The present work is concerned with the extension of modified potential operator splitting methods to specific classes of nonlinear evolution equations. The considered partial differential equations of Schr{\"o}dinger and parabolic type comprise the Laplacian, a potential acting as multiplication operator, and a cubic nonlinearity. Moreover, an invariance principle is deduced that has a significant impact on the efficient realisation of the resulting modified operator splitting methods for the Schr{\"o}dinger case.} Numerical illustrations for the time-dependent Gross--Pitaevskii equation in the physically most relevant case of three space dimensions and for its parabolic counterpart related to ground state and excited state computations confirm the benefits of the proposed fourth-order modified operator splitting method in comparison with standard splitting methods. The presented results are novel and of particular interest from both, a theoretical perspective to inspire future investigations of modified operator splitting methods for other classes of nonlinear evolution equations and a practical perspective to advance the reliable and efficient simulation of Gross--Pitaevskii systems in real and imaginary time.

翻译：本文致力于将修正势算子分裂方法推广至特定类别的非线性发展方程。所考虑的薛定谔型和抛物型偏微分方程包含拉普拉斯算子、作为乘法算子作用的势能项以及三次非线性项。此外，推导出一个不变性原理，这对于有效实现薛定谔情形下修正算子分裂方法具有重要影响。在物理上最具相关性的三维空间情况下，针对含时Gross–Pitaevskii方程及其用于基态与激发态计算的抛物型对应方程进行的数值算例表明，与标准分裂方法相比，所提出的四阶修正算子分裂方法具有显著优势。所呈现的结果具有新颖性，兼具理论与实际意义：从理论层面可启发未来针对其他非线性发展方程类别的修正算子分裂方法研究，从实践层面可推动实时间与虚时间下Gross–Pitaevskii系统的可靠高效模拟。