  We consider the propagation of wave packets for a nonlinear Schr\"odinger
equation, with a matrix-valued potential, in the semi-classical limit. For a
matrix-valued potential, Strichartz estimates are available under long range
assumptions. Under these assumptions, for an initial coherent state polarized
along an eigenvector, we prove that the wave function remains in the same
eigenspace, in a scaling such that nonlinear effects cannot be neglected. We
also prove a nonlinear superposition principle for these nonlinear wave
packets.
