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 where describes the wave function at position and time . Note that and , and that is the mass of the particle and is Planck's constant over . The formal solution to this equation is a complex exponential, so we have that . Since and are operators, they do not in general commute. However, the Baker-Hausdorff formula can be applied to show that the error from treating them as if they do will be of order if we are taking a small but finite time step . We therefore can write . The part of this equation involving can be computed directly using the wave function at time , but to compute the exponential involving we use the fact that in frequency space, the partial derivative operator can be converted into a number by substituting for , where is the frequency (or more properly, wave number, as we are dealing with a spatial variable and thus transforming to a space of spatial frequencies—i.e. wave numbers) associated with the Fourier transform of whatever is being operated on. Thus, we take the Fourier transform of , recover the associated wave number, compute the quantity , and use it to find the product of the complex exponentials involving and in frequency space as below: , where denotes a Fourier transform. We then inverse Fourier transform this expression to find the final result in physical space, yielding the final expression . A variation on this method is the symmetrized split-step Fourier method, which takes half a time step using one operator, then takes a full-time step with only the other, and then takes a second half time step again with only the first. This method is an improvement upon the generic split-step Fourier method because its error is of order for a time step . 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. References External references Thomas E. Murphy, Software, http://www.photonics.umd.edu/software/ssprop/ Andrés A. Rieznik, Software, http://www.freeopticsproject.org Prof.