Large time blow up for a perturbation of the cubic Szegő equation

Haiyan Xu

Published 2013-07-19, updated 2015-04-27Version 2

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