We develop a high accuracy power series method for solving partial differential equations with emphasis on the nonlinear Schr\"odinger equations. The accuracy and computing speed can be systematically and arbitrarily increased to orders of magnitude larger than those of other methods. Machine precision accuracy can be easily reached and sustained for long evolution times within rather short computing time. In-depth analysis and characterisation for all sources of error are performed by comparing the numerical solutions with the exact analytical ones. Exact and approximate boundary conditions are considered and shown to minimise errors for solutions with finite background. The method is extended to cases with external potentials and coupled nonlinear Schr\"odinger equations.

中文翻译：

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用于求解标量、矢量和非齐次非线性薛定谔方程的高精度幂级数方法

我们开发了一种求解偏微分方程的高精度幂级数方法，重点是非线性Schr\"odinger方程。精度和计算速度可以系统地、任意地提高到比其他方法大几个数量级。机器精度精度可以在相当短的计算时间内很容易达到和维持很长的演化时间。通过将数值解与精确解析解进行比较，对所有误差源进行深入分析和表征。考虑并显示了精确和近似的边界条件，以最大限度地减少有限背景解的误差。该方法扩展到具有外部电位和耦合非线性 Schr\"odinger 方程的情况。