title: Exact solution for the self-induced motion of a vortex filament in the arc-length representation of the local induction approximation.
abstract: We review two formulations of the fully nonlinear local induction equation approximating the self-induced motion of the vortex filament (in the local induction approximation), corresponding to the Cartesian and arc-length coordinate systems. The arc-length representation put forth by Umeki [Theor. Comput. Fluid Dyn. 24, 383 (2010)] results in a type of 1+1 derivative nonlinear Schrödinger (NLS) equation describing the motion of such a vortex filament. We obtain exact stationary solutions to this derivative NLS equation; such exact solutions are a rarity. These solutions are periodic in space and we determine the nonlinear dependence of the period on the amplitude.
authors.affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816-1364, USA. rav@knights.ucf.edu
authors.forename: Robert A
authors.initials: RA
authors.lastname: Van Gorder
mesh_terms: D000465:Algorithms; D003198:Computer Simulation; D008962:Models, Theoretical; D012212:Rheology / Q000379:methods*