  In recognition of the highly non-trivial task of computation of the inverse
Hasimoto transformation mapping the intrinsic geometric parameter space onto
the extrinsic vortex filament coordinate space a reformulation of the Da
Rios-Betchov equations in the latter space is given. The nonlinear localized
vortex filament structure solution given by the present formulation is in
detailed agreement with the Betchov-Hasimoto solution in the small-amplitude
limit and is also in qualitative agreement with laboratory experiment
observations of helical-twist solitary waves propagating on concentrated
vortices in rotating fluids. The present formulation also provides for a
discernible effect of the slipping motion of a vortex filament on the vortex
evolution.
