NUMERICAL INVERSE SCATTERING TRANSFORM FOR THE PERIODIC, DEFOCUSING NONLINEAR SCHRODINGER-EQUATION

The nonlinear Schrodinger (NLS) equation, whose solution in terms of the periodic inverse scattering transform (IST) is well known, describes the integrable space-time dynamics of the complex envelope function of a narrow-banded wave train in 1 + 1 dimensions. Here I introduce two numerical algorithms for the IST of the periodic, defocusing NLS equation (analogous to similar steps for ordinary linear Fourier analysis): (1) for obtaining the spectrum of a given complex wave train of NLS and (2) for reconstructing the input wave train from the computed spectrum. Both steps are discrete approximations to NLS, although for a small discrete spatial interval DELTAx, the reconstructed input wave train may be obtained to high precision. An important ingredient of the algorithm is that for a space series of N discrete points, the number of nonlinear modes is also N (extending to the Nyquist cutoff wavenumber k(N) = pi/DELTAx), although, as with the linear Fourier transform, all the modes may not be energetically important; in this context the possibility of nonlinear aliasing must also be addressed. The algorithms given herein are best described as numerical methods for the nonlinear Fourier analysis (or nonlinear signal processing) of computer generated wave trains or of experimentally measured space or time series whose dynamics are near those of the NLS equation.