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 the σ-Painlevé IV equation with two real parameters. Connection formulae for Painlevé 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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Research Article| August 27 2019
On self-similar solutions of the vortex filament equation
Special Collection: XIXth International Congress on Mathematical Physics
O. Gamayun, O. Lisovyy; On self-similar solutions of the vortex filament equation. J. Math. Phys. 1 August 2019; 60 (8): 083510. https://doi.org/10.1063/1.5096170
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