Nonlinear Fourier Methods for Ocean Waves

Multiperiodic Fourier series solutions of integrable nonlinear wave equations are applied to the study of ocean waves for scientific and engineering purposes. These series can be used to compute analytical formulae for the stochastic properties of nonlinear equations, in analogy to the standard approach for linear equations. Here I emphasize analytically computable results for the correlation functions, power spectra and coherence functions of a nonlinear random process associated with an integrable nonlinear wave equation. The multiperiodic Fourier series have the advantage that the coherent structures of soliton physics are encoded in the formulation, so that solitons, breathers, vortices, etc. are contained in the temporal evolution of the nonlinear power spectrum and phases. I illustrate the method for the Korteweg-deVries and nonlinear SchrÖdinger equations. Applications of the method to the analysis of data are discussed.