Variational model for the delayed collapse of Bose-Einstein condensates

We present an action that can be used to study variationally the collapse of Bose-Einstein condensates. This action is real, even though it includes dissipative terms. It adopts long-range interactions between the atoms, so that there is always a stable minimum of the energy, even if the remaining number of atoms is above the number that in the case of local interactions is the critical one. The proposed action incorporates the time needed for the abrupt and delayed onset of collapse, yielding in fact its dependence on the scattering length. We show that the evolution of the condensate is equivalent to the motion of a particle in an effective potential. The particle begins its motion far from the point of stable equilibrium and it then proceeds to oscillate about that point. We prove that the resulting large oscillations in the shape of the wave function after the collapse have frequencies equal to twice the frequencies of the traps. Our results agree with the experimental observations.