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

TL;DR

This paper analyzes a perturbed cubic Szeg\

Contribution

It proves the complete integrability of the perturbed equation and studies the dynamics of Hankel operators.

Findings

Solutions remain bounded for , but can explode for  with certain initial data.

The system is shown to be completely integrable in the Liouville sense.

Necessary conditions for norm explosion are identified.

Abstract

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 G{\'e}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 $\alpha>0$.