Integrating factor techniques applied to the Schr\"odinger-like equations. Comparison with Split-Step methods

TL;DR

This paper compares the effectiveness of integrating factor techniques and splitting methods in numerically solving Schr"odinger-like equations, revealing that the former often outperforms the latter for short-range potentials, with mixed results for long-range potentials.

Contribution

The study provides a comprehensive numerical comparison of integrating factor and splitting methods for Schr"odinger-like equations, including the Schr"odinger-Newton case, across various conditions.

Findings

Integrating factor techniques outperform splitting methods for short-range potentials.

Performance for long-range potentials depends on the specific system.

Extensive analysis in 1D and 2D scenarios with different boundary conditions.

Abstract

The nonlinear Schr\"odinger and the Schr\"odinger-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 Schr\"odinger equation, but has not been thoroughly investigated for the Schr\"odinger-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 Schr\"odinger equation, the integrating factor technique performs better than splitting algorithms, while, for the long range potential of the Schr\"odinger-Newton equation, it depends on the particular system considered.