Computation of the Direct Scattering Transform for the Nonlinear Schroedinger Equation

The cubic nonlinear Schroedinger equation (NLS) describes the space-time evolution of narrow-banded wave trains in one space and one time (1 + 1) dimensions. The richness of nonlinear wave motions described by NLS is exemplified by the fully nonlinear envelope soliton and “breather” solutions, which are fully understood only in terms of the general solution of the equation as described by the inverse scattering transform (IST); the method may be viewed as a nonlinear generalization of the linear Fourier transform. Herein we develop a numerical algorithm for determining the scattering transform spectrum of a nonlinear wave train described by the NLS equation. The analysis of space or time series data obtained from computer simulations of nonlinear, narrow-band wave trains or from experimental measurements is thus a central point of discussion. In particular we develop a numerical algorithm for computing the direct scattering transform (DST) which may be interpreted as the nonlinear Fourier spectrum of the complex envelope function of a wave train; the fact that the nonlinear Fourier modes are constants of the motion for all time provides a physical basis for the analysis of data. While the nonlinear Fourier method is specifically applied to the NLS equation, the approach is easily generalized to include the class of spectral problems due to Ablowitz, Kaup, Newell, and Segur (AKNS). This class includes several other nonlinear wave equations of physical interest in (1 + 1), including the Kortewegde Vries (KdV), modified KdV, sine-Gordon, and sinh-Gordon equations.