A Fast Chebyshev Spectral Method for Nonlinear Fourier Transform

TL;DR

This paper introduces a rapid, well-conditioned spectral method using Chebyshev polynomials to efficiently compute the continuous spectrum of the nonlinear Fourier transform, significantly reducing computational complexity.

Contribution

It proposes a novel spectral algorithm that improves speed and conditioning for nonlinear Fourier spectrum computation using Chebyshev polynomials.

Findings

Achieves $O(N_{iter.}N ext{log}N)$ complexity per spectral node.

Demonstrates improved computational efficiency over existing methods.

Provides a stable and accurate approach for nonlinear Fourier analysis.

Abstract

In this letter, we present a fast and well-conditioned spectral method based on the Chebyshev polynomials for computing the continuous part of the nonlinear Fourier spectrum. The algorithm achieves a complexity of $O(N_{\text{iter.}}N\log N)$ per spectral node for $N$ samples of the signal at the Chebyshev nodes where $N_{\text{iter.}}$ is the number of iterations of the biconjugate gradient stabilized method.