Integrating factor techniques applied to the Schrödinger-like equations. Comparison with Split-Step methods

The nonlinear Schrödinger and the Schrödinger–Newton equations model many phenomena in various fields. Here, we perform an extensive numerical comparison between splitting methods (often employed to numerically solve these equations) and the integrating factor technique, also called Lawson method. Indeed, the latter is known to perform very well for the nonlinear Schrödinger equation, but has not been thoroughly investigated for the Schrödinger–Newton equation. Comparisons are made in one and two spatial dimensions, exploring different boundary conditions and parameters values. We show that for the short range potential of the nonlinear Schrödinger equation, the integrating factor technique performs better than splitting algorithms, while, for the long range potential of the Schrödinger–Newton equation, it depends on the particular system considered. • Extensive numerical study of Schrödinger-like equations. • Comparisons between splitting algorithms and Lawson methods. • Different physical systems, spatial dimensions and boundary conditions considered. • Significant speed and precision increase, depending on the interaction term.