Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm

We study the long-time behaviour of the focusing cubic NLS on R in the Sobolev norms H^s for 0 < s < 1. We obtain polynomial growth-type upper bounds on the H^s norms, and also limit any orbital H^s instability of the ground state to polynomial growth at worst; this is a partial analogue of the H¹ orbital stability result of Weinstein [27],[26]. In the sequel to this paper we generalize this result to other nonlinear Schrödinger equations. Our arguments are based on the "I-method" from earlier papers [9]-[15] which pushes down from the energy norm, as well as an "upside-down I-method" which pushes up from the L² norm.