New spectral collocation algorithms for one- and two-dimensional Schrödinger equations with a Kerr law nonlinearity

Advances in Difference Equations

Mohamed A Abdelkawy¹  Ali H Bhrawy¹  Fouad Mallawi² 

[1] Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt;Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

关键词: one-dimensional Schrödinger equations;    Kerr law nonlinearity;    two-dimensional space Schrödinger equations;    collocation method;    Gauss-type quadratures;   

DOI  :  10.1186/s13662-016-0752-3

学科分类：数学（综合）

来源: SpringerOpen

【 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.

【 授权许可】

CC BY