Stability of solitons on vortex filaments

► We consider the soliton solutions of the Vortex Filament Equation. ► A method to solve linearized VFE associated to the soliton solutions is presented. ► The method relies on the fact that the linearized Nonlinear Schrödinger Equation can be solved completely for soliton solutions. ► We show the spectral stability of the soliton solutions. ► The same analysis is performed for the soliton solutions of the Planar Filament Equation.