Thomas J. Bridges

Symmetry, Phase Modulation and Nonlinear Waves (eBook, PDF)

• Format: PDF

Produktbeschreibung

Nonlinear waves are pervasive in nature, but are often elusive when they are modelled and analysed. This book develops a natural approach to the problem based on phase modulation. It is both an elaboration of the use of phase modulation for the study of nonlinear waves and a compendium of background results in mathematics, such as Hamiltonian systems, symplectic geometry, conservation laws, Noether theory, Lagrangian field theory and analysis, all of which combine to generate the new theory of phase modulation. While the build-up of theory can be intensive, the resulting emergent partial differential equations are relatively simple. A key outcome of the theory is that the coefficients in the emergent modulation equations are universal and easy to calculate. This book gives several examples of the implications in the theory of fluid mechanics and points to a wide range of new applications.

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Produktdetails

• Verlag: Cambridge University Press
• Erscheinungstermin: 3. Juli 2017
• Englisch
• ISBN-13: 9781108102926
• Artikelnr.: 48866231

Autorenporträt

Thomas J. Bridges is currently Professor of Mathematics at the University of Surrey. He has been researching the theory of nonlinear waves for over 25 years. He is co-editor of the volume Lectures on the Theory of Water Waves (Cambridge, 2016) and he has over 140 published papers on such diverse topics as multisymplectic structures, Hamiltonian dynamics, ocean wave energy harvesting, geometric numerical integration, stability of nonlinear waves, the geometry of the Hopf bundle, theory of water waves and phase modulation.

Inhaltsangabe

1. Introduction
2. Hamiltonian ODEs and relative equilibria
3. Modulation of relative equilibria
4. Revised modulation near a singularity
5. Introduction to Whitham Modulation Theory - the Lagrangian viewpoint
6. From Lagrangians to Multisymplectic PDEs
7. Whitham Modulation Theory - the multisymplectic viewpoint
8. Phase modulation and the KdV equation
9. Classical view of KdV in shallow water
10. Phase modulation of uniform flows and KdV
11. Generic Whitham Modulation Theory in 2+1
12. Phase modulation in 2+1 and the KP equation
13. Shallow water hydrodynamics and KP
14. Modulation of three-dimensional water waves
15. Modulation and planforms
16. Validity of Lagrangian-based modulation equations
17. Non-conservative PDEs and modulation
18. Phase modulation - extensions and generalizations
Appendix A. Supporting calculations - 4th and 5th order terms
Appendix B. Derivatives of a family of relative equilibria
Appendix C. Bk and the spectral problem
Appendix D. Reducing dispersive conservation laws to KdV
Appendix E. Advanced topics in multisymplecticity
References
Index.