A soliton on a vortex filament

The intrinsic equation governing the curvature K and the torsion τ of an isolated very thin vortex filament without stretching in an incompressible inviscid fluid is reduced to a non-linear Schrödinger equation \ li t = ² s²+1{2} (||²+A), \ where t is the time, s the length measured along the filament, ψ is the complex variable \ = (i₀^s \, ds) \ and is a function oft. It is found that this equation yields a solution describing the propagation of a loop or a hump of helical motion along a line vortex, with a constant velocity 2τ. The relation to the system of intrinsic equations derived by Betchov (1965) is discussed.