On the numerical solution of the cubic Schrödinger equation in one space variable
type: Item
instance of: scholarly article
title: On the numerical solution of the cubic Schrödinger equation in one space variable
zbMATH Open document ID: 0766.65112
author: S. Singh
published in: Journal of Computational Physics
publication date: 1993-05-16T00:00:00Z
review text: The authors study some basic issues involving the time and spatial discretizations of the cubic nonlinear Schrödinger equation defined over the entire real line. Two types of spatial discretizations are considered: the \(L^ 2\)-Galerkin method with product approximation and the integrable finite difference scheme of \textit{M. J. Ablowitz} and \textit{J. F. Ladik} [Stud. Appl. Math. 55, 213-229 (1976; Zbl 0338.35002)]. By numerical studies using NAG library routines, it is shown that the Galerkin method suffers from an instability during long time integrations with features similar to those observed for the periodic problem. It is further demonstrated that the instability may be avoided (at the response of phase errors) using the integrable discretization of Ablowitz and Ladik. The authors conclude the paper by stressing that since integrable schemes are designed specifically for individual problems, more general numerical techniques remain as important as ever.
Mathematics Subject Classification ID: 65Z05
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 35Q55
zbMATH DE Number: 167070
zbMATH Keywords: Galerkin method
zbMATH Keywords: cubic nonlinear Schrödinger equation
zbMATH Keywords: fininte difference scheme
zbMATH Keywords: instability
describes a project that uses: NAG
MaRDI profile type: Publication
full work available at URL: jcph.1993.1029
OpenAlex ID: W2021045768
DOI: 10.1006/JCPH.1993.1029
description: scientific article; zbMATH DE number 167070