Turbulent cascades for a family of damped Szegö equations

Abstract

In this paper, we study the transfer of energy from low to high frequencies for a family of damped Szegö equations. The cubic Szegö 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 Szegö 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.

arXiv admin note: text overlap with arXiv:1912.10933

References in corpus (1)

• An inverse problem for Hankel operators and turbulent solutions of the cubic Szegő equation on the line

Cited by in corpus (2)

• Exact solutions for a coherent phenomenon of condensation in conservative Hamiltonian systems
• Energy cascades and condensation via coherent dynamics in Hamiltonian systems