In numerical analysis, the split-step (Fourier) method is a pseudo-spectral numerical method used to solve nonlinear partial differential equations like the nonlinear Schrödinger equation. The name arises for two reasons. First, the method relies on computing the solution in small steps, and treating the linear and the nonlinear steps separately (see below). Second, it is necessary to Fourier transform back and forth because the linear step is made in the frequency domain while the nonlinear step is made in the time domain.
An example of usage of this method is in the field of light pulse propagation in optical fibers, where the interaction of linear and nonlinear mechanisms makes it difficult to find general analytical solutions. However, the split-step method provides a numerical solution to the problem. Another application of the split-step method that has been gaining a lot of traction since the 2010s is the simulation of Kerr frequency comb dynamics in optical microresonators. The relative ease of implementation of the Lugiato–Lefever equation with reasonable numerical cost, along with its success in reproducing experimental spectra as well as predicting soliton behavior in these microresonators has made the method very popular.
Description of the method
Consider, for example, the nonlinear Schrödinger equation
where describes the pulse envelope in time at the spatial position . The equation can be split into a linear part,
and a nonlinear part,
Both the linear and the nonlinear parts have analytical solutions, but the nonlinear Schrödinger equation containing both parts does not have a general analytical solution.
However, if only a 'small' step is taken along , then the two parts can be treated separately with only a 'small' numerical error. One can therefore first take a small nonlinear step,
using the analytical solution. Note that this ansatz imposes and consequently .
The dispersion step has an analytical solution in the frequency domain, so it is first necessary to Fourier transform using , where is the center frequency of the pulse. It can be shown that using the above definition of the Fourier transform, the analytical solution to the linear step, commuted with the frequency domain solution for the nonlinear step, is
By taking the inverse Fourier transform of one obtains ; the pulse has thus been propagated a small step . By repeating the above times, the pulse can be propagated over a length of .
The above shows how to use the method to propagate a solution forward in space; however, many physics applications, such as studying the evolution of a wave packet describing a particle, require one to propagate the solution forward in time rather than in space. The non-linear Schrödinger equation, when used to govern the time evolution of a wave function, takes the form