  It is shown that plane wave solutions to the cubic nonlinear Schr\"odinger
equation on a torus behave orbitally stable under generic perturbations of the
initial data that are small in a high-order Sobolev norm, over long times that
extend to arbitrary negative powers of the smallness parameter. The
perturbation stays small in the same Sobolev norm over such long times. The
proof uses a Hamiltonian reduction and transformation and, alternatively,
Birkhoff normal forms or modulated Fourier expansions in time.
