Growth of Sobolev norms for the analytic NLS on $\mathbb T^2$

Abstract

We consider the completely resonant defocusing non-linear Schr\"odinger 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 $H^s$ Sobolev norms. We also give estimates for the time required to attain this growth.