Solitons of Curve Shortening Flow and Vortex Filament Equation

TL;DR

This paper investigates self-similar solutions of the Curve Shortening Flow and Vortex Filament Equation, highlighting their properties, conservation laws, and specific solutions like circles, with derivations of evolution equations in the Frenet frame.

Contribution

It provides a comprehensive analysis of self-similar solutions, including proofs of uniqueness for planar translating solutions and derivation of governing ODEs for vortex filaments.

Findings

Circles are the only planar translating self-similar solutions.

Derived evolution equations for Frenet frame vectors, curvature, and torsion.

Presented classifications and properties of self-similar solutions in both flows.

Abstract

In this paper we explore the nature of self-similar solutions of the Curve Shortening Flow and the Vortex Filament Equation, also known as the Binormal Flow. We explore some of their fundamental conservation properties and describe the behavior of their self-similar solutions. For Curve Shortening Flow we mainly expose the results of Huisken, Grayson, and Halldorsson concerning the equation's basic properties and self-similar solutions in the plane. For the Vortex Filament Equation we present the results by Banica and Vega, Arms and Hama, and Hasimoto. We also derive the evolution equations of the normal, binormal and tangent vectors in the Frenet frame for the vortex filament as well as those of curvature and torsion. We give a proof that circles are the only planar translating self-similar solutions and also derive a system of ordinary differential equations that govern the behavior for rotating