Compressive Split-Step Fourier Method

TL;DR

This paper introduces the compressive split-step Fourier method (CSSFM), which reduces computational effort in spectral simulations by leveraging signal sparsity and compressive sampling, demonstrated on nonlinear Schrödinger equations.

Contribution

The paper presents a novel approach combining compressive sampling with the split-step Fourier method to efficiently simulate differential equations with sparse spectral signals.

Findings

Significantly reduces computational effort compared to classical SSFM.

Accurately reconstructs sparse signals using fewer spectral components.

Effective for nonlinear Schrödinger equation simulations.

Abstract

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.