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.

翻译：非线性薛定谔方程和薛定谔-牛顿方程模拟了多个领域中的许多现象。本文对分裂方法（常用于这些方程的数值求解）与积分因子技术（亦称Lawson方法）进行了广泛的数值比较。事实上，后一种方法已知对非线性薛定谔方程表现优异，但尚未被深入研究其对薛定谔-牛顿方程的适用性。我们在一个和两个空间维度上进行了比较，探索了不同边界条件和参数取值。结果表明，对于非线性薛定谔方程的短程势，积分因子技术优于分裂算法；而对于薛定谔-牛顿方程的长程势，其性能则取决于所考虑的具体系统。