We study the quintic nonlinear Schrödinger 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 discussed by Colliander et al. for the case of the cubic NLS. This requires an accurate combinatorial analysis.
"Growth of Sobolev norms for the quintic NLS on $T^2$." Anal. PDE 8 (4) 883 - 922, 2015. https://doi.org/10.2140/apde.2015.8.883