  The nonlinear dynamics of a quasi-1D BEC loaded in a double-well potential is
studied. The beyond mean-field corrections to the energy in the form of the
Lee-Huang-Yang term are taken into account. One-dimensional geometry is
considered. The problem is described in the scalar approximation by the
extended Gross-Pitaevskii (EGP) equation with the attractive quadratic
nonlinearity, due to the Lee-Huang-Yang correction, and the effective cubic
mean-field nonlinearity describing the residual intra- and inter-species
interactions. To describe tunneling and localization phenomena, a two-mode
model was obtained. The frequencies of the Josephson oscillations are found and
confirmed by the full numerical simulations of the EGP equation. The parametric
resonance in the Josephson oscillations, when the height of the barrier is
periodically modulated, is studied. The predictions of the dimer model,
including the case of the one-dimensional Lee-Huang-Yang superfluid, have been
proven.
