On Backlund transformation and vortex filament equation for null Cartan curve in Minkowski 3-space

In this paper we introduce Backlund transformation of a null Cartan curve in Minkowski 3-space as a transformation which maps a null Cartan helix to another null Cartan helix, congruent to the given one. We also give the sufficient conditions for a transformation between two null Cartan curves in the Minkowski 3-space such that these curves have equal constant torsions. By using the Da Rios vortex filament equation, based on localized induction approximation, we derive the vortex filament equation for a null Cartan curve and obtain evolution equation for it's torsion. As an application, we show that Cartan's frame vectors generate new solutions of the Da Rios vortex filament equation.