  Extensions of the split-step Fourier method (SSFM) for Schr\"odinger-type
pulse propagation equations for simulating femto-second pulses in single- and
two-mode optical communication fibers are developed and tested for Gaussian
pulses. The core idea of the proposed numerical methods is to adopt an operator
splitting approach, in which the nonlinear sub-operator, consisting of Kerr
nonlinearity, the self-steepening and stimulated Raman scattering terms, is
reformulated using Madelung transformation into a quasilinear first-order
system of signal intensity and phase. A second-order accurate upwind numerical
method is derived rigorously for the resulting system in the single-mode case;
a straightforward extension of this method is used to approximate the
four-dimensional system resulting from the nonlinearities of the chosen
two-mode model. Benchmark SSFM computations of prototypical ultra-fast
communication pulses in idealized single- and two-mode fibers with homogeneous
and alternating dispersion parameters and also high nonlinearity demonstrate
the reliable convergence behavior and robustness of the proposed approach.

Numerical Analysis