We consider the completely resonant non-linear Schrödinger equation on the two dimensional torus with any analytic gauge invariant nonlinearity. Fix s>1. We show the existence of solutions of this equation which achieve arbitrarily large growth of Hs Sobolev norms. We also give estimates for the time required to attain this growth.
Guardia, M., Sandhaus, E., Procesi, M. Growth of Sobolev norms for the analytic NLS on T-2. "Advances in mathematics", 1 Octubre 2016, vol. 301, p. 615-692.