Solitons of Discrete Curve Shortening

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.