A Fast Chebyshev Spectral Method for Nonlinear Fourier Transform

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.