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 $\mathbb{R}$ 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 $H^1$ orbital stability result of Weinstein. In the sequel to this paper we generalize this result to other nonlinear Schr"odinger 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 $L^2$ norm.