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. This book 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. It presents techniques and methods of the inverse scattering transform for data analysis.
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
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