A conservative exponential time differencing method for the nonlinear cubic Schrödinger equation

ABSTRACT

A mass and energy conservative exponential time differencing scheme using the method of lines is proposed for the numerical solution of a certain family of first-order time-dependent PDEs. The resulting nonlinear system is solved with an unconditionally stable modified predictor–corrector method using a second-order explicit scheme. The efficiency of the method introduced is analyzed and discussed by applying it to the nonlinear cubic Schrödinger equation. The results arising from the experiments for the single, the double soliton waves and the system of two Schrödinger equations are compared with relevant known ones.

KEYWORDS:

• Exponential time differencing
• method of lines
• modified predictor–corrector
• cubic Schrödinger equation

AMS SUBJECT CLASSIFICATIONS:

• 35K55
• 35Q41
• 65M06
• 65M20

Acknowledgements

The authors are grateful to the referees for their constructive comments and suggestions, which have helped to improve the quality and presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).