Higher Order Convergent Fast Nonlinear Fourier Transform

TL;DR

This paper introduces a family of fast nonlinear Fourier transform algorithms using linear multistep methods, achieving higher orders of convergence and potential applications in optical fiber communication.

Contribution

It develops a new class of fast NFT algorithms based on multistep methods with higher convergence orders, improving computational efficiency and accuracy.

Findings

Error decreases as $O{N^{-p}}$ with $p$ up to 4

Implicit Adams method yields highest accuracy

Algorithm complexity is $O{KN+C_pN ext{log}^2N}$

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 $O{KN+C_pN\log^2N}$ 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.