Symmetry, Phase Modulation and Nonlinear Waves

Bridges studies the origin of Korteweg–de Vries equation using phase modulation and its implications in dynamical systems and nonlinear waves.

Thomas J. Bridges (Author)

9781107188846, Cambridge University Press

Hardback, published 3 July 2017

236 pages, 12 b/w illus.
23.5 x 15.7 x 1.7 cm, 0.46 kg

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

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.

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.

Subject Areas: Fluid mechanics [PHDF], Nonlinear science [PBWR]