESENTATION FORMULA FOR N - 2: CAUCHY FORMULA 24 3.1.3
REPRESENTATION FORMULA FOR N = 3: GREEN FORMULA 25 3.1.4 REPRESENTATION
FORMULA IN GENERAL: STOKES THEOREM 26 3.2 COMMENTS AND EXAMPLES 28
4 VECTOR FIELDS, DIFFERENTIAL FORMS, AND DERIVATIVES 31
4.1 MANIFOLDS AND MAPS 32
4.2 DIFFERENTIAL AND VECTOR FIELDS 35
XI
HTTP://D-NB.INFO/1013327195
IMAGE 2
XII
CONTENTS
4.3 EXISTENCE AND UNIQUENESS THEOREMS: DIFFERENTIAL
EQUATION APPROACH 40
4.4 EXISTENCE AND UNIQUENESS THEOREMS: FLOW BOX APPROACH 45 4.5 COMPACT
SUPPORTED VECTOR FIELDS 47
4.6 DIFFERENTIAL FORMS AND THE LIE DERIVATIVE 48
4.7 DIFFERENTIAL SYSTEMS, INTEGRABILITY AND INVARIANTS 54
4.8 POINCARE LEMMA 56
4.9 FIBER BUNDLES AND COVARIANT DERIVATIVE 57
4.9.1 PRINCIPAL BUNDLE AND FRAMES 59
4.9.2 CONNECTION FORM AND COVARIANT DERIVATIVE 61
4.10 TENSOR ANALYSIS 66
4.11 THE MIXED COVARIANT DERIVATIVE 69
4.12 CURVILINEAR ORTHOGONAL COORDINATES 71
4.13 SPECIAL TWO-DIMENSIONAL NONLINEAR ORTHOGONAL COORDINATES . 75
4.14 PROBLEMS 76
5 GEOMETRY OF CURVES 79
5.1 ELEMENTS OF DIFFERENTIAL GEOMETRY OF CURVES 79
5.2 CLOSED CURVES 86
5.3 CURVES LYING ON A SURFACE 91
5.4 PROBLEMS 94
6 GEOMETRY O F SURFACES 97
6.1 ELEMENTS OF DIFFERENTIAL GEOMETRY OF SURFACES 99
6.2 COVARIANT DERIVATIVE AND CONNECTIONS 107
6.3 GEOMETRY OF PARAMETERIZED SURFACES EMBEDDED IN K 3 110
6.3.1 CHRISTOFFEL SYMBOLS AND COVARIANT DIFFERENTIATION FOR HYBRID
TENSORS 113
6.4 COMPACT SURFACES 115
6.5 SURFACE DIFFERENTIAL OPERATORS 117
6.5.1 SURFACE GRADIENT 117
6.5.2 SURFACE DIVERGENCE 118
6.5.3 SURFACE LAPLACIAN 120
6.5.4 SURFACE CURL 121
6.5.5 INTEGRAL RELATIONS FOR SURFACE DIFFERENTIAL OPERATORS 124 6.5.6
APPLICATIONS 125
6.6 PROBLEMS 129
7 MOTION OF CURVES AND SOLITONS 131
7.1 KINEMATICS OF TWO-DIMENSIONAL CURVES 132
7.2 MAPPING TWO-DIMENSIONAL CURVE MOTION INTO NONLINEAR INTEGRABLE
SYSTEMS 136
7.3 THE TIME EVOLUTION OF LENGTH AND AREA 144
7.4 CARTAN THEORY OF THREE-DIMENSIONAL CURVE MOTION 150
7.5 KINEMATICS OF THREE-DIMENSIONAL CURVES 152
IMAGE 3
CONTENTS X I I I
7.6 MAPPING THREE-DIMENSIONAL CURVE MOTION
INTO NONLINEAR INTEGRABLE SYSTEMS 156
7.7 PROBLEMS 157
8 THEORY O F MOTION OF SURFACES 159
8.1 DIFFERENTIAL GEOMETRY OF SURFACE MOTION 159
8.2 COORDINATES AND VELOCITIES ON A FLUID SURFACE 162
8.3 KINEMATICS OF MOVING SURFACES 168
8.4 DYNAMICS O F MOVING SURFACES 170
8.5 BOUNDARY CONDITIONS FOR MOVING FLUID INTERFACES 173
8.6 DYNAMICS O F THE FLUID INTERFACES 174
8.7 PROBLEMS 176
PART II SOLITONS AND NONLINEAR WAVES ON CLOSED CURVES AND SURFACES
9 KINEMATICS O F HYDRODYNAMICS 179
9.1 LAGRANGIAN VS. EULERIAN FRAMES 179
