Large-time blowup for a perturbation of the cubic Szegő equation

Abstract

We consider the following Hamiltonian equation on a special manifold of rational functions:

$$i{\partial }_{t}u=\Pi \left(|u{|}^{2}u\right)+\alpha \left(u|1\right),\alpha \in ℝ,$$

where $\Pi$ denotes the Szegő projector on the Hardy space of the circle ${\mathbb{S}}^1$. The equation with $\alpha =0$ was first introduced by Gérard and Grellier as a toy model for totally nondispersive 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 in time.