Nonlinear Ocean Waves and the Inverse Scattering Transform

Alfred Osborne

• Publisher: Academic Press
• Year: 2010
• Print ISBN: 9780125286299
• eBook ISBN: 9780080925103
• Topic: Physical Sciences
• Subtopic: Oceanography

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 asthe 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.- Presents techniques and methods of the inverse scattering transform for data analysis- Geared toward both the introductory and advanced reader venturing further into mathematical and numerical analysis- Suitable for classroom teaching as well as research

Table of contents

• Cover image
• Title page
• Table of Contents
• Series Page
• Copyright Page
• Title Page II
• Preface
• Part 1 Introduction
• 1 Brief History and Overview of Nonlinear Water Waves
• Chapter 2 Nonlinear Water Wave Equations
• Chapter 3 The Infinite-Line Inverse Scattering Transform
• Chapter 4 The Infinite-Line Hirota Method
• Part 2 Periodic Boundary Conditions
• Chapter 5 Periodic Boundary Conditions
• 6 The Periodic Hirota Method
• Part 3 Multidimensional Fourier Analysis
• 7 Multidimensional Fourier Series
• 8 Riemann Theta Functions
• Chapter 9 Riemann Theta Functions as Ordinary Fourier Series
• Part 4 Nonlinear Shallow-Water Spectral Theory
• Chapter 10 The Periodic Korteweg-DeVries Equation
• Chapter 11 The Periodic Kadomtsev-Petviashvili Equation
• Part 5 Nonlinear Deep-Water Spectral Theory
• Chapter 12 The Periodic Nonlinear Schrödinger Equation
• 13 The Hilbert Transform
• Part 6 Theoretical Computation of the Riemann Spectrum
• Chapter 14 Algebraic-Geometric Loop Integrals
• Chapter 15 Schottky Uniformization
• Chapter 16 Nakamura-Boyd Approach
• Part 7 Nonlinear Numerical and Time Series Analysis Algorithms
• 17 Automatic Algorithm for the Spectral Eigenvalue Problem for the KdV Equation
• Chapter 18 The Spectral Eigenvalue Problem for the NLS Equation
• Chapter 19 Computation of Algebraic-Geometric Loop Integrals for the KdV Equation
• Chapter 20 Simple, Brute-Force Computation of Theta Functions and Beyond
• 21 The Discrete Riemann Theta Function
• Chapter 22 Summing Riemann Theta Functions over the N-Ellipsoid
• Chapter 23 Determining the Riemann Spectrum from Data and Simulations
• Part 8 Theoretical and Experimental Problems in Nonlinear Wave Physics
• 24 Nonlinear Instability Analysis of Deep-Water Wave Trains
• Chapter 25 Internal Waves and Solitons
• 26 Underwater Acoustic Wave Propagation
• 27 Planar Vortex Dynamics
• 28 Nonlinear Fourier Analysis and Filtering of Ocean Waves
• 29 Laboratory Experiments of Rogue Waves
• 30 Nonlinearity in Duck Pier Data
• 31 Harmonic Generation in Shallow-Water Waves
• Part 9 Nonlinear Hyperfast Numerical Modeling
• 32 Hyperfast Modeling of Shallow-Water Waves: The KdV and KP Equations
• 33 Modeling the 2 + 1 Gardner Equation
• 34 Modeling the Davey-Stewartson (DS) Equations
• References
• International Geophysics Series
• Index