A classical problem of hydrodynamics is the motion of Rankine vortex, or a vortex patch. This is a domain with a fixed vorticity in an incompressible two-dimensional flow. What if the domain consists of a dense assembly of individual vortices? Each vortex as a non-distructable topological characterization. Does it disappear in an approximation when the f vortices matter is treated as a uniform vorticity?
We will show that on the edge of the vortex patch there is a non-linear wave localized within the boundary layer, the edge wave. The edge wave is solely an effect of the topological nature of vortices. It disappears in the approximation that treats the vortex matter as a regular vorticity patch. The edge wave is described by the integrable Benjamin-Davis-Ono equation and exhibits solitons with a quantized total vorticity, revealing the topological characteristics of vortices.
The edge wave is a classical analog of the edge states of the fractional quantum Hall effect.