THE HYPERELLIPTIC INVERSE SCATTERING TRANSFORM FOR THE PERIODIC, DEFOCUSING NONLINEAR SCHROEDINGER EQUATION

The nonlinear Schroedinger (NLS) equation describes the spatio-temporal evolution of the complex envelope function of a narrow-banded, nonlinear wave train. Here I exploit the nonlinear Fourier structure of NLS, known as the inverse scattering transform, to study nonlinear periodic modulations in the spectral domain. Numerical algorithms are presented for both the direct and inverse scattering transforms of the “defocusing” NLS equation with periodic boundary conditions. A discrete algorithm is given for computing the monodromy matrix of periodic spectral theory and the direct scattering transform is then computed from the elements of this matrix. A fast algorithm for the inverse scattering transform in terms of the hyperelliptic function representation is also given; solutions to the defocusing NLS equation are determined by a linear superposition of these nonlinear oscillation modes. The direct algorithm uses computer time proportional to M2, where M is the number of points in the discrete envelope function of the wave train. The inverse scattering algorithm is "fast" in the sense that it is M3; this contrasts to the periodic theta-function inverse problem for NLS which is M4 and to the Gelfan'd-Levitan-Marchenko integral equation for infinite-line boundary conditions which is also M4. Several wave train solutions of NLS are considered and their inverse scattering transform spectra and nonlinear Fourier decompositions are discussed. Application of the method to the analysis of computer generated or experimentally measured space or time series is the major motivation for this work. © 1993 Academic Press, Inc.