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:

itu = Π(|u|2u) + α(u|1),α ,

where Π denotes the Szegő projector on the Hardy space of the circle S1. The equation with α = 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 α < 0, any compact subset of initial data leads to a relatively compact subset of trajectories. For α > 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
Milestones
Received: 19 July 2013
Accepted: 28 April 2014
Published: 18 June 2014
Authors
Haiyan Xu
Laboratoire de Mathématique d’Orsay
Université Paris-Sud (XI)
91405 Paris Orsay
France