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

Robust split-step Fourier methods for simulating the propagation of ultra-short pulses in single- and two-mode optical communication fibers