In this paper, we study the inverse scattering problem for a class of signals that have a compactly supported reflection coefficient. The problem boils down to the solution of the Gelfand–Levitan–Marchenko (GLM) integral equations with a kernel that is bandlimited. By adopting a sampling theory approach to the associated Hankel operators in the Bernstein spaces, a constructive proof of existence of a solution of the GLM equations is obtained under various restrictions on the nonlinear impulse response (NIR). The formalism developed in this article also lends itself well to numerical computations yielding algorithms that are shown to have algebraic rates of convergence. In particular, the use Whittaker–Kotelnikov–Shannon sampling series yields an algorithm that converges as O(N ^{− 1} / ² ) whereas the use of Helms and Thomas (HT) version of the sampling expansion yields an algorithm that converges as O(N ⁻ m ^{− 1} / ² ) for any m > 0 provided the regularity conditions are fulfilled. The complexity of the algorithms depend on the linear solver used. The use of conjugate-gradient (CG) method yields an algorithm of complexity O(N _iter. N ² ) per sample of the signal where N is the number of sampling basis functions used and N _iter. is the number of CG iterations involved. The HT version of the sampling expansions facilitates the development of algorithms of complexity O(N _iter. N log N) (per sample of the signal) by exploiting the special structure as well as the (approximate) sparsity of the matrices involved. The algorithms are numerically validated using Schwartz class functions as NIRs that are either bandlimited or effectively bandlimited. The results suggest that the HT variant of our algorithm is spectrally convergent for an input of the aforementioned class.

We explore two classes of exponential integrators, in this letter, to design the nonlinear Fourier transform (NFT) algorithms with a convergence order of four on an equispaced grid. The integrating factor-based method in the class of the Runge-Kutta methods yields algorithms with complexity O(N\log²N) (where N is the number of samples of the signal), which have superior accuracy-complexity tradeoff than any of the fast methods known currently. The integrators based on Magnus series expansion, namely, standard and commutator-free Magnus methods yield the algorithms of complexity O(N²) that have superior error behavior than that of the fast methods.

Of the two main objectives we pursue in this paper, the first one consists in the studying operators of the form (∂t−i△Γ)α, α=1/2,−1/2,−1,…, where △_Γ is the Laplace-Beltrami operator. These operators arise in the context of nonreflecting boundary conditions in the pseudo-differential approach for the general Schrödinger equation. The definition of such operators is discussed in various settings, and a formulation in terms of fractional operators is provided. The second objective consists in deriving corner conditions for a rectangular domain in order to make such domains amenable to the pseudo-differential approach. The stability and uniqueness of the solution is investigated for each of these novel boundary conditions.

It is demonstrated in this letter that linear multistep methods for integrating ordinary differential equations can be used to develop a family of fast forward scattering algorithms with higher orders of convergence. Excluding the cost of computing the discrete eigenvalues, the nonlinear Fourier transform (NFT) algorithm thus obtained has a complexity of O(KN+C_pNlog²N) such that the error vanishes as mathop O(N^-p) where p ϵ {1,2,3,4} and K is the number of eigenvalues. Such an algorithm can be potentially useful for the recently proposed NFT-based modulation methodology for optical fiber communication. The exposition considers the particular case of the backward differentiation formula (C_p=p³) and the implicit Adams method (C_p=(p-1³,p>1) of which the latter proves to be the most accurate family of methods for fast NFT.

This paper considers the non-Hermitian Zakharov-Shabat 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 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+Nlog2N) 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(Nlog2N)] together with a new fast Darboux transformation (FDT) algorithm [O(KN+Nlog2N)] previously developed by V. Vaibhav [Phys. Rev. E 96, 063302 (2017)2470-004510.1103/PhysRevE.96.063302]. 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. The second step involves addition of bound states using the FDT algorithm. Finally, the performance of these algorithms is demonstrated through exhaustive numerical tests.

In this article, we study the properties of the nonlinear Fourier spectrum in order to gain better control of the temporal support of the signals synthesized using the inverse nonlinear Fourier transform. In particular, we provide necessary and sufficient conditions satisfied by the nonlinear Fourier spectrum such that the generated signal has a prescribed support. In our exposition, we assume that the support is a simply connected domain that is either a bounded interval or the half-line, which amounts to studying the class of signals which are either time-limited or one-sided, respectively. Further, it is shown that the analyticity properties of the scattering coefficients of the aforementioned classes of signals can be exploited to improve the numerical conditioning of the differential approach of inverse scattering. Here, we also revisit the integral approach of inverse scattering and provide the correct derivation of the so called Töplitz inner-bordering algorithm. Finally, we conduct extensive numerical tests in order to verify the analytical results presented in the article. These tests also provide us an opportunity to compare the performance of the two aforementioned numerical approaches in terms of accuracy and complexity of computations.

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 OKN+Nlog2N such that the error in the computed N-samples of the K-soliton vanishes as ON-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 OK2N. 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.

In optical fiber communication, the nonlinear frequency division multiplexing scheme requires a fast inverse nonlinear Fourier transform (NFT). We present two algorithms with O(N(K + log² N)) complexity for N samples of a signal comprising K eigenvalues.

Recently, there has been much interest in using (inverse) nonlinear Fourier transforms (NFTs) to (de-)modulate data in multi-solitonic signals. In this paper, a Newton-type NFT with a reduced complexity order is investigated.