A nonlinear adiabatic theorem for coherent states

We consider the propagation of wave packets for a one-dimensional nonlinear Schrödinger equation with a matrix-valued potential, in the semi-classical limit. For an initial coherent state polarized along an 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.