Nonlinear Ocean Waves & the Inverse Scattering Transform


  • Alfred Osborne, University of Torino, Italy

For more than 200 years, the Fourier Transform has been one of the most important mathematical tools for understanding the dynamics of linear wave trains. Nonlinear Ocean Waves and the Inverse Scattering Transform presents the development of the nonlinear Fourier analysis of measured space and time series, which can be found in a wide variety of physical settings including surface water waves, internal waves, and equatorial Rossby waves. This revolutionary development will allow hyperfast numerical modelling of nonlinear waves, greatly advancing our understanding of oceanic surface and internal waves. Nonlinear Fourier analysis is based upon a generalization of linear Fourier analysis referred to as the inverse scattering transform, the fundamental building block of which is a generalized Fourier series called the Riemann theta function. Elucidating the art and science of implementing these functions in the context of physical and time series analysis is the goal of this book.
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Academia, government, and industry in the fields of ocean sciences, geophysics, and engineering mathematics.


Book information

  • Published: March 2010
  • ISBN: 978-0-12-528629-9


"The book is an encyclopaedia of mathematical models of nonlinear ocean waves and methods of their investigation."--Zentralblatt MATH 1250

"The book should be of great interest not only  to oceanographers, but to others interested in the intriguing physics of nonlinear waves. Much of the theory developed in the book applies to other branches of physics, such as plasma physics and nonlinear optics. Because of the high quality of writing, this book may make an ideal focus for graduate-level seminars as well as being a comprehensive reference text on nonlinear water waves."--Pure and Applied Geophysics

Table of Contents



1. Brief History of Nonlinear Water Waves, Fourier analysis and Integrability

Part I: Theoretical and Physical Considerations

2. Theories of Water Waves

3. Introduction to Linear and Nonlinear Fourier analysis

4. The Infinite-Line Inverse Scattering Transform

5. The Periodic Inverse Scattering Transform

6. Additional Theoretical Considerations

7. Physical Perspective and Exotic Solutions

8. Extending the Inverse Scattering Transform to Higher Order

Part II: Numerical and Time Series Analysis Algorithms

9. Numerical Methods for Infinite-Line Boundary Conditions

10. Numerical Algorithms for the Analysis of Nonlinear Space and Time Series

11. Procedures for the Analysis of Nonlinear Wave Data

12. Nonlinear Power Spectral Analysis

Part II: Time Series Analysis of Experimental Data

13. Surface Waves in the Laboratory

14. Surface Waves in the Adriatic Sea

15. Surface Waves at Duck Pier

16. Surface Waves in the North Sea

17. Internal Waves in the Andaman Sea

Part IV: Future Directions

18. Engineering Applications of IST

19. Chaotic Dynamics Nonlinear Wave Equations and Water Waves