2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/95028We continue the study (initiated in \cite{ckstt:7}) of the orbital stability of the ground state cylinder for focussing non-linear Schrödinger equations in the $H^s(\R^n)$ norm for $1-\eps < s < 1$, for small $\eps$. In the $L^2$-subcritical case we obtain a polynomial bound for the time required to move away from the ground state cylinder. If one is only in the $H^1$-subcritical case then we cannot show this, but for defocussing equations we obtain global well-posedness and polynomial growth of $H^s$ norms for $s$ sufficiently close to 1.20 pages, no figures. Some typos correctedAnalysis of PDEs35Q53, 42B35, 37K10text