In this paper an approach for decreasing the computational effort required
for the split-step Fourier method (SSFM) is introduced. It is shown that using
the sparsity property of the simulated signals, the compressive sampling
algorithm can be used as a very efficient tool for the split-step spectral
simulations of various phenomena which can be modeled by using differential
equations. The proposed method depends on the idea of using a smaller number of
spectral components compared to the classical split-step Fourier method with a
high number of components. After performing the time integration with a smaller
number of spectral components and using the compressive sampling technique with
l1 minimization, it is shown that the sparse signal can be reconstructed with a
significantly better efficiency compared to the classical split-step Fourier
method. Proposed method can be named as compressive split-step Fourier method
(CSSFM). For testing of the proposed method the Nonlinear Schrodinger Equation
and its one-soliton and two-soliton solutions are considered
