#### Vol. 7, No. 3, 2014

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

### Haiyan Xu

Vol. 7 (2014), No. 3, 717–731
##### Abstract

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

$i{\partial }_{t}u=\Pi \left(|u{|}^{2}u\right)+\alpha \left(u|1\right),\phantom{\rule{1em}{0ex}}\alpha \in ℝ,$

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.

##### Keywords
Szegő equation, integrable Hamiltonian systems, Lax pair, large-time blowup
##### Mathematical Subject Classification 2010
Primary: 37J35, 47B35, 35B44