Double spiral singularities for a flow of regular planar curves

TL;DR

This paper investigates the formation of finite-time singularities in a geometric flow of planar curves, specifically those composed of two rotating logarithmic spirals, using Painlevé II transcendents for exact solutions.

Contribution

It provides the first exact construction of singular solutions for this flow, linking them to Painlevé II transcendents and analyzing their asymptotic behavior near singularities.

Findings

Finite-time singularities occur for curves made of two rotating logarithmic spirals.

Exact solutions are constructed using Painlevé II transcendents.

Asymptotic expansions describe the behavior near singularities.

Abstract

In this paper we study the singularity formation for the geometric flow of complex curves $$z_t = -z_{xxx} + \frac{3}{2}\o z_{x} z_{xx}^2,$$ that was derived [R. E. Goldstein and D. M. Petrich, {\em Phys. Rev. Lett.}, 69 (1992), pp. 555--558] while considering the vortex patch dynamics for the incompressible 2D Euler equation. We prove that arbitrary curve, consisting of two rotating logarithmic spirals, is a finite time singularity developed by a smooth solution of the flow. We provide exact construction of the solution in the terms of appropriate Painlev\'e II transcendents and furthermore we establish its asymptotic expansion in the vicinity of the singularity.