A course on nonlinear waves:
This volume presents a carefully written introduction to nonlinear waves in the natural sciences and engineering. It not only contains many classical results but also includes more recent results, dealing with topics such as the forced Korteweg-de Vries equation and material relating to X-ray crysta...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Dordrecht u.a.
Kluwer
1993
|
| Schriftenreihe: | Nonlinear topics in the mathematical sciences
3 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | This volume presents a carefully written introduction to nonlinear waves in the natural sciences and engineering. It not only contains many classical results but also includes more recent results, dealing with topics such as the forced Korteweg-de Vries equation and material relating to X-ray crystallography The book contains nine chapters. Chapter 1 concerns asymptotics and nonlinear ordinary differential equations. Conservation laws are discussed in Chapter 2, and Chapter 3 considers water waves. The scattering and inverse scattering method is described in Chapter 4, which also contains a full explanation of using the inverse scattering method for finding 1-, 2- and 3-soliton solutions of the Korteweg-de Vries equation Chapter 5 studies the Burgers' equation and Chapter 6 discusses the forced Korteweg-de Vries equations. Here the emphasis is on steady-state bifurcations and unsteady-state periodic soliton generation. The Sine-Gordon and nonlinear Schrodinger equations are the subject of Chapter 7. The final two chapters consider wave instability and resonance. Every chapter contains problems and exercises, together with guidance for their solution. The volume concludes with some appendices which describe symbolic derivations of certain results on solitons. Several user-friendly MATHEMATICA packages are included |
| Beschreibung: | XIV, 327 S. Ill., graph. Darst. |
| ISBN: | 0792322924 |
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| 100 | 1 | |a Shen, Samuel S. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a A course on nonlinear waves |c by Samuel S. Shen |
| 264 | 1 | |a Dordrecht u.a. |b Kluwer |c 1993 | |
| 300 | |a XIV, 327 S. |b Ill., graph. Darst. | ||
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| 490 | 1 | |a Nonlinear topics in the mathematical sciences |v 3 | |
| 520 | 3 | |a This volume presents a carefully written introduction to nonlinear waves in the natural sciences and engineering. It not only contains many classical results but also includes more recent results, dealing with topics such as the forced Korteweg-de Vries equation and material relating to X-ray crystallography | |
| 520 | |a The book contains nine chapters. Chapter 1 concerns asymptotics and nonlinear ordinary differential equations. Conservation laws are discussed in Chapter 2, and Chapter 3 considers water waves. The scattering and inverse scattering method is described in Chapter 4, which also contains a full explanation of using the inverse scattering method for finding 1-, 2- and 3-soliton solutions of the Korteweg-de Vries equation | ||
| 520 | |a Chapter 5 studies the Burgers' equation and Chapter 6 discusses the forced Korteweg-de Vries equations. Here the emphasis is on steady-state bifurcations and unsteady-state periodic soliton generation. The Sine-Gordon and nonlinear Schrodinger equations are the subject of Chapter 7. The final two chapters consider wave instability and resonance. Every chapter contains problems and exercises, together with guidance for their solution. The volume concludes with some appendices which describe symbolic derivations of certain results on solitons. Several user-friendly MATHEMATICA packages are included | ||
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-005870497 | ||
Datensatz im Suchindex
| _version_ | 1804123230367121408 |
|---|---|
| adam_text | Contents
Preface xi
Chapter 1
Asymptotic Expansion 1
1.1 Concepts of Asymptoticity 2
1.1.1 Is a divergent series useful? 2
1.1.2 The symbols ~, o and O 5
1.1.3 Asymptotic sequences 6
1.2 Method of Multiple Scales 10
1.2.1 Introduction 10
1.2.2 Description of the method of multiple scales 12
1.2.3 The van der Pol oscillator 15
1.2.4 A forced oscillator and its stability 18
Chapter 2
Hyperbolic Waves 25
2.1 Conservation Laws 26
2.1.1 The traffic flow problem 26
2.1.2 Conservation laws in a continuum media 28
2.1.3 Jump boundary conditions 33
2.2 Characteristic Method 35
2.2.1 Linear initial value problem 35
2.2.2 Nonlinearity and wave breaking 41
2.2.3 Shocks 45
2.2.4 Entropy condition 46
2.2.5 Shock structure 49
Chapter 3
Water Waves 53
3.1 Governing Equations for Water Waves 53
3.1.1 Euler equations 53
3.1.2 Potential flow 54
3.2 Shallow Water Equations 58
3.2.1 Shallow water equations 58
3.2.2 Wave breaking on a beach 62
viii Contents
3.3 Dispersive Water Waves 64
3.3.1 Dispersive waves 64
3.3.2 Boussinesq equations and the KdV equation 66
3.3.3 Solutions to Korteweg de Vries equations 70
Chapter 4
Scattering and Inverse Scattering 75
4.1 Scattering Method 77
4.1.1 String spring scattering 77
4.1.2 Linear Schrodinger equation 78
4.2 Inverse Scattering for the KdV 82
4.2.1 Inverse scattering method 82
4.2.2 KdV solitons 85
4.2.3 KdV solitons with a wake 99
4.3 Lax Pair and KdV Hierarchy* 102
4.4 Backlund Transform* 105
4.5 Derivation of Inverse Scattering Method* 115
4.6 Soliton Fission 119
Chapter 5
Burgers Equation 123
5.1 Viscous Fluid on an Inclined Plate 124
5.2 Cole Hopf Transformation 131
5.3 Stability of the Burgers Shock Wave 140
5.4 Interfacial Boundary Conditions of Two Viscous Fluids* .... 142
Chapter 6
Forced KdV Equation 147
6.1 An Ideal Flow Over a Small Bump 148
