Split-step method

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In numerical analysis, the split-step (Fourier) method is a general computational technique 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.

[edit] Description of the method

Consider, for example, the nonlinear Schrödinger equation^[1]

${\part A \over \part z} = - {i\beta_2 \over 2} {\part^2 A \over \part t^2} + i \gamma | A |^2 A = [\hat D + \hat N]A,$

where $A (t, z)$ describes the pulse envelope in time $t$ at the spatial position $z$ . The equation can be split in a linear part,

${\part A_L \over \part z} = - {i\beta_2 \over 2} {\part^2 A \over \part t^2} = \hat D A,$

and a nonlinear part,

${\part A_N \over \part z} = i \gamma | A |^2 A = \hat N A.$

Both the linear and the nonlinear parts have analytical solutions, but the nonlinear Schrödinger equation containing both parts does not have a general solution.

However, if only a 'small' step $h$ is taken along $z$ , then the two parts can be treated separately with only a 'small' numerical error. One can therefore first take a small nonlinear step,

$A_N(t, z+h) = \exp\left[i \gamma |A|^2 h \right] A(t, z),$

using the analytical solution. The linear step has an analytical solution in the frequency domain, so it is first necessary to Fourier transform $A N$ using

$\tilde A_N(\omega, z+h) = \int_{-\infty}^\infty A_N(t,z+h) \exp[i(\omega-\omega_0)] dt$ ,

where $ω 0$ 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

$\tilde{A}(\omega, z+h) = \exp\left[{i \beta_2 \over 2} (\omega-\omega_0)^2 h \right] \tilde{A}_N(t, z+h).$

By taking the inverse Fourier transform of $\tilde{A}(\omega, z+h)$ one obtains $A\left(t, z+h\right)$ ; the pulse has thus been propagated a small step $h$ . By repeating the above $N$ times, the pulse can be propagated over a length of $N h$ .

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^[2].

[edit] References

^ Agrawal, Govind P. (2001). Nonlinear Fiber Optics, 3rd ed., San Diego, CA, USA: Academic Press. ISBN 0-12-045143-3.
^ T. R. Taha and M. J. Ablowitz (1984). "Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation". J. Comput. Phys. 55 (2): 203-230.