Nonlinear Fourier Analysis with Sine Wave, Stokes Wave and Rogue Wave Basis Functions: A Paradigm Change in the Understanding of Nonlinear Waves

I give a new perspective for the description of nonlinear water wave trains using mathematical methods I refer to as nonlinear Fourier analysis (NLFA). I discuss how this approach holds for one-space and one time dimensions (1+1) and for two-space and one time dimensions (2+1) to all orders of approximation. I begin with the nonlinear Schroedinger (NLS) equation in 1+1 dimensions: Here the NLFA method is derived from the complete integrability of the equation by the periodic inverse scattering transform. I show how to compute the nonlinear Fourier series that exactly solve 1+1 NLS. I then show how to extend the order of 1+1 NLS to the Dysthe and the extended Dysthe equations. I also show how to include directional spreading in the formulation so that I can address the 2+1 NLS, the 2+1 Dysthe and the 2+1 Trulsen-Dysthe equations. This hierarchy of equations extends formally all the way to the Zakharov equations in the infinite order limit. Each order and extension from 1+1 to 2+1 dimensions is characterized by its own modulational dispersion relation that is required at each order of the NLFA formalism. NLFA is characterized by its own fundamental nonlinear Fourier series, which has particular nonlinear Fourier modes: sine waves, Stokes waves and breather trains. We are all familiar with sine waves (known for centuries) and Stokes waves (known since the Stokes paper in 1847). Breather trains have become known over the past three decades as a major source of rogue or freak waves in the ocean: Breather packets are known to pulse up and down during their evolution. At the moment of the maximum amplitude the largest wave in a breather packet is often referred to as a “rogue” or “freak” wave. Such extreme packets are known to be “coherent structures" so that pure linear dispersion does not occur as in a linear packet. Instead the breather packets have components that are phase locked with each other and hence remain coherent and are “long lived” just as vortices do in classical turbulence. Because the breathers live for a long time, the notion of risk based upon linear dispersion, as used in the oil and shipping industries, must be revised upwards. I discuss how to apply NLFA to (1) nonlinearly Fourier analyze time series, (2) to analyze wave fields from radar, lidar and synthetic aperture radar measurements, (3) how to treat NLFA to describe nonlinear, random wave trains using a kind of nonlinear random phase approximation and (4) how to compute the nonlinear power spectrum in terms of the parameters used to describe the rogue wave Fourier modes in a random wave train. Thus the emphasis here is to discuss a number of new tools for nonlinear Fourier analysis in a wide range of problems in the field of ocean surface waves.