Orbital Stability for Rotating Planar Vortex Filaments in the Cartesian and Arclength Forms of the Local Induction Approximation

Abstract

The local induction approximation (LIA) is commonly used to study the motion of a vortex filament in a fluid. The fully nonlinear form of the LIA is equivalent to a type of derivative nonlinear Schrödinger (NLS) equation, and stationary solutions of this equation correspond to rotating planar vortex filaments. Such solutions were first discussed in the plane by Hasimoto [J. Phys. Soc. Jpn. 31 (1971) 293], and have been described both in Cartesian three-space and in the arclength formulation in subsequent works. Despite their interest, fully analytical stability results have been elusive. In the present paper, we present elegant and simple proofs of the orbital stability for the stationary solutions to the derivative nonlinear Schrödinger equations governing the self-induced motion of a vortex filament under the LIA, in both the extrinsic (Cartesian) and intrinsic (arclength) coordinate representations. Such results constitute an exact criterion for the orbital stability of rotating planar vortex filament solutions for the vortex filament problem under the LIA.