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arXiv · 2111.05247

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

Patrick Gérard·Sandrine Grellier·Zihui He
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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.

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Patrick Gérard, Sandrine Grellier, Zihui He. 2021-11-09. Turbulent cascades for a family of damped Szeg\"o equations. https://doi.org/10.1088/1361-6544%2Fac7e13

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