Pérez García, Victor M.. 

In this paper, we study a general nonlinear Schrödinger equation with a time-dependent harmonic potential. Despite the lack of translational invariance, we find a symmetry trasformation that, up from any solution, produces infinitely many others that 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 behavior of the center of the wave. Our findings have implications on the dynamics of Bose-Einstein condensates in magnetic traps.