Solitons of Discrete Curve Shortening

For a polygon x = ( x j ) j ∈ z in R n we consider the polygon ( T ( x ) ) j = x j - 1 + 2 x j + x j + 1 / 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 c ¨ ( t ) = B c ( 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 ∈ R the polygon x ( a , s ) = ( x j ( a , s ) j ) j ∈ z ; x j ( a , s ) = c ( a + s j ) is a soliton of T . For example we obtain solitons lying on spiral curves which under the transformation T rotate and shrink.