Fast inverse nonlinear Fourier transformation using exponential one-step methods: Darboux transformation

Abstract

This paper considers the non-Hermitian Zakharov-Shabat (ZS) scattering problem which forms the basis for defining the SU(2) nonlinear Fourier transformation (NFT). The theoretical underpinnings of this generalization of the conventional Fourier transformation are quite well established in the Ablowitz-Kaup-Newell-Segur formalism; however, efficient numerical algorithms that could be employed in practical applications are still unavailable. In this paper, we present a unified framework for the forward and inverse NFT using exponential one-step methods which are amenable to FFT-based fast polynomial arithmetic. Within this discrete framework, we propose a fast Darboux transformation (FDT) algorithm having an operational complexity of O(KN+Nlog^{2}N) such that the error in the computed N-samples of the K-soliton vanishes as O(N^{-p}) where p is the order of convergence of the underlying one-step method. For fixed N, this algorithm outperforms the classical DT (CDT) algorithm which has a complexity of O(K^{2}N). We further present an extension of these algorithms to the general version of DT which allows one to add solitons to arbitrary profiles that are admissible as scattering potentials in the ZS problem. The general CDT and FDT algorithms have the same operational complexity as that of the K-soliton case and the order of convergence matches that of the underlying one-step method. A comparative study of these algorithms is presented through exhaustive numerical tests.