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

where $Π$ denotes the Szegő projector on the Hardy space of the circle $S^{1}$ . 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

Vol. 7, No. 3, 2014

Haiyan Xu

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors