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

Abstract

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.