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 \[ {{ l}}{i}∂ ψ/∂ t = ∂^2ψ/∂ s^2+{1/2}(|ψ|^2+A)ψ, \] where t is the time, s the length measured along the filament, ψ is the complex variable \[ ψ = κexp(i∫_0ˢτ \,ds) \] 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.