9.1.1 INTRODUCTION 180
9.1.2 GEOMETRICAL PICTURE FOR LAGRANGIAN VS. EULERIAN 181 9.2 FLUID
FIBER BUNDLE 183
9.2.1 INTRODUCTION 183
9.2.2 MOTIVATION FOR A GEOMETRICAL APPROACH 186
9.2.3 THE FIBER BUNDLE 189
9.2.4 FIXED FLUID CONTAINER 190
9.2.5 FREE SURFACE FIBER BUNDLE 193
9.2.6 HOW DOES THE TIME DERIVATIVE O F TENSORS TRANSFORM FROM EULER TO
LAGRANGE FRAME? 196
9.3 PATH LINES, STREAM LINES, AND PARTICLE CONTOURS 199
9.4 EULERIAN-LAGRANGIAN DESCRIPTION FOR MOVING CURVES 203
9.5 THE FREE SURFACE 206
9.6 EQUATION O F CONTINUITY 207
9.6.1 INTRODUCTION 208
9.6.2 SOLUTIONS O F THE CONTINUITY EQUATION ON COMPACT INTERVALS 214
9.7 PROBLEMS 220
10 DYNAMICS OF HYDRODYNAMICS 223
10.1 MOMENTUM CONSERVATION: EULER AND NAVIER-STOKES EQUATIONS 223
10.2 BOUNDARY CONDITIONS 226
10.3 CIRCULATION THEOREM 228
10.4 SURFACE TENSION 234
10.4.1 PHYSICAL PROBLEM 234
10.4.2 MINIMAL SURFACES 236
10.4.3 APPLICATION 238
IMAGE 4
XIV CONTENTS
10.4.4 ISOTHERMAL PARAMETRIZATION 241
10.4.5 TOPOLOGICAL PROPERTIES OF MINIMAL SURFACES 244
10.4.6 GENERAL CONDITION FOR MINIMAL SURFACES 246
10.4.7 SURFACE TENSION FOR ALMOST ISOTHERMAL PARAMETRIZATION 247
10.5 SPECIAL FLUIDS 250
10.6 REPRESENTATION THEOREMS IN FLUID DYNAMICS 250
10.6.1 HELMHOLTZ DECOMPOSITION THEOREM IN M 3 250
10.6.2 DECOMPOSITION FORMULA FOR TRANSVERSAL ISOTROPIC VECTOR FIELDS 253
10.6.3 SOLENOIDAL-TOROIDAL DECOMPOSITION FORMULAS 256 10.7 PROBLEMS 257
11 NONLINEAR SURFACE WAVES IN ONE DIMENSION 259
11.1 KDV EQUATION DEDUCTION FOR SHALLOW WATERS 259
11.2 SMOOTH TRANSITIONS BETWEEN PERIODIC AND APERIODIC SOLUTIONS 264
11.3 MODIFIED KDV EQUATION AND GENERALIZATIONS 269
11.4 HYDRODYNAMIC EQUATIONS INVOLVING HIGHER-ORDER NONLINEARITIES 270
11.4.1 A COMPACT VERSION FOR KDV 271
11.4.2 SMALL AMPLITUDE APPROXIMATION 273
11.4.3 DISPERSION RELATIONS 276
11.4.4 THE FULL EQUATION 277
11.4.5 REDUCTION OF GKDV TO OTHER EQUATIONS AND SOLUTIONS 279
11.4.6 THE FINITE DIFFERENCE FORM 283
11.5 BOUSSINESQ EQUATIONS ON A CIRCLE 286
12 NONLINEAR SURFACE WAVES IN TWO DIMENSIONS 289
12.1 GEOMETRY OF TWO-DIMENSIONAL FLOW 289
12.2 TWO-DIMENSIONAL NONLINEAR EQUATIONS 296
12.3 TWO-DIMENSIONAL FLUID SYSTEMS WITH BOUNDARY 299
12.4 OSCILLATIONS IN TWO-DIMENSIONAL LIQUID DROPS 302
12.5 CONTOURS DESCRIBED BY QUARTIC CLOSED CURVES 304
12.6 SURFACE NONLINEAR WAVES IN TWO-DIMENSIONAL LIQUID NITROGEN DROPS
305
13 NONLINEAR SURFACE WAVES IN THREE DIMENSIONS 309
13.1 OSCILLATIONS OF INVISCID DROPS: THE LINEAR MODEL 311
13.1.1 DROP IMMERSED IN ANOTHER FLUID 313
13.1.2 DROP WITH RIGID CORE 315
13.1.3 MOVING CORE 321
