# Edge wave and boundary layer of vortex matter

## TL;DR

This paper investigates the boundary layer and localized edge wave phenomena in dense vortex matter, revealing their topological origin, governed by the Benjamin-Davis-Ono equation, and their analogy to quantum Hall edge states.

## Contribution

It introduces the concept of an edge wave and vorticity boundary layer in vortex matter, linking them to topological properties and odd viscosity, and models the edge wave with the integrable Benjamin-Davis-Ono equation.

## Key findings

- Edge wave is governed by the Benjamin-Davis-Ono equation.
- Vorticity layer and edge wave are due to odd viscosity.
- Edge wave dynamics relate to the Virasoro-Bott group.

## Abstract

We show that a vortex matter, that is a dense assembly of vortices in an incompressible two-dimensional flow, such as a fast rotating superfluid or turbulent flows with sign-like eddies, exhibits (i) a boundary layer of vorticity (vorticity layer), and (ii) a nonlinear wave localized within the vorticity layer, the edge wave. Both are solely an effect of the topological nature of vortices. Both are lost if the vortex matter is approximated as a continuous vorticity patch. The edge wave is governed by the integrable Benjamin-Davis-Ono equation exhibiting solitons with a quantized total vorticity. Quantized solitons reveal the topological nature of the vortices through their dynamics. The edge wave and the vorticity layer are due to odd viscosity of the vortex matter. We also identify the dynamics with the action of the Virasoro-Bott group of diffeomorphisms of the circle, where odd viscosity parametrizes the central extension. Our edge wave is a hydrodynamic analog of the edge states of the fractional quantum Hall effect.