Efficient dissipation-preserving scheme for the damped nonlinear Schrödinger equation in three dimensions

Abstract

We develop an efficient prediction–correction scheme which preserves the exact decay rate of normalization for the damped nonlinear Schrödinger equation in three dimensions. The prediction step only requires solving a scalar nonlinear equation at each grid point and the correction step needs solving one scalar quadratic equation which costs negligible CPU time. Further, compact representation of the prediction–correction scheme saves computational cost and storage requirement greatly. Numerical results are presented to validate the effectiveness and accuracy of the proposed scheme.