Turbulent cascades for a family of damped Szeg\"o equations

TL;DR

This paper investigates how damping influences energy transfer and turbulence in a family of Szeg"o equations, revealing that damping can promote turbulent cascades and exploring their dynamics.

Contribution

It introduces a two-parameter family of damped Szeg"o equations and demonstrates that damping can induce turbulent cascades, expanding understanding of energy transfer in these systems.

Findings

Damping promotes turbulent cascades in Szeg"o equations.

Existence of turbulent cascades is linked to damping effects.

Dynamics on a six-dimensional submanifold are characterized.

Abstract

In this paper, we study the transfer of energy from low to high frequencies for a family of damped Szeg\"o equations. The cubic Szeg\"o equation has been introduced as a toy model for a totally non-dispersive degenerate Hamiltonian equation. It is a completely integrable system which develops growth of high Sobolev norms, detecting transfer of energy and hence cascades phenomena. Here, we consider a two-parameter family of variants of the cubic Szeg\"o equation and prove that adding a damping term unexpectedly promotes the existence of turbulent cascades. Furthermore, we give a panorama of the dynamics for such equations on a six-dimensional submanifold.