Geometric resonances in Bose?Einstein condensates with two- and three-body interactions

We investigate geometric resonances in Bose?Einstein condensates by solving the underlying time-dependent Gross?Pitaevskii equation for systems with two- and three-body interactions in an axially symmetric harmonic trap. To this end, we use a recently developed analytical method (Vidanovi? et al 2011 Phys. Rev. A 84 013618), based on both a perturbative expansion and a Poincar??Lindstedt analysis of a Gaussian variational approach, as well as a detailed numerical study of a set of ordinary differential equations for variational parameters. By changing the anisotropy of the confining potential, we numerically observe and analytically describe strong nonlinear effects: shifts in the frequencies and mode coupling of collective modes, as well as resonances. Furthermore, we discuss in detail the stability of a Bose?Einstein condensate in the presence of an attractive two-body interaction and a repulsive three-body interaction. In particular, we show that a small repulsive three-body interaction is able to significantly extend the stability region of the condensate.