Solitons: an introduction
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
1996
|
| Edition: | Reprinted |
| Series: | Cambridge texts in applied mathematics
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 226 S. graph. Darst. |
| ISBN: | 052133389X 0521336554 |
Staff View
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Record in the Search Index
| _version_ | 1804129168545284096 |
|---|---|
| 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
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| any_adam_object | 1 |
| author | Drazin, Philip G. 1934-2002 Johnson, R. S. 1944- |
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| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| edition | Reprinted |
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| language | English |
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| spelling | Drazin, Philip G. 1934-2002 Verfasser (DE-588)128929146 aut Solitons an introduction P. G. Drazin ; R. S. Johnson Reprinted Cambridge [u.a.] Cambridge Univ. Press 1996 XII, 226 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge texts in applied mathematics Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Soliton (DE-588)4135213-0 gnd rswk-swf Algebraische Gleichung (DE-588)4001162-8 gnd rswk-swf Störungstheorie (DE-588)4128420-3 gnd rswk-swf Approximationsalgorithmus (DE-588)4500954-5 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content Soliton (DE-588)4135213-0 s DE-604 Approximationsalgorithmus (DE-588)4500954-5 s Algebraische Gleichung (DE-588)4001162-8 s Partielle Differentialgleichung (DE-588)4044779-0 s Störungstheorie (DE-588)4128420-3 s Johnson, R. S. 1944- Verfasser (DE-588)122078942 aut HEBIS Datenaustausch application/pdf 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 Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| 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 |
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