Beating effects in cubic Schr\"odinger systems and growth of Sobolev   norms
  We consider a system of coupled cubic Schr\"odinger equations. We prove that there exists a beating effect, i.e. an energy exchange between different modes. This construction may be transported to the linear time-dependent Schr\"odinger equation: we build solutions such that their Sobolev norms grow logarithmically. All these results are stated for large but finite times. 
Beno\^it Gr\'ebert, \'Eric Paturel and Laurent Thomann
math.AP