THE CUBIC SZEG\H{O} EQUATION WITH A LINEAR PERTURBATION

Haiyan Xu (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We consider the following Hamiltonian equation on the $L^2$ Hardy space on the circle $S^1$ , $$i\partial_ t u = \Pi(|u|^ 2 u) + \alpha(u|1) , \alpha \in\mathbb{R} ,$$ where $\Pi$ is the Szeg\H{o} projector. The above equation with $\alpha= 0$ 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 $\alpha>0$.

Mots clés: cubic Szeg\H{o} equation; integrable system; Hamiltonian system; energy cascade

Domaines: Equations aux dérivées partielles [math.AP]

Haiyan Xu. THE CUBIC SZEG\H{O} EQUATION WITH A LINEAR PERTURBATION. 2015. ⟨hal-01182835⟩