Numerical algorithms for the direct spectral transform with applications to nonlinear Schrödinger type systems
type: Item
instance of: scholarly article
title: Numerical algorithms for the direct spectral transform with applications to nonlinear Schrödinger type systems
zbMATH Open document ID: 0924.65076
author: K. Appert
published in: Journal of Computational Physics
publication date: 1999-05-02T00:00:00Z
review text: In the context of the nonlinear Schrödinger equation which describes propagation of light pulses in an optical fiber, the direct Zakharov-Shabat spectral problem plays an important role. It reads \[ \Phi_1'= q\Phi_2- i\zeta\Phi_1,\quad \Phi_2'= -q^*\Phi_1+ i\zeta\Phi_2 \] together with some asymptotic conditions. Here, \(\Phi_1\) and \(\Phi_2\) are scalar wave functions, \(q\) is a potential, and \(\zeta\) an eigenvalue parameter. Two different algorithms were implemented for solving numerically the eigenvalue problem on the infinite line. The first algorithm replaces the potential by a piecewise constant approximation, which allows one to solve analytically the corresponding ordinary differential equation. The second algorithm uses the fourth-order Runge-Kutta method. The authors find that even though the Runge-Kutta method is of higher-order, this accuracy can be lost because of the additional dependence of its numerical error on the eigenvalue, whereas the first scheme converges uniformly for both continuous and discrete spectrum components.
Mathematics Subject Classification ID: 65L15
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34L40
zbMATH DE Number: 1282745
zbMATH Keywords: convergence
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: optical fiber
zbMATH Keywords: Zakharov-Shabat spectral problem
zbMATH Keywords: wave functions
zbMATH Keywords: eigenvalue
zbMATH Keywords: algorithms
zbMATH Keywords: Runge-Kutta method
reviewed by: Waldemar Velte
MaRDI profile type: Publication
OpenAlex ID: W2096695668
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DOI: 10.1006/JCPH.1998.6087
description: scientific article; zbMATH DE number 1282745