Discrete nonlinear Fourier transforms and their inverses

TL;DR

This paper develops recursive algorithms for the inverse of two discretizations of the nonlinear Fourier transform, with a focus on distributions involving delta functions, and explores symmetry properties to improve computational efficiency.

Contribution

It introduces recursive algorithms for inverting two discretized nonlinear Fourier transforms and analyzes their properties, including symmetry, for efficient computation.

Findings

Recursive algorithms for ${ m ({ m F}^E)}^{-1}$ and ${ m ({ m F}^D)}^{-1}$ are constructed.

${ m ({ m F}^D)}^{-1}$ is more computationally demanding than ${ m ({ m F}^E)}^{-1}$.

Symmetry property of ${ m F}^D$ enables reduction in nonlinear Fourier analysis for certain distributions.

Abstract

We study two discretisations of the nonlinear Fourier transform of AKNS-ZS type, ${\cal F}^E$ and ${\cal F}^D$. Transformation ${\cal F}^D$ is suitable for studying the distributions of the form $u = \sum_{n = 1}^N u_n \, \delta_{x_n}$, where $\delta _{x_n}$ are delta functions. The poles $x_n$ are not equidistant. The central result of the paper is the construction of recursive algorithms for inverses of these two transformations. The algorithm for $({\cal F}^D)^{- 1}$ is numerically more demanding than that for $({\cal F}^E)^{- 1}$. We describe an important symmetry property of ${\cal F}^D$. It enables the reduction of the nonlinear Fourier analysis of the constant mass distributions $u = \sum_{n = 1}^N u_c \, \delta _{x_n}$ for the numerically more efficient ${\cal F}^E$ and its inverse.