Vortex Filament Equation for a Regular Polygon

Abstract

In this paper, we study the evolution of the vortex filament equation (VFE), $${\mathbf{X}}_t={\mathbf{X}}_s\wedge {\mathbf{X}}_{ss},$$ with $\mathbf{X}(s,0)$ being a regular planar polygon. Using algebraic techniques, supported by full numerical simulations, we give strong evidence that $\mathbf{X}(s,t)$ is also a polygon at any rational time; moreover, it can be fully characterized, up to a rigid movement, by a generalized quadratic Gauß sum. We also study the fractal behavior of $\mathbf{X}(0,t)$, relating it with the so-called Riemann's non-differentiable function, that was proved by Jaffard to be a multifractal.

31 pages, 15 figures (27 pages in the final version of Nonlinearity)

References in corpus (2)

• Stability of the selfsimilar dynamics of a vortex filament
• Talbot effect for the cubic nonlinear Schrödinger equation on the torus

Cited by in corpus (14)

• On the vortex filament conjecture for Euler flows
• Domain wall problem in the quantum XXZ chain and semiclassical behavior close to the isotropic point
• Riemann's non-differentiable function and the binormal curvature flow
• On the Relationship between the One-Corner Problem and the $M$-Corner Problem for the Vortex Filament Equation
• Geometric differentiability of Riemann's non-differentiable function
• Convergence over fractals for the periodic Schrödinger equation
• About the quantum Talbot effect on the sphere
• New Revival Phenomena for Bidirectional Dispersive Hyperbolic Equations
• Some geometric properties of Riemann's non-differentiable function
• The vortex filament equation as a pseudorandom generator
• Intermittency of Riemann's non-differentiable function through the fourth-order flatness
• The Talbot effect as the fundamental solution to the free Schrödinger equation
• Multifractality and intermittency in the limit evolution of polygonal vortex filaments
• An analytical study of flatness and intermittency through Riemann's non-differentiable functions