A new second-order accurate time discretization method for the nonlinear cubic Schrödinger equation

Abstract

A new second-order accurate semi-analytical time discretization method is introduced for the numerical solution of the one-dimensional nonlinear cubic Schrödinger equation. This method is based on the combination of the method of lines, Crank–Nicolson method, Newton method and Lanczos’ Tau method. It is a self-starting averaged two-time-level scheme that has proved to be stable, accurate and energy conservative for long time integration periods. At each time level, approximate solutions are sought on a segmented spatial interval as finite expansions in terms of a given orthogonal polynomial basis mapped appropriately onto each spatial subsegment. We have carried out numerical simulation concerning several cases for the propagation, collision and the bound states of solitons. Accurate results have been obtained using Chebyshev and Legendre polynomials. These results are well comparable with other published results obtained by the use of various standard numerical methods.

Keywords:

• nonlinear cubic Schrödinger equation
• the method of lines
• the Lanczos’ Tau method
• the Crank–Nicolson method
• the Newton method

2000 AMS Subject Classifications:

• 65D30
• 74G15
• 74S30

Computing Classification System:

• G.1
• G.1.2
• G.1.8

Acknowledgements

The author would like to thank the referee for drawing his attention to additional vital references and for the useful comments and suggestions on an earlier version of this article. He would also like to thank his colleague I. Osman at the AUB for providing some vital references.