Direct nonlinear Fourier transform algorithms for the computation of solitonic spectra in focusing nonlinear Schrödinger equation

A. Vasylchenkova, J.e. Prilepsky, D. Shepelsky, A. Chattopadhyay

Research output: Contribution to journalArticle

Abstract

Starting from a comparison of some established numerical algorithms for the computation of the eigenvalues (discrete or solitonic spectrum) of the non-Hermitian version of the Zakharov–Shabat spectral problem, this article delivers new algorithms that combine the best features of the existing ones and thereby allays their relative weaknesses. Our algorithm is modelled within the remit of the so-called direct nonlinear Fourier transform (NFT) associated with the focusing nonlinear Schrödinger equation. First, we present the data for the calibration of existing methods comparing the relative errors associated with the computation of the continuous NF spectrum. Then each method is paired with different numerical algorithms for finding zeros of a complex-valued function to obtain the eigenvalues. Next we describe a new class of methods based on the contour integrals evaluation for the efficient search of eigenvalues. After that we introduce a new hybrid method, one of our main results: the method combines the advances of contour integral approach and makes use of the iterative algorithms at its second stage for the refined eigenvalues search. The veracity of our new hybrid algorithm is established by estimating the convergence speed and accuracy across three independent test profiles. Along with the development of a new approach for the computation of the eigenvalues, our study also addresses the problem of computation of the so-called norming constants associated with the eigenvalues. We show that our formalism effectively amounts to accurate and fast enough computation of residues of the reflection coefficient in the upper complex half-plane of the spectral parameter.

Original languageEnglish
Pages (from-to)347-371
Number of pages25
JournalCommunications in Nonlinear Science and Numerical Simulation
Volume68
Early online date6 Sep 2018
DOIs
Publication statusPublished - 1 Mar 2019

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Nonlinear equations
Fourier transform
Fourier transforms
Nonlinear Equations
Eigenvalue
Contour integral
Numerical Algorithms
Continuous Spectrum
Reflection Coefficient
Spectral Problem
Convergence Speed
Relative Error
Hybrid Algorithm
Hybrid Method
Half-plane
Iterative Algorithm
Calibration
Evaluation
Zero

Bibliographical note

© 2018, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/

Funding: JEP acknowledges the support from the UK EPSRC Programme Grant UNLOC EP/J017582/1. JEP and DS are thankful to the Erasmus+ ERC mobility programme between the Aston University and Kharkiv National University that helped us to launch the collaborative activity. AC acknowledges the RISE-FRAMED grant.

Keywords

  • Inverse scattering method
  • Nonlinear Schrödinger equation
  • Numerical algorithms
  • Signal processing

Cite this

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title = "Direct nonlinear Fourier transform algorithms for the computation of solitonic spectra in focusing nonlinear Schr{\"o}dinger equation",
abstract = "Starting from a comparison of some established numerical algorithms for the computation of the eigenvalues (discrete or solitonic spectrum) of the non-Hermitian version of the Zakharov–Shabat spectral problem, this article delivers new algorithms that combine the best features of the existing ones and thereby allays their relative weaknesses. Our algorithm is modelled within the remit of the so-called direct nonlinear Fourier transform (NFT) associated with the focusing nonlinear Schr{\"o}dinger equation. First, we present the data for the calibration of existing methods comparing the relative errors associated with the computation of the continuous NF spectrum. Then each method is paired with different numerical algorithms for finding zeros of a complex-valued function to obtain the eigenvalues. Next we describe a new class of methods based on the contour integrals evaluation for the efficient search of eigenvalues. After that we introduce a new hybrid method, one of our main results: the method combines the advances of contour integral approach and makes use of the iterative algorithms at its second stage for the refined eigenvalues search. The veracity of our new hybrid algorithm is established by estimating the convergence speed and accuracy across three independent test profiles. Along with the development of a new approach for the computation of the eigenvalues, our study also addresses the problem of computation of the so-called norming constants associated with the eigenvalues. We show that our formalism effectively amounts to accurate and fast enough computation of residues of the reflection coefficient in the upper complex half-plane of the spectral parameter.",
keywords = "Inverse scattering method, Nonlinear Schr{\"o}dinger equation, Numerical algorithms, Signal processing",
author = "A. Vasylchenkova and J.e. Prilepsky and D. Shepelsky and A. Chattopadhyay",
note = "{\circledC} 2018, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/ Funding: JEP acknowledges the support from the UK EPSRC Programme Grant UNLOC EP/J017582/1. JEP and DS are thankful to the Erasmus+ ERC mobility programme between the Aston University and Kharkiv National University that helped us to launch the collaborative activity. AC acknowledges the RISE-FRAMED grant.",
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AU - Vasylchenkova, A.

AU - Prilepsky, J.e.

AU - Shepelsky, D.

AU - Chattopadhyay, A.

N1 - © 2018, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/ Funding: JEP acknowledges the support from the UK EPSRC Programme Grant UNLOC EP/J017582/1. JEP and DS are thankful to the Erasmus+ ERC mobility programme between the Aston University and Kharkiv National University that helped us to launch the collaborative activity. AC acknowledges the RISE-FRAMED grant.

PY - 2019/3/1

Y1 - 2019/3/1

N2 - Starting from a comparison of some established numerical algorithms for the computation of the eigenvalues (discrete or solitonic spectrum) of the non-Hermitian version of the Zakharov–Shabat spectral problem, this article delivers new algorithms that combine the best features of the existing ones and thereby allays their relative weaknesses. Our algorithm is modelled within the remit of the so-called direct nonlinear Fourier transform (NFT) associated with the focusing nonlinear Schrödinger equation. First, we present the data for the calibration of existing methods comparing the relative errors associated with the computation of the continuous NF spectrum. Then each method is paired with different numerical algorithms for finding zeros of a complex-valued function to obtain the eigenvalues. Next we describe a new class of methods based on the contour integrals evaluation for the efficient search of eigenvalues. After that we introduce a new hybrid method, one of our main results: the method combines the advances of contour integral approach and makes use of the iterative algorithms at its second stage for the refined eigenvalues search. The veracity of our new hybrid algorithm is established by estimating the convergence speed and accuracy across three independent test profiles. Along with the development of a new approach for the computation of the eigenvalues, our study also addresses the problem of computation of the so-called norming constants associated with the eigenvalues. We show that our formalism effectively amounts to accurate and fast enough computation of residues of the reflection coefficient in the upper complex half-plane of the spectral parameter.

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