Construction of exact solutions by spatial traslations in inhomogeneous Nonlinear Schrodinger equations. Applications to Bose-Einstein condensation

In this paper we study a general nonlinear Schr\"odinger equation with a time dependent harmonic potential. Despite the lack of traslational invariance we find a symmetry trasformation which, up from any solution, produces infinitely many others which are centered on classical trajectories. The results presented here imply that, not only the center of mass of the wave-packet satisfies the Ehrenfest theorem and is decoupled from the dynamics of the wave-packet, but also the shape of the solution is independent of the behaviour of the center of the wave. Our findings have implications on the dynamics of Bose-Einstein condensates in magnetic traps