On the numerical solution and dynamical laws of nonlinear fractional Schrödinger/Gross–Pitaevskii equations

ABSTRACT

The purpose of this paper is to discuss some recent developments concerning the numerical simulation of space and time fractional Schrödinger and Gross–Pitaevskii equations. In particular, we address some questions related to the discretization of the models (order of accuracy and fast implementation) and clarify some of their dynamical properties. Some numerical simulations illustrate these points.

KEYWORDS:

• Fractional Schrödinger equation
• fractional Gross–Pitaevskii equations
• dynamical laws
• conservation properties
• discretization schemes
• accuracy order
• fast implementation

2010 AMS SUBJECT CLASSIFICATIONS:

• 35Q41
• 34A08
• 65M06
• 65M70

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Xavier Antoine  http://orcid.org/0000-0002-6501-7757

Additional information

Funding

Xavier Antoine thanks the support from the LIASFMA (University of Lorraine), Agence Nationale de la Recherche (grant no. ANR-12-MONU-0007-02 BECASIM) and the hosting of the Beijing CSRC. Qinglin Tang thanks the support from the Zheng Ge Ru Foundation and the Singapore Ministry of Education Academic Research Fund Tier 2 (grant no. R-146-000-223-112). Jiwei Zhang gratefully acknowledges the support of National Natural Science Foundation of China (grant nos. 91430216 and U1530401).