Squared eigenfunctions and linear stability properties of closed vortex filaments

Abstract

We develop a general framework for studying the linear stability of closed solutions of the vortex filament equation (VFE), based on the correspondence between the VFE and the nonlinear Schrödinger (NLS) equation provided by the Hasimoto map, and on the construction of solutions of the linearized equations in terms of NLS squared eigenfunctions. In particular, we show that the differential of the Hasimoto map is a one-to-one correspondence between curve variations and perturbations of NLS potentials induced by squared eigenfunctions. We apply this framework to vortex filaments associated with periodic finite-gap NLS potentials in the genus one case, and for cnoidal potentials we characterize the stability of the associated filaments in terms of their knot type.