Efficient Nonlinear Fourier Transform Algorithms of Order Four on Equispaced Grid

Abstract

We explore two classes of exponential integrators in this letter to design nonlinear Fourier transform (NFT) algorithms with a desired accuracy-complexity trade-off and a convergence order of $4$ on an equispaced grid. The integrating factor based method in the class of Runge-Kutta methods yield algorithms with complexity $O(N{\log }^{2}N)$ (where $N$ is the number of samples of the signal) which have superior accuracy-complexity trade-off than any of the fast methods known currently. The integrators based on Magnus series expansion, namely, standard and commutator-free Magnus methods yield algorithms of complexity $O(N^2)$ that have superior error behavior even for moderately small step-sizes and higher signal strengths.

4 pages

References in corpus (4)

• Fast Inverse Nonlinear Fourier Transformation using Exponential One-Step Methods, Part I: Darboux Transformation
• Fast Inverse Nonlinear Fourier Transform
• Higher Order Convergent Fast Nonlinear Fourier Transform
• Novel Numerical Algorithm with Fourth-Order Accuracy for the Direct Zakharov-Shabat Problem

Cited by in corpus (3)

• Exponential Fourth Order Schemes for Direct Zakharov-Shabat problem
• Successive Eigenvalue Removal for Multi-Soliton Spectral Amplitude Estimation
• Discrete Darboux Transformation for Ablowitz-Ladik Systems Derived from Numerical Discretization of Zakharov-Shabat Scattering Problem