Large time blow up for a perturbation of the cubic Szeg\H{o} equation

We consider the following Hamiltonian equation on a special manifold of rational functions, $$i\text{p}\mathrm{\_}tu=\mathrm{\Pi }(\mathrm{\mid }u{\mathrm{\mid }}^{2}u)+\text{al}(u\mathrm{\mid }1),\text{al}\in \mathbb{R},$$ where $\mathrm{\Pi }$ denotes the Szeg\H{o} projector on the Hardy space of the circle ${\text{SS}}^1$. The equation with $\text{al}=0$ 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 $\text{al}\text{textless}0$, any compact subset of initial data leads to a relatively compact subset of trajectories. For $\text{al}\text{textgreater}0$, there exist trajectories on which high Sobolev norms exponentially grow with time.