The Cubic Szego Equation with a Linear Perturbation

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We consider the following Hamiltonian equation on the L² Hardy space on the circle S¹, i∂_ t u = Π(|u| 2 u) + α(u|1), α, where Π is the Szegő projector. The above equation with α= 0 was introduced by Gérard 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.