Bose–Einstein condensates in optical lattices: the cubic–quintic nonlinear Schrödinger equation with a periodic potential

Abstract

The cubic–quintic nonlinear Schrödinger equation with a periodic potential, which is expressed in terms of the Jacobian function dn, is used to model a Bose–Einstein condensate trapped in an optical lattice (in most cases, the dn potential is very close to the ordinary sinusoidal one). The quintic term in the equation represents either effects of three-body collisions (which is relevant in the case when the ordinary two-body collisions are very weak), or an effective deviation of the trapped condensate from the one-dimensionality. The cases of both repulsive and attractive two-body interactions are considered. The same model governs the planar propagation of light in waveguides with the cubic–quintic optical nonlinearity, the periodic potential being induced by transverse modulation of the refractive index. In the case of the attractive quintic term (which is always the case if it accounts for the residual non-one-dimensionality of the tightly trapped condensate), a new family of exact periodic solutions is constructed in terms of Jacobian elliptic functions. Using semi-analytical and numerical methods, a stability region for these periodic states is identified. The effect of boundaries is shown to be very weak.E Kengne dedicates this work to Taa Bah Deffo Takotue Innocent, the Chairman of the Bandjoun Community in Douala (Cameroon).