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. Table Of Contents Preface 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