The cubic Szegö equation has been introduced as a toy model for totally non dispersive evolution equations. It turned out that it is a complete integrable Hamiltonian system for which we built a non linear Fourier transform giving an explicit expression of the solutions.
This explicit formula allows to study the dynamics of the solutions. We will explain different aspects of it: almost-periodicity of the solutions in the energy space, uniform analyticity for a large set of initial data, turbulence phenomenon for a dense set of smooth initial data in large Sobolev spaces.
From joint works with Patrick Gérard.[-]

It has been observed by physicists (Isaacson, Burnett, Green-Wald) that metric perturbations of a background solution, which are small amplitude but with high frequency, yield at the limit to a non trivial contribution which corresponds to the presence of an energy impulsion tensor in the equation for the background metric. This non trivial contribution is of due to the nonlinearities in Einstein equations, which involve products of derivatives of the metric. It has been conjectured by Burnett that the only tensors which can be obtained this way are massless Vlasov, and it has been proved by Green and Wald that the limit tensor must be traceless and satisfy the dominant energy condition. The known exemples of this phenomena are constructed under symmetry reductions which involve two Killing fields and lead to an energy impulsion tensor which consists in at most two dust fields propagating in null directions. In this talk, I will explain our construction, under a symmetry reduction involving one Killing field, which leads to an energy impulsion tensor consisting in N dust fields propagating in arbitrary null directions. This is a joint work with Jonathan Luk (Stanford).[-]