Extending Integrability of Nonlinear Water Wave Equations: Nonlinear Fourier Analysis of Breather Packets and Rogue Waves at Higher Order

I suggest a formulation to give approximate spectral solutions of nonintegrable, nonlinear wave equations in 2+1 dimensions. Nonintegrable systems such as the 2+1 NLS, Dysthe and extended Dysthe equations can be approximately integrated by selecting a nearby theta function formulation. I study the subclass of wave equations that are in the form of nonlinear envelope equations for which all members can be reduced to a particular Hirota bilinear form. To find the approximately integrable formulation associated with a nonintegrable equation, I first study the one and two soliton solutions and subsequently extend these to larger numbers of solitons to obtain the Hirota N-soliton solution (for infinite-plane boundary conditions). Subsequently, I address the one and two periodic solutions from the bilinear form, so that I can develop the associated Riemann theta function solution to a nearby integrable case. I discuss how to obtain the higher order breather packets from the point of view of the theta functions. This work is being developed for real time analysis of shipboard radar analysis of ocean waves. Further applications include real time analysis of lidar and synthetic aperture radar (SAR) data taken by airplanes flying over high sea states.