English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

This paper considers the non-Hermitian Zakharov-Shabat (ZS) scatteringproblem which forms the basis for defining the SU$(2)$-nonlinear Fouriertransformation (NFT). The theoretical underpinnings of this generalization ofthe conventional Fourier transformation is quite well established in theAblowitz-Kaup-Newell-Segur (AKNS) formalism; however, efficient numericalalgorithms that could be employed in practical applications are stillunavailable.  In this paper, we present a unified framework for the forward and inverse NFTusing exponential one-step methods which are amenable to FFT-based fastpolynomial arithmetic. Within this discrete framework, we propose a fastDarboux 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. Forfixed $N$, this algorithm outperforms the the classical DT (CDT) algorithmwhich has a complexity of $\mathscr{O}\left(K^2N\right)$. We further presentextension of these algorithms to the general version of DT which allows one toadd solitons to arbitrary profiles that are admissible as scattering potentialsin the ZS-problem. The general CDT/FDT algorithms have the same operationalcomplexity as that of the $K$-soliton case and the order of convergence matchesthat of the underlying one-step method. A comparative study of these algorithmsis presented through exhaustive numerical tests.

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