  We study the energy cascade problematic for some nonlinear Schr\"odinger
equations on the torus $\T^2$ in terms of the growth of Sobolev norms. We
define the notion of long-time strong instability and establish its connection
to the existence of unbounded Sobolev orbits. This connection is then explored
for a family of cubic Schr\"odinger nonlinearities that are equal or closely
related to the standard polynomial one $|u|^2u$. Most notably, we prove the
existence of unbounded Sobolev orbits for a family of Hamiltonian cubic
nonlinearities that includes the resonant cubic NLS equation (a.k.a. the first
Birkhoff normal form).
