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The Gross-Pitaevskii equation is a widely used model in physics, inparticular in the context of Bose-Einstein condensates. However, it only takesinto account local interactions between particles. This paper demonstrates thevalidity of using a nonlocal formulation as a generalization of the localmodel. In particular, the paper demonstrates that the solution of the nonlocalmodel approaches in norm the solution of the local model as the nonlocal modelapproaches the local model. The nonlocality and potential used for theGross-Pitaevskii equation are quite general, thus this paper shows that one caneasily add nonlocal effects to interesting classes of Bose-Einstein condensatemodels. Based on a particular choice of potential for the nonlocalGross-Pitaevskii equation, we establish the orbital stability of a class ofparameter-dependent solutions to the nonlocal problem for certain parameterregimes. Numerical results corroborate the analytical stability results andlead to predictions about the stability of the class of solutions for parametervalues outside of the purview of the theory established in this paper.

On Nonlocal Gross-Pitaevskii Equations with Periodic Potentials