Large time blow up for a perturbation of the cubic Szeg\H{o} equation

TL;DR

This paper investigates a Hamiltonian equation related to the cubic Szegő equation, revealing that for positive perturbation parameters solutions can exhibit exponential growth in Sobolev norms, indicating blow-up behavior.

Contribution

It introduces and analyzes a perturbed Szegő equation on a rational function manifold, demonstrating contrasting dynamics depending on the sign of the perturbation parameter.

Findings

For , solutions remain relatively compact.

For \u00b1, solutions can exhibit exponential Sobolev norm growth.

The equation models non-dispersive evolution phenomena.

Abstract

We consider the following Hamiltonian equation on a special manifold of rational functions, \[i\p\_tu=\Pi(|u|^2u)+\al (u|1),\ \al\in\R,\] where $\Pi $ denotes the Szeg\H{o} projector on the Hardy space of the circle $\SS^1$. The equation with $\al=0$ was first introduced by G{\'e}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.