Solitons an introduction
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| Hauptverfasser: | , |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
1996
|
| Ausgabe: | Reprinted |
| Schriftenreihe: | Cambridge texts in applied mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
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Datensatz im Suchindex
| _version_ | 1819575916738641920 |
|---|---|
| adam_text | Contents
Preface page xi
1 The Korteweg-de Vries equation 1
1 1 Preliminaries 1
1 2 The discovery of solitary waves 7
1 3 The discovery of soliton interactions 12
1 4 Applications of the KdV equation 15
Further reading 16
Exercises 17
2 Elementary solutions of the Korteweg-de Vries equation 20
2 1 Travelling-wave solutions 20
2 2 Solitary waves 21
2 3 General waves of permanent form 22
2 4 Description in terms of elliptic functions 26
2 5 Limiting behaviours of the cnoidal wave 29
2 6 Other solutions of the KdV equation 30
Further reading 32
Exercises 33
3 The scattering and inverse scattering problems 39
3 1 Preamble 39
3 2 The scattering problem 40
Example (i): the delta function 44
Example (ii): the sech2 function 45
3 3 The inverse scattering problem 48
3 4 The solution of the Marchenko equation 56
Example (i): reflection coefficient with one pole 57
Example (ii): zero reflection coefficient 58
Further reading 60
Exercises 61
Contents
viii
4 The initial-value problem for the Korteweg-de Vries equation 64
4 1 Recapitulation 64
4 2 Inverse scattering and the KdV equation 65
4 3 Time evolution of the scattering data 67
431 Discrete spectrum 68
432 Continuous spectrum 70
4 4 Construction of the solution: summary 71
4 5 Reflectionless potentials 72
Example (i): solitary wave 73
Example (ii): two-soliton solution 74
Example (iii): /V-soliton solution 78
4 6 Description of the solution when b{k) # 0 81
Example (i): delta-function initial profile 81
Example (ii): a negative sech2 initial profile 83
Example (iii): a positive sech2 initial profile 83
Further reading 86
Exercises 86
5 Further properties of the Korteweg-de Vries equation 89
5 1 Conservation laws 89
511 Introduction 89
512 An infinity of conservation laws 92
513 Of Lagrangians and Hamiltonians 95
5 2 Lax formulation and its KdV hierarchy 97
521 Description of the method: operators 97
522 The Lax KdV hierarchy 99
5 3 Hirota’s method: the bilinear form 102
531 The bilinear operator 103
532 The solution of the bilinear equation 106
5 4 Bäcklund transformations 109
541 Introductory ideas 110
542 Bäcklund transformation for the KdV equation 112
543 The KdV Bäcklund transformation: an algebraic relation 114
544 Bäcklund transformations and the bilinear form 116
Further reading 118
Exercises 118
6 More general inverse methods 127
6 1 The AKNS scheme 128
611 The 2x2 eigenvalue problem 128
Contents ix
612 The inverse scattering problem 131
613 An example: r = — q, q=/ sech Xx 134
614 Time evolution of the scattering data 137
615 The evolution equations for q and r 140
(a) Quadratic in ( 141
(b) Polynomial in (“1 142
(c) General function of ( 143
6 2 The ZS scheme 144
621 The integral operators 144
622 The differential operators 146
623 Scalar operators 149
(a) The KdV equation 149
(b) The two-dimensional KdV equation 151
624 Matrix operators 152
(a) The nonlinear Schrödinger equation 152
(b) The sine-Gordon equation 154
6 3 Two examples 157
Example (i): The nonlinear Schrödinger equation 157
Example (ii): The sine-Gordon equation 160
Further reading 162
Exercises 162
7 The Painleve property, perturbations and numerical methods 169
7 1 The Painleve property 169
711 Painleve equations 169
712 The Painleve conjecture 171
713 Linearisation of the Painleve equations 172
7 2 Perturbation theory 174
721 Perturbation theory: an example 177
7 3 Numerical methods 180
731 Spectral methods 182
732 Finite-difference methods 183
733 Long-wave equations 184
734 Nonlinear Klein-Gordon equations 186
735 The nonlinear Schrödinger equation 187
Further reading 187
Exercises 187
8 Epilogue 190
8 1 Some numerical solutions of nonlinear evolution equations 190
X
Contents
8 2 Applications of nonlinear evolution equations 196
Further reading 200
Exercises 201
Answers and hints 205
Bibliography and author index 213
Motion picture index 220
|
| any_adam_object | 1 |
| author | Drazin, Philip G. 1934-2002 Johnson, R. S. 1944- |
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| author_sort | Drazin, Philip G. 1934-2002 |
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| dewey-ones | 515 - Analysis |
| dewey-raw | 515.355 |
| dewey-search | 515.355 |
| dewey-sort | 3515.355 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| edition | Reprinted |
| format | Book |
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| genre_facet | Einführung |
| id | DE-604.BV014278004 |
| illustrated | Illustrated |
| indexdate | 2024-12-23T15:51:25Z |
| institution | BVB |
| isbn | 052133389X 0521336554 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009791412 |
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| physical | XII, 226 S. graph. Darst. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series2 | Cambridge texts in applied mathematics |
| spellingShingle | Drazin, Philip G. 1934-2002 Johnson, R. S. 1944- Solitons an introduction Partielle Differentialgleichung (DE-588)4044779-0 gnd Soliton (DE-588)4135213-0 gnd Algebraische Gleichung (DE-588)4001162-8 gnd Störungstheorie (DE-588)4128420-3 gnd Approximationsalgorithmus (DE-588)4500954-5 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4135213-0 (DE-588)4001162-8 (DE-588)4128420-3 (DE-588)4500954-5 (DE-588)4151278-9 |
| title | Solitons an introduction |
| title_auth | Solitons an introduction |
| title_exact_search | Solitons an introduction |
| title_full | Solitons an introduction P. G. Drazin ; R. S. Johnson |
| title_fullStr | Solitons an introduction P. G. Drazin ; R. S. Johnson |
| title_full_unstemmed | Solitons an introduction P. G. Drazin ; R. S. Johnson |
| title_short | Solitons |
| title_sort | solitons an introduction |
| title_sub | an introduction |
| topic | Partielle Differentialgleichung (DE-588)4044779-0 gnd Soliton (DE-588)4135213-0 gnd Algebraische Gleichung (DE-588)4001162-8 gnd Störungstheorie (DE-588)4128420-3 gnd Approximationsalgorithmus (DE-588)4500954-5 gnd |
| topic_facet | Partielle Differentialgleichung Soliton Algebraische Gleichung Störungstheorie Approximationsalgorithmus Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009791412&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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