Large time blow up for a perturbation of the cubic Szeg\H{o} equation

Haiyan Xu (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We consider the following Hamiltonian equation on a special manifold of rational functions, \[i\p_tu=\Pi(|u|^2u)+\al (u|1),\ \al\in\R,\] where $\Pi $ denotes the Szeg\H{o} projector on the Hardy space of the circle $\SS^1$. The equation with $\al=0$ was first introduced by Gérard and Grellier in \cite{GG1} as a toy model for totally non dispersive evolution equations. We establish the following properties for this equation. For $\al<0$, any compact subset of initial data leads to a relatively compact subset of trajectories. For $\al>0$, there exist trajectories on which high Sobolev norms exponentially grow with time.

Mots clés: Szeg\H{o} equation; integrable Hamiltonian systems; Lax pair; large time blow up

Domaines: Equations aux dérivées partielles [math.AP]

Haiyan Xu. Large time blow up for a perturbation of the cubic Szeg\H{o} equation. Analysis & PDE, 2014, pp.717-731. ⟨hal-00846626v2⟩