Nonlinear Resonance Analysis: Theory, Computation, Applications
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.
1111388643
Nonlinear Resonance Analysis: Theory, Computation, Applications
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.
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Nonlinear Resonance Analysis: Theory, Computation, Applications

Nonlinear Resonance Analysis: Theory, Computation, Applications

by Elena Kartashova
Nonlinear Resonance Analysis: Theory, Computation, Applications
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Nonlinear Resonance Analysis: Theory, Computation, Applications

Nonlinear Resonance Analysis: Theory, Computation, Applications

by Elena Kartashova

Overview

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.

Product Details

ISBN-13: 9780521763608
Publisher: Cambridge University Press
Publication date: 10/21/2010
Pages: 240
Product dimensions: 6.90(w) x 9.80(h) x 0.60(d)

About the Author

Elena Kartashova is a Professor at the Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz. Her research interests include nonlinear resonances in various application areas, from fluid mechanics and meteorology to engineering. She was awarded the V. I. Vernadsky medal by the Russian Academy of Natural Sciences in 2009.

Table of Contents

Preface ix

Glossary xiv

1 Exposition 1

1.1 An easy start 1

1.2 Two classifications of PDEs 11

1.3 Hamiltonian formalism 19

2 Kinematics: Wavenumbers 30

2.1 An easy start 30

2.2 Irrational dispersion function, analytical results 32

2.3 q-class decomposition 45

2.4 Rational dispersion function 57

2.5 General form of dispersion function 60

3 Kinematics: Resonance clusters 64

3.1 An easy start 64

3.2 Topological structure vs dynamical system 66

3.3 Three-wave resonances 69

3.4 Four-wave resonances 76

3.5 NR-diagrams 82

3.6 What is beyond kinematics? 88

4 Dynamics 90

4.1 An easy start 90

4.2 Decay instability 93

4.3 A triad 96

4.4 Clusters of triads 106

4.5 A quartet 116

4.6 Explosive instability 117

4.7 NR-reduced numerical models 122

4.8 What is beyond dynamics? 127

5 Mechanical playthings 130

5.1 Linear pendulum 130

5.2 Elastic pendulum 138

6 Wave turbulent regimes 144

6.1 An easy start 144

6.2 Quasi-resonances vs approximate interactions 147

6.3 Model of laminated turbulence 150

6.4 Energy cascades: dynamic vs kinetic 158

6.5 Rotational capillary waves 161

6.6 Discrete regimes in various wave systems 170

6.7 Open problems 176

7 Epilogue 182

Appendix Software 185

References 209

Index 221

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