Double Spiral Singularities for a Flow of Regular Planar Curves

In this paper, we study the singularity formation for the geometric flow of complex curves $z_t = -z_{xxx} + \frac{3}{2}\overline z_{x} z_{xx}^2,$ which was derived in [R. E. Goldstein and D. M. Petrich, 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 Painlevé II transcendents, and furthermore we establish its asymptotic expansion in the vicinity of the singularity.