Growth of Sobolev norms for the quintic NLS on $\mathbb T^2$
  We study the quintic Non Linear Schr\"odinger equation on a two dimensional torus and exhibit orbits whose Sobolev norms grow with time. The main point is to reduce to a sufficiently simple toy model, similar in many ways to the one used in the case of the cubic NLS. This requires an accurate combinatorial analysis. 
Emanuele Haus and Michela Procesi
math.AP