Nonlinear resonance analysis is a unique mathematical tool that can be used to study resonances in relation to, but independently of, any single area of application. This is the first book to present the theory of nonlinear resonances as a new scientific field, with its own theory, computational methods, applications and open questions. The book includes several worked examples, mostly taken from fluid dynamics, to explain the concepts discussed. Each chapter demonstrates how nonlinear resonance analysis can be applied to real systems, including large-scale phenomena in the Earth's atmosphere and novel wave turbulent regimes, and explains a range of laboratory experiments. The book also contains a detailed description of the latest computer software in the field. It is suitable for graduate students and researchers in nonlinear science and wave turbulence, along with fluid mechanics and number theory. Colour versions of a selection of the figures are available at www.cambridge.org/9780521763608.
Author(s): Elena Kartashova
Edition: 1
Publisher: Cambridge University Press
Year: 2010
Language: English
Pages: 241
Tags: Математика;Математическая физика;
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 11
Glossary......Page 16
1.1 An easy start......Page 19
Superposition principle......Page 22
Resonance conditions......Page 25
Mathematical classification......Page 29
Physical classification......Page 30
Multi-scale method......Page 34
Notion of Hamiltonian......Page 37
Dynamical equations......Page 40
2.1 An easy start......Page 48
2.2 Irrational dispersion function, analytical results......Page 50
Complete solution of (2.21)......Page 54
Structural properties of q-classes......Page 55
Multiplicative properties of solutions......Page 56
Parametric series of solutions......Page 62
One q-class......Page 63
Two q-classes......Page 69
Implementation results......Page 72
2.4 Rational dispersion function......Page 75
2.5 General form of dispersion function......Page 78
3.1 An easy start......Page 82
3.2 Topological structure vs dynamical system......Page 84
3.3 Three-wave resonances......Page 87
Hypergraph presentation......Page 89
Incidence matrix......Page 90
Multigraph construction......Page 92
3.4 Four-wave resonances......Page 94
Noncollinear quartets......Page 95
Clusters......Page 97
Mixed cascades......Page 99
3.5 NR-diagrams......Page 100
3.6 What is beyond kinematics?......Page 106
4.1 An easy start......Page 108
Decay instability of a triad......Page 111
Decay instability of a quartet......Page 112
Reduction types......Page 113
Time-dependent conservation laws......Page 114
Dynamical invariant for a triad......Page 115
Solutions for amplitudes......Page 117
Degenerate initial conditions......Page 118
Solutions for phases......Page 121
Integrable clusters......Page 124
Solution trajectory......Page 129
Amplitudes’ evolution......Page 130
4.5 A quartet......Page 134
4.6 Explosive instability......Page 135
Explosive instability of a quartet......Page 138
4.7 NR-reduced numerical models......Page 140
4.8 What is beyond dynamics?......Page 145
5.1 Linear pendulum......Page 148
5.2 Elastic pendulum......Page 156
6.1 An easy start......Page 162
6.2 Quasi-resonances vs approximate interactions......Page 165
Interaction scales......Page 168
Inertial interval vs k-space......Page 170
Main conclusions from the model of laminated turbulence......Page 171
Characteristic wavelengths......Page 172
Characteristic resonance broadening......Page 174
Discrete layer......Page 176
6.5 Rotational capillary waves......Page 179
Dispersion function......Page 180
Dimension of flows with constant vorticity......Page 184
Resonance clustering......Page 186
6.6 Discrete regimes in various wave systems......Page 188
6.7 Open problems......Page 194
7 Epilogue......Page 200
Three-wave resonances......Page 203
Four-wave resonances......Page 208
Benchmarks and evaluation......Page 212
Topological structure of the solution set......Page 213
Coupling coefficient......Page 217
A.2 Mathematical services......Page 220
A.3 A web portal......Page 224
References......Page 227
Index......Page 239
