High accuracy power series method for solving scalar, vector, and inhomogeneous nonlinear Schr\"odinger equations

TL;DR

This paper introduces a high-accuracy power series method for solving nonlinear Schrödinger equations, achieving machine precision and outperforming existing methods in speed and accuracy for various problem setups.

Contribution

The paper presents a novel power series approach that significantly improves accuracy and efficiency in solving nonlinear Schrödinger equations, including cases with external potentials and coupling.

Findings

Achieves machine precision accuracy for long-time evolution

Systematically increases accuracy and speed over existing methods

Effectively handles boundary conditions and external potentials

Abstract

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.