Spectral splitting method for nonlinear Schr\"odinger equations with quadratic potential

TL;DR

This paper introduces a modified spectral splitting method for solving one-dimensional nonlinear Schrödinger equations with quadratic potentials, providing theoretical error estimates and demonstrating improved efficiency through numerical experiments.

Contribution

It proposes a new spectral splitting approach tailored for quadratic potentials, with rigorous error analysis and empirical validation of enhanced performance.

Findings

The modified method accurately approximates solutions with a rigorous error bound.

Numerical experiments show the method is more efficient than standard approaches.

The approach effectively separates linear and nonlinear components for better computation.

Abstract

In this paper we propose a modified Lie-type spectral splitting approximation where the external potential is of quadratic type. It is proved that we can approximate the solution to a one-dimensional nonlinear Schroedinger equation by solving the linear problem and treating the nonlinear term separately, with a rigorous estimate of the remainder term. Furthermore, we show by means of numerical experiments that such a modified approximation is more efficient than the standard one.