6.2 Supercritical Solitary Waves 153
6.2.1 Locally forced supercritical waves 154
6.2.2 Non locally forced supercritical waves 156
6.3 Subcritical Cnoidal Waves and Hydraulic Fall 161
6.3.1 Locally forced subcritical flows 162
6.3.2 Non locally forced subcritical flows 165
6.4 Transcritical Periodic Soliton Radiation 167
6.4.1 Approximate solutions and mass momentum energy work
relationship 168
6.4.2 Spectral method for finding locally forced solitons . . . 174
6.4.3 Program of the spectral scheme in Mathematica .... 178
6.5 Stability of Solitary Waves 183
Contents ix
Chapter 7
Sine Gordon and Nonlinear Schrodinger 189
7.1 Physical Models of the Sine Gordon Equation 190
7.1.1 Coupled torsional pendulums 191
7.1.2 One dimensional crystal dislocation 192
7.1.3 Magnetic flux in a long one dimensional Josephson junction 193
7.2 Solutions of the Sine Gordon Equation 196
7.3 Optical Self focusing 206
7.3.1 Pulse broadening due to dispersion 206
7.3.2 Optical self focusing 209
7.4 A Simple Solution of the NLS 212
7.5 Arctan Type of Solutions of the sG 213
Chapter 8
Selected Examples of Flow Instabilities 219
8.1 Concept of Stability 220
8.2 Kelvin Helmholtz: Gravitational Instability 225
8.3 Benard Problem: Thermal Instability 230
8.4 Taylor Problem: Centrifugal Instability 237
8.5 Benjamin Feir: Side Band Instability 242
Chapter 9
Wave Interactions and X Ray Crystallography 247
9.1 Wave Interactions 248
9.1.1 Introduction 248
9.1.2 Forced harmonic motion 249
9.1.3 Resonance conditions for nonlinear systems 250
9.1.4 Four wave interactions 255
9.1.5 Nonlinear wave interactions in other systems 258
9.2 Phase Problem in X ray Crystallography 258
9.2.1 Bragg s law of X ray diffraction 258
9.2.2 Fourier representation of electron density 259
9.2.3 Coordinates in crystal cells 263
9.2.4 The phase problem 265
9.2.5 Structure Invariants 267
9.2.6 Neighborhood principle 270
9.2.7 Probability distributions of structure invariants 270
Appendix A
KdV Solitons via Inverse Scattering 277
Appendix B
KdV Solitons via Backlund Transform 283
B.I Backlund Transform Program 283
X Contents
B.2 TwoSolitons 289
B.3 Three Solitons 289
B.4 Four Solitons 290
B.5 Five Solitons 291
B.6 Six Solitons 293
B.7 Seven Solitons 297
Appendix C
Derivation of the Stationary KdV 309
Bibliography 317
Index 323
|
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| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
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| id | DE-604.BV008875736 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:26:28Z |
| institution | BVB |
| isbn | 0792322924 |
| language | English |
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| physical | XIV, 327 S. Ill., graph. Darst. |
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| series | Nonlinear topics in the mathematical sciences |
| series2 | Nonlinear topics in the mathematical sciences |
| spelling | Shen, Samuel S. Verfasser aut A course on nonlinear waves by Samuel S. Shen Dordrecht u.a. Kluwer 1993 XIV, 327 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Nonlinear topics in the mathematical sciences 3 This volume presents a carefully written introduction to nonlinear waves in the natural sciences and engineering. It not only contains many classical results but also includes more recent results, dealing with topics such as the forced Korteweg-de Vries equation and material relating to X-ray crystallography The book contains nine chapters. Chapter 1 concerns asymptotics and nonlinear ordinary differential equations. Conservation laws are discussed in Chapter 2, and Chapter 3 considers water waves. The scattering and inverse scattering method is described in Chapter 4, which also contains a full explanation of using the inverse scattering method for finding 1-, 2- and 3-soliton solutions of the Korteweg-de Vries equation Chapter 5 studies the Burgers' equation and Chapter 6 discusses the forced Korteweg-de Vries equations. Here the emphasis is on steady-state bifurcations and unsteady-state periodic soliton generation. The Sine-Gordon and nonlinear Schrodinger equations are the subject of Chapter 7. The final two chapters consider wave instability and resonance. Every chapter contains problems and exercises, together with guidance for their solution. The volume concludes with some appendices which describe symbolic derivations of certain results on solitons. Several user-friendly MATHEMATICA packages are included Mecanica, elasticidade e reologia larpcal Nonlinear waves Nichtlineare Welle (DE-588)4042102-8 gnd rswk-swf Nichtlineare Welle (DE-588)4042102-8 s DE-604 Nonlinear topics in the mathematical sciences 3 (DE-604)BV004223528 3 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005870497&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Shen, Samuel S. A course on nonlinear waves Nonlinear topics in the mathematical sciences Mecanica, elasticidade e reologia larpcal Nonlinear waves Nichtlineare Welle (DE-588)4042102-8 gnd |
| subject_GND | (DE-588)4042102-8 |
| title | A course on nonlinear waves |
| title_auth | A course on nonlinear waves |
| title_exact_search | A course on nonlinear waves |
| title_full | A course on nonlinear waves by Samuel S. Shen |
| title_fullStr | A course on nonlinear waves by Samuel S. Shen |
| title_full_unstemmed | A course on nonlinear waves by Samuel S. Shen |
| title_short | A course on nonlinear waves |
| title_sort | a course on nonlinear waves |
| topic | Mecanica, elasticidade e reologia larpcal Nonlinear waves Nichtlineare Welle (DE-588)4042102-8 gnd |
| topic_facet | Mecanica, elasticidade e reologia Nonlinear waves Nichtlineare Welle |
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