  We study self-similar solutions of the binormal curvature flow which governs
the evolution of vortex filaments and is equivalent to the Landau-Lifshitz
equation. The corresponding dynamics is described by the real solutions of
$\sigma$-Painlev\'{e} IV equation with two real parameters. Connection formulae
for Painlev\'{e} IV transcendents allow for a complete characterization of the
asymptotic properties of the curvature and torsion of the filament. We also
provide compact hypergeometric expressions for self-similar solutions
corresponding to corner initial conditions.
