  We have constructed one-phase quasi-periodic solutions of the curve equation
induced by the mKdV equation. The solution is expressed in terms of the
elliptic functions of Weierstrass. This solution can describe curve dynamics
such as a vortex filament with axial velocity embedded in an incompressible
inviscid fluid. There exist two types of curves (type-A, type-B) according to
the form of the main spectra of the finite-band integrated solution. Our
solution includes various filament shapes such as the Kelvin-type wave, the
rigid vortex, plane curves, closed curves, and the Hasimoto one-solitonic
filament.
