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.
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