Nonlinear Fourier Transformation Research Articles

Characterization of sidebands in fiber lasers based on nonlinear Fourier transformation.

Phase evolution of soliton and that of first-order sidebands in a fiber laser are investigated by using nonlinear Fourier transform (NFT). Development from dip-type sidebands to peak-type (Kelly) sidebands is presented. The phase relationship between the soliton and the sidebands calculated by the NFT are in good agreement with the average soliton theory. Our results suggest that NFT can be an effective tool for the analysis of laser pulses.

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

Equivalent Integral Equations of the Static Schrödinger Equation by Variation of Constants Method

As a milestone method, the inverse scattering transformation is also known as the nonlinear Fourier transformation for solving nonlinear partial differential equations analytically. The equivalent integral equations play a crucial role for the inverse scattering transformation. In this paper, the equivalent integral equations of the static Schrödinger equation are derived by the variation of constants method. The derivation of the equivalent integral equations provides with a necessary help for the beginners to study the inverse scattering transformation method in soliton theory.

On deconvolving spectra

A simple method to deconvolve spectra combines aspects of nonlinear least-squares curve fitting and Fourier transformation to reduce the effects of noise. The method fits data to smooth analytical functions, then deconvolves the latter analytically. For example, when the data can be written as a sum of Gaussians gr=ar exp[−(x−cr)2/2br2], and the transfer function as a single Gaussian gt, then the deconvolved signal is the corresponding sum of Gaussians gs, with cs=cr, bs2=br2−bt2, and as=arbr/bs.