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Abstract
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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 as a toy model for totally non dispersive evolution equations by P. Gerard (Paris-Sud University, France) and S. Grellier (University of Orleans, France). This equation admits a Lax pair structure due to the Hankel operator. We establish the following properties for this equation. For $\al<0$, any compact subset of initial data leads to a relatively compact subset of trajectories. For $\al>0$, there exist trajectories on which high Sobolev norms exponentially grow with time.
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