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The main purpose of this talk is to introduce some recent results on the long time behavior of solutions to some Hamiltonian PDEs, including global existence, growth of high Sobolev norms, scattering and long time approximation by resonant dynamics.
The mathematical models we study are related to the so--called the cubicSzegőequation, which was first introduced by Gérard and Grellier as an important toy model for totally non dispersive evolution equations and a completely integrable system. | 
