文章基本信息

标题：New spectral collocation algorithms for one- and two-dimensional Schrödinger equations with a Kerr law nonlinearity
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作者：Ali H Bhrawy ; Fouad Mallawi ; Mohamed A Abdelkawy 等
期刊名称：Advances in Difference Equations
印刷版ISSN：1687-1839
电子版ISSN：1687-1847
出版年度：2016
卷号：2016
期号：1
页码：18
DOI：10.1186/s13662-016-0752-3
语种：English
出版社：Hindawi Publishing Corporation
摘要：A shifted Jacobi collocation method in two stages is constructed and used to numerically solve nonlinear Schrödinger equations (NLSEs) with a Kerr law nonlinearity, subject to initial-boundary conditions. An expansion in a series of spatial shifted Jacobi polynomials with temporal coefficients for the approximate solution is considered. The first stage, collocation at the shifted Jacobi Gauss-Lobatto (SJ-GL) nodes, is applied for a spatial discretization; its spatial derivatives occur in the NLSE with a treatment of the boundary conditions. This in all will produce a system of ordinary differential equations (SODEs) for the coefficients. The second stage is to collocate at the shifted Jacobi Gauss-Radau (SJ-GR-C) nodes in the temporal discretization to reduce the SODEs to a system of algebraic equations which is solved by an iterative method. Both stages can be extended to solve the two-dimensional NLSEs. Numerical examples are carried out to confirm the spectral accuracy and the efficiency of the proposed algorithms.
关键词：one-dimensional Schrödinger equations ; Kerr law nonlinearity ; two-dimensional space Schrödinger equations ; collocation method ; Gauss-type quadratures