On Nonlocal Gross-Pitaevskii Equations with Periodic Potentials

TL;DR

This paper validates the use of nonlocal Gross-Pitaevskii equations as a generalization of the local model for Bose-Einstein condensates, demonstrating convergence of solutions and establishing stability for certain parameters.

Contribution

It introduces a nonlocal formulation of the Gross-Pitaevskii equation, proves solution convergence to the local model, and establishes stability results for specific parameter regimes.

Findings

Nonlocal solutions approach local solutions in norm.

Orbital stability of certain solutions is established.

Numerical results support analytical stability predictions.

Abstract

The Gross-Pitaevskii equation is a widely used model in physics, in particular in the context of Bose-Einstein condensates. However, it only takes into account local interactions between particles. This paper demonstrates the validity of using a nonlocal formulation as a generalization of the local model. In particular, the paper demonstrates that the solution of the nonlocal model approaches in norm the solution of the local model as the nonlocal model approaches the local model. The nonlocality and potential used for the Gross-Pitaevskii equation are quite general, thus this paper shows that one can easily add nonlocal effects to interesting classes of Bose-Einstein condensate models. Based on a particular choice of potential for the nonlocal Gross-Pitaevskii equation, we establish the orbital stability of a class of parameter-dependent solutions to the nonlocal problem for certain parameter regimes. Numerical results corroborate the analytical stability results and lead to predictions about the stability of the class of solutions for parameter values outside of the purview of the theory established in this paper.