Collective oscillations of a quasi one dimensional Bose condensate under damping

Influence of the damping on collective oscillations of a one-dimensional trapped Bose gas in the mean field regime has been studied. Using the phenomenological damping approach developed by L.P. Pitaevskii, modified variational equations for the parameters of the condensate wave function is derived. Analytical expressions for the condensate parameters in equilibrium state have been obtained. Bistability in nonlinear oscillations of the condensate under periodic variations of the trap potential is predicted. The predictions of the modified variational approach are confirmed by full numerical simulations of the 1D GP equation with the damping.