The dynamics of modulated wave trains:
Saved in:
| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Providence, R.I.
American Math. Soc.
2009
|
| Series: | Memoirs of the American Mathematical Society
934 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VII, 105 S. |
| ISBN: | 9780821842935 |
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Record in the Search Index
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| adam_text | Titel: The dynamics of modulated wave trains
Autor: Doelman, Arjen
Jahr: 2009
Contents
Notation 1
Chapter 1. Introduction 3
1.1. Grasshopper s guide 3
1.2. Slowly-varying modulations of nonlinear wave trains 4
1.3. Predictions from the Burgers equation 7
1.4. Verifying the predictions made from the Burgers equation 8
1.5. Related modulation equations 12
1.6. References to related works 13
Chapter 2. The Burgers equation 15
2.1. Decay estimates 15
2.2. Fronts in the Burgers equation 17
Chapter 3. The complex cubic Ginzburg-Landau equation 19
3.1. Set-up 19
3.2. Slowly-varying modulations of the k = 0 wave train: Results 20
3.3. Derivation of the Burgers equation 23
3.4. The construction of higher-order approximations 24
3.5. The approximation theorem for the wave numbers 25
3.6. Mode filters, and separation into critical and noncritical modes 25
3.7. Estimates of the linear semigroups 29
3.8. Estimates of the residual 30
3.9. Estimates of the errors 31
3.10. Proofs of the theorems from §3.2 34
Chapter 4. Reaction-diffusion equations: Set-up and results 39
4.1. The abstract set-up 39
4.2. Expansions of the linear and nonlinear dispersion relations 41
4.3. Formal derivation of the Burgers equation 43
4.4. Validity of the Burgers equation 45
4.5. Existence and stability of weak shocks 48
Chapter 5. Validity of the Burgers equation in reaction-diffusion equations 53
5.1. From phases to wave numbers 53
5.2. Bloch-wave analysis 55
5.3. Mode filters, and separation into critical and noncritical modes 58
5.4. Estimates for residuals and errors 61
5.5. Proofs of the theorems from §4.4 63
V
vi CONTENTS
Chapter 6. Validity of the inviscid Burgers equation in reaction-diffusion
systems 65
6.1. An illustration: The Ginzburg—Landau equation 65
6.2. Formal derivation of the conservation law 66
6.3. Validity of the inviscid Burgers equation 67
6.4. Proof of the theorems from §6.3 68
Chapter 7. Modulations of wave trains near sideband instabilities 73
7.1. Introduction 73
7.2. An illustration: The Ginzburg-Landau equation 74
7.3. Validity of the Korteweg-de Vries and the Kuramoto-Sivashinsky
equation 75
7.4. Proof of Theorem 7.2 78
7.5. Proof of Theorem 7.5 79
Chapter 8. Existence and stability of weak shocks 83
8.1. Proof of Theorem 4.10 83
8.2. Proof of Theorem 4.12 88
Chapter 9. Existence of shocks in the long-wavelength limit 93
9.1. A lattice model for weakly interacting pulses 93
9.2. Proof of Theorem 9.2 95
Chapter 10. Applications 99
10.1. The FitzHugh-Nagumo equation 99
10.2. The weakly unstable Taylor-Couette problem 100
Bibliography 103
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| indexdate | 2024-07-09T21:36:10Z |
| institution | BVB |
| isbn | 9780821842935 |
| language | English |
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| physical | VII, 105 S. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
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| series | Memoirs of the American Mathematical Society |
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| spelling | The dynamics of modulated wave trains Arjen Doelman ... Providence, R.I. American Math. Soc. 2009 VII, 105 S. txt rdacontent n rdamedia nc rdacarrier Memoirs of the American Mathematical Society 934 Aproximação (teoria) larpcal Equações diferenciais da física larpcal Equações diferenciais parciais parabólicas larpcal Reaction-diffusion equations Approximation theory Burgers equation Reaktions-Diffusionsgleichung (DE-588)4323967-5 gnd rswk-swf Burgers-Gleichung (DE-588)4529578-5 gnd rswk-swf Reaktions-Diffusionsgleichung (DE-588)4323967-5 s Burgers-Gleichung (DE-588)4529578-5 s DE-604 Doelman, Arjen Sonstige (DE-588)1294378279 oth Memoirs of the American Mathematical Society 934 (DE-604)BV008000141 934 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017397346&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | The dynamics of modulated wave trains Memoirs of the American Mathematical Society Aproximação (teoria) larpcal Equações diferenciais da física larpcal Equações diferenciais parciais parabólicas larpcal Reaction-diffusion equations Approximation theory Burgers equation Reaktions-Diffusionsgleichung (DE-588)4323967-5 gnd Burgers-Gleichung (DE-588)4529578-5 gnd |
| subject_GND | (DE-588)4323967-5 (DE-588)4529578-5 |
| title | The dynamics of modulated wave trains |
| title_auth | The dynamics of modulated wave trains |
| title_exact_search | The dynamics of modulated wave trains |
| title_full | The dynamics of modulated wave trains Arjen Doelman ... |
| title_fullStr | The dynamics of modulated wave trains Arjen Doelman ... |
| title_full_unstemmed | The dynamics of modulated wave trains Arjen Doelman ... |
| title_short | The dynamics of modulated wave trains |
| title_sort | the dynamics of modulated wave trains |
| topic | Aproximação (teoria) larpcal Equações diferenciais da física larpcal Equações diferenciais parciais parabólicas larpcal Reaction-diffusion equations Approximation theory Burgers equation Reaktions-Diffusionsgleichung (DE-588)4323967-5 gnd Burgers-Gleichung (DE-588)4529578-5 gnd |
| topic_facet | Aproximação (teoria) Equações diferenciais da física Equações diferenciais parciais parabólicas Reaction-diffusion equations Approximation theory Burgers equation Reaktions-Diffusionsgleichung Burgers-Gleichung |
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