Automatic algorithm for the numerical inverse scattering transform of the Korteweg–de Vries equation

The inverse scattering transform (IST) for the periodic Kortweg—de Vries (KdV) equation in the μ-function representation is considered and numerical formulations are given (1) for determining the direct scattering transform spectrum of an input discrete wave train and (2) for reconstructing the wave train from the spectrum via the inverse scattering problem. The advantage of the present method over previous approaches is that the numerical computations are automatic, i.e. one is guaranteed that the algorithm will search out and find all the nonlinear modes (to within the input numerical precision) of a given arbitrary, N degree-of-freedom wave train. The algorithms are most appropriate for the time series analysis of measured and computed data. One is thus numerically able to analyze an input time series with M discrete points: (1) to construct the IST spectrum, (2) to determine the N = M/2 hyperelliptic-function oscillation modes, (3) to reconstruct the input wave train by a linear superposition law and (4) to nonlinearly filter the input wave train. Numerical details of the algorithm are discussed and an example for which N = 128 is given.