We consider the following Hamiltonian equation on a special manifold of rational functions, \[i\p\_tu=\Pi(|u|^2u)+\al (u|1),\ \al\in\R,\] where $\Pi $ denotes the Szeg\H{ o} projector on the Hardy space of the circle $\SS^1$. The equation with $\al=0$ was first introduced by G{ \'e}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.
Categories: math.AP