We studied the efficiency of different implementations of the split-step Fourier method for solving the nonlinear Schrödinger equation that employ different step-size selection criteria. We compared the performance of the different implementations for a variety of pulse formats and systems,including higher order solitons, collisions of soliton pulses, a single-channel periodically stationary dispersion-managed soliton system, and chirped return to zero systems with single and multiple channels. We introduce a globally third-order accurate split-step scheme, in which a bound on the local error is used to select the step size. In many cases, this method is the most efficient when compared with commonly used step-size selection criteria, and it is robust for a wide range of systems providing a system-independent rule for choosing the step sizes. We find that a step-size selection method based on limiting the nonlinear phase rotation of each step is not efficient for many optical-fiber transmission systems, although it works well for solitons. We also tested a method that uses a logarithmic step-size distribution to bound the amount of spurious four-wave mixing. This method is as efficient as other second-order schemes in the single-channel dispersion-managed soliton system, while it is not efficient in other cases including multichannel simulations. We find that in most cases, the simple approach in which the step size is held constant is the least efficient of all the methods. Finally, we implemented a method in which the step size is inversely proportional to the largest group velocity difference between channels. This scheme performs best in multichannel optical communications systems for the values of accuracy typically required in most transmission simulations.
Oleg V. Sinkin, Ronald Holzlöhner, John Zweck, and Curtis R. Menyuk, "Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems," J. Lightwave Technol. 21, 61- (2003)