A vortex filament moving without change of form

Abstract

The motion of a very thin vortex filament is investigated using the localized induction equation. A family of vortex filaments which move without change of form are obtained. They are expressed in terms of elliptic integrals of the first, second and third kinds. In general they do not close and have infinite lengths. In some particular cases they take the form of closed coils which wind a doughnut. There exist a family of closed vortex filaments which do not travel in space but only rotate around a fixed axis. Our solutions include various well-known shapes such as the circular vortex ring, the helicoidal filament, the plane sinusoidal filament, Euler's elastica and the solitary-wave-type filament. It is shown that they correspond to the travelling wave solution of a nonlinear Schrödinger equation.