  We consider the propagation of wave packets for a one-dimensional nonlinear
Schrodinger equation with a matrix-valued potential, in the semi-classical
limit. For an initial coherent state polarized along some eigenvector, we prove
that the nonlinear evolution preserves the separation of modes, in a scaling
such that nonlinear effects are critical (the envelope equation is nonlinear).
The proof relies on a fine geometric analysis of the role of spectral
projectors, which is compatible with the treatment of nonlinearities. We also
prove a nonlinear superposition principle for these adiabatic wave packets.
