2020/10/06 by Bialy, Misha, Bor, Gil, Tabachnikov, Serge · 1 citation
#Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2010.02719
Twenty five years ago U. Pinkall discovered that the Korteweg-de Vries equation can be realized as an evolution of curves in centoraffine geometry. Since then, a number of authors interpreted various properties of KdV and its generalizations in terms of centoraffine geometry. In particular, the Bäcklund transformation of the Korteweg-de Vries equation can be viewed as a relation between centroaffine curves. Our paper concerns self-Bäcklund centroaffine curves. We describe general properties of these curves and provide a detailed description of them in terms of elliptic functions. Our work is a centroaffine counterpart to the study done by F. Wegner of a similar problem in Euclidean geometry, related to Ulam's problem of describing the (2-dimensional) bodies that float in equilibrium in all positions and to bicycle kinematics. We also consider a discretization of the problem where curves are replaced by polygons. This is related to discretization of KdV and the cross-ratio dynamics on ideal polygons.