On the ground states and dynamics of space fractional nonlinear Schr\"{o}dinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions

TL;DR

This paper develops efficient numerical methods, including a Gaussian-sum solver and a time-splitting scheme, to compute ground states and simulate dynamics of fractional Schrödinger equations with rotation and nonlocal interactions.

Contribution

It introduces a novel combination of Gaussian-sum solver with Krylov methods and a rotating coordinate transformation for improved simulation of fractional Schrödinger equations.

Findings

Effective computation of ground states using combined methods.

Accurate simulation of dynamics with rotation via coordinate transformation.

Validation of methods through numerical experiments.

Abstract

In this paper, we propose some efficient and robust numerical methods to compute the ground states and dynamics of Fractional Schr\"{o}dinger Equation (FSE) with a rotation term and nonlocal nonlinear interactions. In particular, a newly developed Gaussian-sum (GauSum) solver is used for the nonlocal interaction evaluation \cite{EMZ2015}. To compute the ground states, we integrate the preconditioned Krylov subspace pseudo-spectral method \cite{AD1} and the GauSum solver. For the dynamics simulation, using the rotating Lagrangian coordinates transform \cite{BMTZ2013}, we first reformulate the FSE into a new equation without rotation. Then, a time-splitting pseudo-spectral scheme incorporated with the GauSum solver is proposed to simulate the new FSE.