THE NUMERICAL INVERSE SCATTERING TRANSFORM FOR THE PERIODIC KORTEWEG-DEVRIES EQUATION

We introduce inverse scattering transform (IST) algorithms for obtaining the spectrum of complex wave trains governed by the periodic Korteweg-de Vries (KdV) equation and for constructing solutions to this equation. A scattering matrix formulation is implemented to determine the Floquet spectrum and an iterated similarity transformation is exploited to compute the hyperelliptic oscillation modes. A linear superposition law of the hyperelliptic modes, a generalization of ordinary Fourier series, is used to construct general wave train solutions to the KdV equation. The algorithms should be useful for nonlinear Fourier analysis of computer generated wave trains and experimentally measured space or time series.