# Efficient Nonlinear Fourier Transform Algorithms of Order Four on Equispaced Grid

## TL;DR

This paper introduces two classes of exponential integrators to develop nonlinear Fourier transform algorithms with fourth-order accuracy on equispaced grids, achieving improved efficiency and error performance compared to existing methods.

## Contribution

The paper presents novel NFT algorithms based on Runge-Kutta and Magnus integrators that offer a better accuracy-complexity trade-off and higher convergence order.

## Key findings

- Runge-Kutta based algorithms achieve O(N log^2 N) complexity with superior accuracy.
- Magnus series expansion methods achieve O(N^2) complexity with improved error behavior.
- Algorithms perform well for moderate step-sizes and higher signal strengths.

## 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 $O(N\log^2N)$ (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.