Solitons of Discrete Curve Shortening

Abstract

For a polygon ${x=(x_{j})_{j \in \mathbb{z}}}$ in ${\mathbb{R}^{n}}$ we consider the polygon ${(T(x))_j=\left\{x_{j-1}+2x_j+x_{j+1}\right\}/4.}$ This transformation is obtained by applying the midpoints polygon construction twice. For a closed polygon or a polygon with finite vertices this is a curve shortening process. We call a polygon x a soliton of the transformation T if the polygon T(x) is an affine image of x. We describe a large class of solitons for the transformation T by considering smooth curves c which are solutions of the differential equation ${\ddot{c}(t)=Bc(t)+d}$ for a real matrix B and a vector d. The solutions of this differential equation can be written in terms of power series in the matrix B. For a solution c and for any ${s > 0,a\in \mathbb{R}}$ the polygon ${x(a,s)=(x_j(a,s)_j)_{j \in \mathbb{z}}; x_j(a,s)=c(a+sj)}$ is a soliton of T. For example we obtain solitons lying on spiral curves which under the transformation T rotate and shrink.