Growth of Sobolev norms for the analytic NLS on T2T2

We consider the completely resonant non-linear Schrödinger equation on the two dimensional torus with any analytic gauge invariant nonlinearity. Fix s>1s>1. We show the existence of solutions of this equation which achieve arbitrarily large growth of HsHs Sobolev norms. We also give estimates for the time required to attain this growth.