New second- and fourth-order accurate numerical schemes for the nonlinear cubic Schrödinger equation

Abstract

New second- and fourth-order accurate semianalytical methods are introduced for the numerical solution of the one-dimensional nonlinear cubic Schrödinger equation. These methods are based on the combination of the Method of Lines and the Lanczos's Tau Method. The methods are self-starting and proved to be stable, accurate, and energy conservative for long time-integration periods. Approximate solutions are sought, on segmented sets of parallel lines, as finite expansions in terms of a given orthogonal polynomial basis. We have carried out numerical application concerning several cases for the propagation, collision and the bound states of N solitons, 2 ≤ N ≤ 5. Accurate results have been obtained using both shifted Chebyshev and Legendre polynomials. These results compare competitively with some other published results obtained using different methods.

Dedicated to the memory of Professor David J. Evans Former Editor of International Journal of Computer Mathematics, October 2005

Keywords:

• Nonlinear Schrödinger equation
• Bound states of solitons
• Finite difference approxi-mations
• Method of lines
• Lanczos' Tau method

Acknowledgements

The author would like to thank the referees for drawing his attention to additional vital references and for their useful comments and suggestions on an earlier version of this paper. He would also like to thank his colleagues M. Bartuccelli at Surrey, UK and P. G. Estévez at Salamanca, Spain for providing some related references. Thanks are also sent to his colleagues Dee Holmes and Peter Kelly from AVS/UK for providing the Gsharp software package that was used for the generation of the presented graphical results.

Notes

Dedicated to the memory of Professor David J. Evans Former Editor of International Journal of Computer Mathematics, October 2005