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arXiv · 1903.02105

On self-similar solutions of the vortex filament equation

O. Gamayun·O. Lisovyy
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Abstract

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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O. Gamayun, O. Lisovyy. 2019-03-05. On self-similar solutions of the vortex filament equation. https://doi.org/10.1063/1.5096170

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