The Spectral Transform: Methods for the Fourier Analysis of Nonlinear Wave Data

Abstract

The Fourier transform has historically been the preferred mathematical method for determining analytic solutions to linear wave equations and for analyzing wave data assumed to behave approximately linearly. However, in recent years there has been remarkable progress in the search for exact analytical solutions to certain classes of partial differential equations which describe nonlinear wave evolution. Probably the most successful mathematical approach in this regard has been the spectral or scattering transform which parallels the Fourier method in many important respects. The purpose of this paper is to review the use of newly developed techniques for the analysis of nonlinear data; these methods are based upon the concept of the spectral transform rather than upon the more traditional Fourier transform. We restrict our attention to wave motion governed by the Korteweg-deVries equation on the infinite interval. We give examples of the application of our methods to the analysis of computer generated waveforms, laboratory data and large amplitude internal wave signals obtained in the Andaman Sea.