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

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 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 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 $\mathscr{O}\left(KN+N\log^2N\right)$ such that the error in the computed $N$-samples of the $K$-soliton vanishes as $\mathscr{O}\left(N^{-p}\right)$ where $p$ is the order of convergence of the underlying one-step method. For fixed $N$, this algorithm outperforms the the classical DT (CDT) algorithm which has a complexity of $\mathscr{O}\left(K^2N\right)$. We further present 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/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.