  We consider a linear second order parabolic system with a third order
dispersion term. This type of system arises when considering a nonlinear model
equation describing the motion of a vortex filament with axial flow immersed in
an incompressible and inviscid fluid. We prove the solvability of an
initial-boundary value problem of the parabolic-dispersive system which allows
application to the motion of a vortex filament. To do so, we propose a new
regularization technique by adding a space-time derivative term.
