On self-similar solutions of the vortex filament equation

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 $\sigma$-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.

15 pages, 1 figure

References in corpus (1)

• Domain wall dynamics in the Landau--Lifshitz magnet and the classical-quantum correspondence for spin transport

Cited by in corpus (3)

• Superdiffusion in spin chains
• Recent results for the Landau-Lifshitz equation
• A note on N-soliton solutions for the viscid incompressible Navier-Stokes differential equation