The solution of nonlinear Schrödinger equations using orthogonal spline collocation

Abstract

An orthogonal spline collocation semidiscretization, previously applied in the numerical solution of the cubic Schrödinger equation, is extended to the solution of Schrödinger equations which possess general power nonlinearities and nonlinear terms that contain derivatives. The accuracy of the method is examined, as well as its ability to conserve approximations to certain invariants associated with the theoretical solution. The results of numerical experiments are presented in which the integration in time is performed using a routine from a software library. Particular attention is paid to the value and limitations of conservation as an indicator of accuracy.