We develop a hydrodynamic description of the collective modes of interacting liquids in a quasi-one-dimensional confining potential. By solving Navier-Stokes equations we determine analytically excitation spectrum of sloshing oscillations. For parabo
lic confinement, the lowest frequency eigenmode is not renormalized by interactions and is protected from decay by the Kohn theorem, which states that center of mass motion decouples from internal dynamics. We find that the combined effect of potential anharmonicity and interactions results in the depolarization shift and final lifetime of the Kohn mode. All other excited modes of sloshing oscillations thermalize with the parametrically faster rates. Our results are significant for the interpretation of recent experiments with trapped Fermi gases that observed weak violation of the Kohn theorem.
The results of experimental and theoretical studies of the parametric decay instability of capillary waves on the surface of superfluid helium He-II are reported. It is demonstrated that in a system of turbulent capillary waves low-frequency waves ar
e generated along with the direct Kolmogorov-Zakharov cascade of capillary turbulence. The effects of low-frequency damping and the discreteness of the wave spectrum are discussed.
