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Symmetry, Phase Modulation and Nonlinear Waves

Symmetry, Phase Modulation and Nonlinear Waves

$87.99 (C)

Part of Cambridge Monographs on Applied and Computational Mathematics

  • Date Published: July 2017
  • availability: In stock
  • format: Hardback
  • isbn: 9781107188846

$ 87.99 (C)

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About the Authors
  • 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.

    • Leads to a deeper understanding of the underlying theory of phase modulation
    • Encourages new interpretations of well-known examples that point to new applications of model equations
    • Provides a new tool for fluids and waves applications that opens the door to new simplified models
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    Reviews & endorsements

    'This book has been written by a well-established researcher in the field. His expertise is evidenced by the deft exposition of relatively challenging material. In that regard, one of the very useful functions of this book is its provision of a number of background mathematical techniques in Hamiltonians systems, symplectic geometry, Noether theory and Lagrangian field theory.' K. Alan Shore, Contemporary Physics

    'The book is clearly written, and only the most basic knowledge of Hamiltonian and Lagrangian theories is required.' Wen-Xiu Ma, MathSciNet

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    Product details

    • Date Published: July 2017
    • format: Hardback
    • isbn: 9781107188846
    • length: 236 pages
    • dimensions: 235 x 157 x 17 mm
    • weight: 0.46kg
    • contains: 12 b/w illus.
    • availability: In stock
  • Table of Contents

    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

  • Author

    Thomas J. Bridges, University of Surrey
    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.

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