13.1.4 DROP VOLUME 325
13.2 OSCILLATIONS OF VISCOUS DROPS: THE LINEAR MODEL 327
13.2.1 MODEL 1 328
IMAGE 5
CONTENTS X V
13.3 NONLINEAR THREE-DIMENSIONAL OSCILLATIONS
OF AXISYMMETRIC DROPS 341
13.3.1 NONLINEAR RESONANCES IN DROP OSCILLATION 351
13.4 OTHER NONLINEAR EFFECTS IN DROP OSCILLATIONS 362
13.5 SOLITONS ON THE SURFACE OF LIQUID DROPS 363
13.6 PROBLEMS 372
14 OTHER SPECIAL NONLINEAR COMPACT SYSTEMS 373
14.1 NONLINEAR COMPACT SHAPES AND COLLECTIVE MOTION 373
14.2 THE HAMILTONIAN STRUCTURE FOR FREE BOUNDARY PROBLEMS ON COMPACT
SURFACES 378
PART III PHYSICAL NONLINEAR SYSTEMS AT DIFFERENT SCALES
15 FILAMENTS, CHAINS, AND SOLITONS 385
15.1 VORTEX FILAMENTS 385
15.1.1 GAS DYNAMICS FILAMENT MODEL AND SOLITONS 391
15.1.2 SPECIAL SOLUTIONS 394
15.1.3 INTEGRATION OF SERRET-FRENET EQUATIONS FOR FILAMENTS 395 15.1.4
THE RICCATI FORM OF THE SERRET-FRENET EQUATIONS 397 15.2 SOLITON
SOLUTIONS ON THE VORTEX FILAMENT 400
15.2.1 CONSTANT TORSION VORTEX FILAMENTS 400
15.2.2 VORTEX FILAMENTS AND THE NONLINEAR SCHRODINGER EQUATION 403
15.3 CLOSED CURVES SOLITONS 406
15.4 NONLINEAR DYNAMICS OF STIFF CHAINS 408
15.5 PROBLEMS 410
16 SOLITONS ON THE BOUNDARIES OF MICROSCOPIC SYSTEMS 411
16.1 SOLITONS AS ELEMENTARY PARTICLES 412
16.2 QUANTIZATION OF SOLITONS ON A CLOSED CONTOUR AND INSTANTONS 414
16.3 CLUSTERS AS SOLITARY WAVES ON THE NUCLEAR SURFACE 417
16.4 SOLITONS AND QUASIMOLECULAR STRUCTURE 426
16.5 SOLITON MODEL FOR HEAVY EMITTED NUCLEAR CLUSTERS 428
16.5.1 QUINTIC NONLINEAR SCHRODINGER EQUATION FOR NUCLEAR CLUSTER DECAY
430
16.6 CONTOUR SOLITONS IN THE QUANTUM HALL LIQUID 433
16.6.1 PERTURBATIVE APPROACH 436
16.6.2 GEOMETRIC APPROACH 438
17 NONLINEAR CONTOUR DYNAMICS IN MACROSCOPIC SYSTEMS 445
17.1 PLASMA VORTEX 445
17.1.1 EFFECTIVE SURFACE TENSION IN MAGNETOHYDRODYNAMICS AND PLASMA
SYSTEMS 445 17.1.2 TRAJECTORIES IN MAGNETIC FIELD CONFIGURATIONS 446
17.1.3 MAGNETIC SURFACES IN STATIC EQUILIBRIUM 455
IMAGE 6
XVI CONTENTS
17.2 ELASTIC SPHERES 462
17.3 CURVATURE DEPENDENT NONLINEAR DIFFUSION ON CLOSED SURFACES- 464
17.4 NONLINEAR EVOLUTION O F OSCILLATION MODES IN NEUTRON STARS 465
18 MATHEMATICAL ANNEX 467
18.1 DIFFERENTIABLE MANIFOLDS 467
18.2 RICCATI EQUATION 468
18.3 SPECIAL FUNCTIONS 469
18.4 ONE-SOLITON SOLUTIONS FOR THE KDV, MKDV, AND THEIR COMBINATION 470
18.5 SCALING AND NONLINEAR DISPERSION RELATIONS 472
REFERENCES 475
INDEX 485 |
| any_adam_object | 1 |
| author | Ludu, Andrei |
| author_facet | Ludu, Andrei |
| author_role | aut |
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| ctrlnum | (OCoLC)796230080 (DE-599)DNB1013327195 |
