Internal waves in 2D domains with ergodic classical dynamics

TL;DR

This paper investigates internal wave behavior in 2D domains with ergodic classical dynamics, proving energy boundedness under certain conditions and providing explicit spectral descriptions for specific geometries.

Contribution

It establishes energy boundedness for internal waves in ergodic 2D domains and explicitly describes the spectrum of related operators for rectangular and elliptic shapes.

Findings

Energy of internal waves remains bounded in ergodic domains.

Explicit spectral descriptions are provided for rectangular and elliptic domains.

The spectrum analysis involves pseudodifferential operators at specific spectral parameters.

Abstract

We study a model of internal waves in an effectively 2D aquarium under periodic forcing. In the case when the underlying classical dynamics has sufficiently irrational rotation number, we prove that the energy of the internal waves remains bounded. This involves studying the spectrum of a related 0-th order pseudodifferential operator at spectral parameters corresponding to such dynamics. For the special cases of rectangular and elliptic domains, we give an explicit spectral description of that operator.