  Coherent states play an important role in quantum mechanics because of their
unique properties under time evolution. Here we explore this concept for
one-dimensional repulsive nonlinear Schr\"odinger equations, which describe
weakly interacting Bose-Einstein condensates or light propagation in a
nonlinear medium. It is shown that the dynamics of phase-space translations of
the ground state of a harmonic potential is quite simple: the centre follows a
classical trajectory whereas its shape does not vary in time. The parabolic
potential is the only one that satisfies this property. We study the time
evolution of these nonlinear coherent states under perturbations of their
shape, or of the confining potential. A rich variety of effects emerges. In
particular, in the presence of anharmonicities, we observe that the packet
splits into two distinct components. A fraction of the condensate is
transferred towards uncoherent high-energy modes, while the amplitude of
oscillation of the remaining coherent component is damped towards the bottom of
the well.
