A Chebyshev Spectral Method for Nonlinear Fourier Transform: Norming Constants

TL;DR

This paper introduces a Chebyshev spectral method for computing the nonlinear Fourier spectrum, specifically focusing on an algorithm to accurately determine norming constants using a minimum total variation principle.

Contribution

The paper develops a novel Chebyshev spectral approach for the Zakharov--Shabat problem and introduces an MTV-based algorithm for stable norming constant computation.

Findings

Accurate computation of norming constants achieved.

Spectral method improves numerical conditioning.

Algorithm demonstrates robustness in complex spectral parameters.

Abstract

In this paper, we present a Chebyshev based spectral method for the computation of the Jost solutions corresponding to complex values of the spectral parameter in the Zakharov--Shabat scattering problem. The discrete framework is then used to devise a new algorithm based on a minimum total variation (MTV) principle for the computation of the norming constants which comprise the discrete part of the nonlinear Fourier spectrum. The method relies on the MTV principle to find the points where the expressions for norming constants are numerically well-conditioned.