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

Abstract

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

References in corpus (1)

• Multiple singular values of Hankel operators

Cited by in corpus (1)

• Turbulent cascades in a truncation of the cubic Szego equation and related systems