Higher Order Convergent Fast Nonlinear Fourier Transform

Abstract

It is demonstrated is this letter that linear multistep methods for integrating ordinary differential equations can be used to develop a family of fast forward scattering algorithms with higher orders of convergence. Excluding the cost of computing the discrete eigenvalues, the nonlinear Fourier transform (NFT) algorithm thus obtained has a complexity of $OKN+C_{p}N{\log }^{2}N$ such that the error vanishes as $O{N}^{-p}$ where $p\in \{1,2,3,4\}$ and $K$ is the number of eigenvalues. Such an algorithm can be potentially useful for the recently proposed NFT based modulation methodology for optical fiber communication. The exposition considers the particular case of the backward differentiation formula ($C_p=p^3$) and the implicit Adams method ($C_p=(p-1{)}^3$) of which the latter proves to be the most accurate family of methods for fast NFT.

References in corpus (1)

• Fast Inverse Nonlinear Fourier Transformation using Exponential One-Step Methods, Part I: Darboux Transformation

Cited by in corpus (10)

• Fast Nonlinear Fourier Transform Algorithms Using Higher Order Exponential Integrators
• Exponential Fourth Order Schemes for Direct Zakharov-Shabat problem
• Direct scattering transform: catch soliton if you can
• Introducing phase jump tracking -- a fast method for eigenvalue evaluation of the direct Zakharov-Shabat problem
• Novel Numerical Algorithm with Fourth-Order Accuracy for the Direct Zakharov-Shabat Problem
• Solitons in a box-shaped wavefield with noise: perturbation theory and statistics
• Nonlinear Fourier Transform of Time-Limited and One-sided Signals
• Efficient Nonlinear Fourier Transform Algorithms of Order Four on Equispaced Grid
• Data Transmission based on Exact Inverse Periodic Nonlinear Fourier Transform, Part II: Waveform Design and Experiment
• Discrete Darboux Transformation for Ablowitz-Ladik Systems Derived from Numerical Discretization of Zakharov-Shabat Scattering Problem