  We study the asymptotic behavior of large data solutions to Schr\"odinger
equations $i u_t + \Delta u = F(u)$ in $\R^d$, assuming globally bounded
$H^1_x(\R^d)$ norm (i.e. no blowup in the energy space), in high dimensions $d
\geq 5$ and with nonlinearity which is energy-subcritical and
mass-supercritical. In the spherically symmetric case, we show that as $t \to
+\infty$, these solutions split into a radiation term that evolves according to
the linear Schr\"odinger equation, and a remainder which converges in
$H^1_x(\R^d)$ to a compact attractor, which consists of the union of
spherically symmetric almost periodic orbits of the NLS flow in $H^1_x(\R^d)$.
This is despite the total lack of any dissipation in the equation. This
statement can be viewed as weak form of the "soliton resolution conjecture". We
also obtain a more complicated analogue of this result for the
non-spherically-symmetric case. As a corollary we obtain the "petite
conjecture" of Soffer in the high dimensional non-critical case.
