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We have analyzed the nonlinear propagation characteristics and wave mixing of picosecond pulses in semiconductor optical amplifiers (SOAs) by the finite- difference beam propagation method (FD-BPM). From the simulation results, it was found in the output waveform that the pulses peak positions are shifted to the leading edge. The output spectra are red-shifted and the dips are observed at the higher frequency side in the frequency spectra. We have analyzed all-optical demultiplexing (DEMUX) based on four-wave mixing (FWM) in SOAs. It is clear from the simulation results that the SOAs with wider gain bandwidth are required for the faster DEMUX operation. We have simulated pattern effects in the FWM signal. We have also obtained DEMUX from the time-multiplexed signals by repetitive pump pulses. FWM signal intensity decreases with the strong energy pump pulses. However, there is no pattern effect due to gain saturation because the pump pulses are injected continuously.