In
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
, the split-step (Fourier) method is a
pseudo-spectral numerical method used to solve nonlinear
partial differential equations like the
nonlinear Schrödinger equation
In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlin ...
. 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
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
while the nonlinear step is made in the
time domain
Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the c ...
.
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 Kerr frequency combs (also known as microresonator frequency combs) are optical frequency combs which are generated from a continuous wave pump laser by the Kerr nonlinearity. This coherent conversion of the pump laser to a frequency comb takes p ...
dynamics in
optical microresonators.
The relative ease of implementation of the
Lugiato–Lefever equation
The model usually designated as Lugiato–Lefever equation (LLE) was formulated in 1987 by Luigi Lugiato and René Lefever
as a paradigm for spontaneous pattern formation in nonlinear optical systems. The patterns originate from the interaction ...
with reasonable numerical cost, along with its success in reproducing experimental spectra as well as predicting
soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medi ...
behavior in these microresonators has made the method very popular.
Description of the method
Consider, for example, the
nonlinear Schrödinger equation
In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlin ...
:
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
In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlin ...
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
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
, 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 In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information a ...
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
:
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:
: