Instability of the split-step method for a signal with nonzero central frequency

We obtain analytical conditions for the occurrence of numerical instability (NI) of a split-step method when the simulated solution of the nonlinear Schrodinger equation is close to a plane wave with nonzero carrier frequency. We also numerically study such an instability when the solution is a sequence of pulses rather than a plane wave. The plane-wave-based analysis gives reasonable predictions for the frequencies of the numerically unstable Fourier modes but overestimates the instability growth rate. The latter is found to be strongly influenced by the randomness of the signal's profile: The more randomly it varies during the propagation, the weaker is the NI. Using an example of a realistic transmission system, we demonstrate that our single-channel results can be used to predict occurrences of NI in multichannel simulations. We also give an estimate for the integration step size for which NI, while present, will not affect simulation results for such systems. Using that estimate may lead to a significant saving of computational time. (C) 2013 Optical Society of America