| dewey-full | 530.14 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.14 |
| dewey-search | 530.14 |
| dewey-sort | 3530.14 |
| dewey-tens | 530 - Physics |
| discipline | Physik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV040141400 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-21T00:32:56Z |
| institution | BVB |
| isbn | 9783642228940 9783642228957 3642228941 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024998332 |
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| owner | DE-703 |
| owner_facet | DE-703 |
| physical | 520 S. |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer Series in Synergetics |
| spelling | Ludu, Andrei Verfasser aut Nonlinear waves and solitons on contours and closed surfaces Andrei Ludu 2. ed. Berlin [u.a.] Springer 2012 520 S. txt rdacontent n rdamedia nc rdacarrier Springer Series in Synergetics Kompakter Raum (DE-588)4164857-2 gnd rswk-swf Nichtlineare Welle (DE-588)4042102-8 gnd rswk-swf Soliton (DE-588)4135213-0 gnd rswk-swf Nichtlineare Welle (DE-588)4042102-8 s Soliton (DE-588)4135213-0 s Kompakter Raum (DE-588)4164857-2 s DE-604 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3850548&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024998332&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ludu, Andrei Nonlinear waves and solitons on contours and closed surfaces Kompakter Raum (DE-588)4164857-2 gnd Nichtlineare Welle (DE-588)4042102-8 gnd Soliton (DE-588)4135213-0 gnd |
| subject_GND | (DE-588)4164857-2 (DE-588)4042102-8 (DE-588)4135213-0 |
| title | Nonlinear waves and solitons on contours and closed surfaces |
| title_auth | Nonlinear waves and solitons on contours and closed surfaces |
| title_exact_search | Nonlinear waves and solitons on contours and closed surfaces |
| title_full | Nonlinear waves and solitons on contours and closed surfaces Andrei Ludu |
| title_fullStr | Nonlinear waves and solitons on contours and closed surfaces Andrei Ludu |
| title_full_unstemmed | Nonlinear waves and solitons on contours and closed surfaces Andrei Ludu |
| title_short | Nonlinear waves and solitons on contours and closed surfaces |
| title_sort | nonlinear waves and solitons on contours and closed surfaces |
| topic | Kompakter Raum (DE-588)4164857-2 gnd Nichtlineare Welle (DE-588)4042102-8 gnd Soliton (DE-588)4135213-0 gnd |
| topic_facet | Kompakter Raum Nichtlineare Welle Soliton |
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| work_keys_str_mv | AT luduandrei nonlinearwavesandsolitonsoncontoursandclosedsurfaces |