Natural Wave Trains and Scattering Transform

Abstract

The potential of the nonlinear Schroedinger equation and the Zakharov-Shabat scattering transformation as a third generation analysis technique for natural ocean wave trains is considered. Such an approach retains many of the advantages of the second generation Gaussian random wave model but accomodates such fundamentally nonlinear concepts as Benjamin-Feir instability and especially envelope solitons. Wave envelope solitons are naturally occurring and stable wave groups which exist quite separately from uniform wave train solutions. Envelope soliton and uniform wave train solutions interact nonlinearly in physical space and are difficult to separately identify from a wave record. Resolution is possible, however, in scattering transform space. The direct scattering transform leads to the scattering data, a broadly analogous summary of a wave record to the Fourier spectrum. The inverse scattering transform recreates the wave train at the same or a later time, in a similar manner to the inverse Fourier transform. The results of exploratory computations of both analysis and synthesis of wave trains are included, together with a discussion of the potential application of the method to natural wave trains.