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
∂ A ∂ z = − i β 2 2 ∂ 2 A ∂ t 2 + i γ | A | 2 A = [ D ^ + N ^ ] A , {\displaystyle {\partial A \over \partial z}=-{i\beta _{2} \over 2}{\partial ^{2}A \over \partial t^{2}}+i\gamma |A|^{2}A=[{\hat {D}}+{\hat {N}}]A,}
where A ( t , z ) {\displaystyle A(t,z)} describes the pulse envelope in time t {\displaystyle t} at the spatial position z {\displaystyle z} . The equation can be split into a linear part,
∂ A D ∂ z = − i β 2 2 ∂ 2 A ∂ t 2 = D ^ A , {\displaystyle {\partial A_{D} \over \partial z}=-{i\beta _{2} \over 2}{\partial ^{2}A \over \partial t^{2}}={\hat {D}}A,}
and a nonlinear part,
∂ A N ∂ z = i γ | A | 2 A = N ^ A . {\displaystyle {\partial A_{N} \over \partial z}=i\gamma |A|^{2}A={\hat {N}}A.}
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 h {\displaystyle h} is taken along z {\displaystyle z} , then the two parts can be treated separately with only a 'small' numerical error. One can therefore first take a small nonlinear step,
A N ( t , z + h ) = exp [ i γ | A ( t , z ) | 2 h ] A ( t , z ) , {\displaystyle A_{N}(t,z+h)=\exp \left[i\gamma |A(t,z)|^{2}h\right]A(t,z),}
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