The study of solutions of Hamiltonian PDEs undergoing growth of Sobolev norms as time tends to infinity has drawn considerable attention in recent years. The importance of growth of Sobolev norms is that it implies that the solution transfers energy to higher modes as time evolves.
Consider the cubic defocusing nonlinear Schrödinger equation with periodic boundary conditions and fix s>1. Colliander, Keel, Staffilani, Tao and Takaoka (2010) proved the existence of solutions whose s-Sobolev norm grows in time by any given factor R. Refining their methods in several aspects and using Dynamical Systems techniques, jointly with V. Kaloshin we obtain solutions with s-Sobolev norm growing in polynomial time in R. These improvements allow also to show that the growth of Sobolev norms in polynomial time can also be attained by solutions of the cubic defocusing NLS with a convolution potential.