Nonlinear fourier analysis for the infinite-interval Korteweg-de Vries equation I: An algorithm for the direct scattering transform

Abstract

The nonlinear Fourier analysis of wave motion governed approximately by the Korteweg-de Vries (KdV) equation on the infinite line is the central point of discussion. We assume that the wave amplitude is recorded in the form of a discrete space or time series which is determined either by experimental measurement or by computer simulation of the physical system of interest. We develop numerical data analysis procedures based upon the scattering transform solution to the KdV equation as given by Gardner et al. [ 1 ]. We are motivated by the observation that historically the Fourier transform has been ubiquitously used to spectrally analyze linear wave data; here we develop methods for employing the scattering transform as a tool to similarly analyze nonlinear wave data. Specifically we develop numerical methods to evaluate the direct scattering transform (DST) of a space or time series: the approach thus provides a basis for analyzing and interpreting nonlinear wave behavior in the wavenumber or frequency domain. The DST spectrum separates naturally into soliton and radiation components and may be simply interpreted in terms of the large-time asymptotic state of the infinite-line KdV equation.