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The Cubic Szeg\h{o} Equation with a Linear Perturbation

We consider the following Hamiltonian equation on the $L^2$ Hardy space on the circle $S^1$ , $$i\partial\_ t u = Π(|u|^ 2 u) + α(u|1) , α\in\mathbb{R} ,$$ where $Π$ is the Szegő projector. The above equation with $α= 0$ was introduced by G{é}rard and Grellier as an important mathematical model [5, 7, 3]. In this paper, we continue our studies started in [22], and prove our system is completely integrable in the Liouville sense. We study the motion of the singular values of the related Hankel operators and find a necessary condition of norm explosion. As a consequence, we prove that the trajectories of the solutions will stay in a compact subset, while more initial data will lead to norm explosion in the case $α>0$.