High-order splitting integrators for nonlinear Schr\"odinger equations over long times

TL;DR

This paper analyzes the long-time behavior of splitting integrators for nonlinear Schrödinger equations, demonstrating near energy conservation over extended periods using resonant Fourier expansion techniques.

Contribution

It provides a comprehensive proof that all consistent splitting integrators with real coefficients nearly conserve energy over long times in a weakly nonlinear setting.

Findings

Energy is nearly conserved over long times.

All consistent splitting integrators with real coefficients are included.

High-order splitting integrators are also analyzed.

Abstract

The long-time behaviour of splitting integrators applied to nonlinear Schr\"odinger equations in a weakly nonlinear setting is studied. It is proven that the energy is nearly conserved on long time intervals. The analysis includes all consistent splitting integrators with real-valued coefficients, in particular splitting integrators of high order. The proof is based on a completely resonant modulated Fourier expansion in time of the numerical solution.