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

Abstract

We consider the following Hamiltonian equation on a special manifold of rational functions, \[i\p\_tu=Π(|u|^2u)+\al (u|1),\ \al\in\R,\] where $\Pi$ denotes the Szegő projector on the Hardy space of the circle $\SS^1$. The equation with $\al=0$ was first introduced by G{é}rard and Grellier in \cite{GG1} as a toy model for totally non dispersive evolution equations. We establish the following properties for this equation. For $\al\textless{}0$, any compact subset of initial data leads to a relatively compact subset of trajectories. For $\al\textgreater{}0$, there exist trajectories on which high Sobolev norms exponentially grow with time.

page number:15

Cited by in corpus (8)

• Bounds on quantum evolution complexity via lattice cryptography
• Resonant Hamiltonian systems and weakly nonlinear dynamics in AdS spacetimes
• Growth of Sobolev norms for coupled Lowest Landau Level equations
• Turbulent cascades in a truncation of the cubic Szego equation and related systems
• Exact solutions for a coherent phenomenon of condensation in conservative Hamiltonian systems
• Turbulent cascades for a family of damped Szegö equations
• Energy cascades and condensation via coherent dynamics in Hamiltonian systems
• A superintegrable quantum field theory