Solitons of discrete curve shortening (Q511840)

For an infinite polygon \((x_{j})_{j \in \mathbb{Z}}\) given by the vertices \(x_{j}\) in the vector space \(\mathbb{R}^{n}\) the midpoints polygon is defined by \(M(x)\), where \(M(x)_{j} := \frac{1}{2}(x_{j}+x_{j+1})\), \(j \in \mathbb{Z}\). For a polygon \(x=(x_{j})_{ j\in \mathbb Z}\) in \(\mathbb{R}^{n}\) the authors consider the polygon \(T(x) = (T(x)_{j})_{j \in \mathbb{Z}} = ((M^{2}(x))_{j-1})_{j \in \mathbb{Z}}\) defined by the equation \(T(x)_{j} = \frac{1}{4}\{x_{j-1}+2x_{j}+x_{j+1}\}.\) For a closed polygon or a polygon with finite vertices this is a curve shortening process. They call a polygon \(x\) a ``soliton of the transformation \(T\)'' if the polygon \(T(x)\) is an affine image of \(x\). The authors describe a large class of solitons for transformations \(T\) 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 any \(s>0\), \(a \in \mathbb{R}\), the polygon \(x(a,s) = \biggl(\bigl(x_{j}(a,s)\bigr)_{j}\biggr)_{j \in \mathbb{Z}}\) with \( x_{j}(a,s) = c(a+sj)\), is a soliton of \(T\). They obtain for example solitons lying on spiral curves which, under the transformation \(T\), rotate and shrink.