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Large time blow up for a perturbation of the cubic Szegő equation

We consider the following Hamiltonian equation on a special manifold of rational functions, \[i\p\_tu=Π(|u|^2u)+\al (u|1),\ \al\in\R,\] where $Π$ denotes the Szegő projector on the Hardy space of the circle $\SS^1$. The equation with $\al=0$ was first introduced by G{é}rard and Grellier in \cite{GG1} as a toy model for totally non dispersive evolution equations. We establish the following properties for this equation. For $\al\textless{}0$, any compact subset of initial data leads to a relatively compact subset of trajectories. For $\al\textgreater{}0$, there exist trajectories on which high Sobolev norms exponentially grow with time.