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

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