Bose–Einstein condensation under the cubic–quintic Gross–Pitaevskii equation in radial domains

Abstract

We study stationary and quasi-stationary solutions for the cubic–quintic Gross–Pitaevskii equation modeling Bose–Einstein condensates (BECs) in one, two, and three spatial dimensions under the assumption of radial symmetry with the BEC dynamics influenced by a confining potential. We consider both repulsive and attractive cubic interactions – corresponding respectively to repulsive and attractive two-body interactions – under similar frameworks in order to deduce the effects of the potentials in each case. We also carefully consider the role played by the quintic nonlinearity (modeling the strength of inter-atomic coupling) in modifying the solutions arising due to a purely cubic interaction term. In one spatial dimension, we obtain a variety of exact solutions in the zero-potential limit (including new periodic solutions which generalize known soliton solutions) as well as perturbation solutions for small amplitude confining potentials. For more general forms of the confining potential, we rely on numerical simulations, but these agree with the analytical results when the latter are valid. We also consider the limit where the quintic term dominates the cubic term (with such a limit relevant in the study of a Tonks–Girardeau gas). Under the assumption of radial symmetry, we also consider cylindrical (or, cigar-shaped) and spherical BECs. We consider the nonperturbative regime where either the potential or the amplitude of the solutions is large, obtaining various qualitative analytical results. When the kinetic energy term is small (relative to the nonlinearity and the confining potential), we recover the expected Thomas–Fermi approximation for the stationary solutions. Numerical simulations, under a variety of external confining potentials, are then used to understand the role these potentials play on the BEC solution structure for both the attractive and repulsive regimes. This assortment of analytical and numerical results allows us to better understand the structure of BECs modeled via cubic–quintic Gross–Pitaevskii equation under confining potentials with radial symmetry. Some of the results are natural extensions of those for the cubic Gross–Pitaevskii equation, while others appear new.