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 anlytical solutions. However, the split-step method provides a numerical solution to the problem.
Description of the method
Consider, for example, the nonlinear Schrödinger equation
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.
The linear step has an analytical solution in the frequency domain, so it is first necessary to Fourier transform using
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 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 Fourier transforms of this algorithm can be computed relatively fast using the fast Fourier transform (FFT). The split-step Fourier method can therefore be much faster than typical finite difference methods.
- Thomas E. Murphy, Software, http://www.photonics.umd.edu/software/ssprop/
- Andrés A. Rieznik, Software, http://photonics.incubadora.fapesp.br