# On self-similar solutions of the vortex filament equation

## TL;DR

This paper analyzes self-similar solutions of the vortex filament equation, linking their dynamics to Painlevé IV equations, and provides explicit formulas for solutions with corner initial conditions.

## Contribution

It establishes a connection between vortex filament self-similar solutions and Painlevé IV transcendents, offering a complete asymptotic characterization and explicit hypergeometric solutions.

## Key findings

- Characterization of asymptotic properties of curvature and torsion.
- Explicit hypergeometric solutions for corner initial conditions.
- Connection between vortex filament evolution and Painlevé IV equations.

## 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.