Exact solution for the self-induced motion of a vortex filament in the arc-length representation of the local induction approximation

摘要

We review two formulations of the fully nonlinear local induction equation approximating the self-inducednmotion of the vortex filament (in the local induction approximation), corresponding to the Cartesian and arc-lengthncoordinate systems. The arc-length representation put forth by Umeki [Theor. Comput. Fluid Dyn. 24, 383n(2010)] results in a type of 1 + 1 derivative nonlinear Schr¨odinger (NLS) equation describing the motion ofnsuch a vortex filament.We obtain exact stationary solutions to this derivative NLS equation; such exact solutionsnare a rarity. These solutions are periodic in space and we determine the nonlinear dependence of the period onnthe amplitude.