A soliton on a vortex filament

Abstract

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 \[ \frac{{\rm l}}{i}\frac{\partial \psi}{\partial t} = \frac{\partial^2\psi}{\partial s^2}+{\textstyle\frac{1}{2}}(|\psi|^2+A)\psi, \] where t is the time, s the length measured along the filament, ψ is the complex variable \[ \psi = \kappa\exp\left(i\int_0^{s}\tau \,ds\right) \] 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.