Hashimoto Surfaces and Smoke-Ring-Flow
Felix Knöppel
Media
Description
Here we see a certain time-continuous flow on discrete space curves, the smoke-ring-flow.
By a time-continuous flow of a closed discrete space curve $$\gamma\colon \mathbb{Z}/n \mathbb{Z} \rightarrow \mathbb{R}^3$$ we mean the continuous solution of an equation
\[\dot\gamma = X(\gamma)\,.\] Here $X$ is vector field on the space of discrete curves, i.e. a map that assigns to a curve $\gamma$ with $n$ vertices a sequence $$X\left(\gamma\right)\colon \mathbb{Z}/n \mathbb{Z}\rightarrow \mathbb{R}^3\,.$$
In the following we will shortly introduce a discrete analogue of the continuous vortex filament flow (or smoke-ring-flow) \[\dot\gamma = \gamma'\times \gamma''\,, \] where $(.)'$ denotes the derivative with respect to arclength. Given a closed discrete space curve with $n$ vertices, then for each edge we have an edge vector \[S_i = \gamma_{i+1}-\gamma_i\,.\] This provides a canonical (unit) tangent vector along each edge. Though at a vertex itself there is no obvious choice of a tangent vector.
The vortex filament flow is related to soliton theory. It describes how vortex filaments move and is related to the cubic non-linear Schrödinger equation - an equation well-known to soliton theorists.
To motivate the discrete vortex filament flow let us look at the continuous equation and assume for a second that we are dealing with a Frénét curve, i.e. $\gamma '' \neq 0$. Then with $$T = \gamma'\,,\quad N=\gamma''\,,\quad B = T\times N$$ we get $$\gamma'' = \kappa N$$ and the vortex filament flow becomes $$\dot\gamma = \kappa B\,.$$ The function $\kappa$ is the so called Frénét curvature of $\Gamma$ and equals the the inverse radius of the osculating circle.
Certainly a discrete curve is not a Frénét curve. Though unless $S_{i-1}$ and $S_i$ are parallel there is an osculating plane and a corresponding normal $B_i$ perpendicular to that plane. But what is a discrete Frénét curvature? Again there is no canonical choice. We will just stick here to the definition using vertex osculating circles, i.e. \[ \kappa_i = \frac{2\sin(\alpha_i)}{\left|S_{i-1}+S_i\right|} \] where $\alpha_i$ denotes the angle between $S_{i-1}$ and $S_i$. Thus we get $$\kappa_i B_i = \frac{2 S_{i-1}\times S_i}{\left|S_{i-1}\right|\left|S_{i-1}+S_i\right|\left|S_i\right|}\,.$$ The surface swept out by the vortex filament flow are called Hashimoto surfaces.
