Fast Inverse Nonlinear Fourier Transform

Abstract

This paper considers the non-Hermitian Zakharov-Shabat (ZS) scattering problem which forms the basis for defining the SU$(2)$-nonlinear Fourier transform (NFT). The theoretical underpinnings of this generalization of the conventional Fourier transform is quite well established in the Ablowitz-Kaup-Newell-Segur (AKNS) formalism; however, efficient numerical algorithms that could be employed in practical applications are still unavailable. In this paper, we present two fast inverse NFT algorithms with $O(KN+N{\log }^{2}N)$ complexity and a convergence rate of $O(N^{-2})$ where $N$ is the number of samples of the signal and $K$ is the number of eigenvalues. These algorithms are realized using a new fast layer-peeling (LP) scheme ($O(N{\log }^{2}N)$) together with a new fast Darboux transformation (FDT) algorithm ($O(KN+N{\log }^{2}N)$) previously developed by the author. The proposed fast inverse NFT algorithm proceeds in two steps: The first step involves computing the radiative part of the potential using the fast LP scheme for which the input is synthesized under the assumption that the radiative potential is nonlinearly bandlimited, i.e., the continuous spectrum has a compact support and the discrete spectrum is empty. The second step involves addition of bound states using the FDT algorithm. Finally, the performance of these algorithms is demonstrated through exhaustive numerical tests.

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References in corpus (1)

• Fast Inverse Nonlinear Fourier Transformation using Exponential One-Step Methods, Part I: Darboux Transformation

Cited by in corpus (8)

• Efficient Nonlinear Fourier Transform Algorithms of Order Four on Equispaced Grid
• Nonlinear Fourier Transform of Time-Limited and One-sided Signals
• Nonlinearly Bandlimited Signals
• Darboux Transformation: New Identities
• Fast Nonlinear Fourier Transform using Chebyshev Polynomials
• A Fast Chebyshev Spectral Method for Nonlinear Fourier Transform
• Discrete Darboux Transformation for Ablowitz-Ladik Systems Derived from Numerical Discretization of Zakharov-Shabat Scattering Problem
• A Chebyshev Spectral Method for Nonlinear Fourier Transform: Norming Constants