Self-Bäcklund curves in centroaffine geometry and Lamé's equation

Abstract

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.

Symplectically convex and symplectically star-shaped curves -- a variational problem

“…This introduction would not be complete if we failed to mention another reason for our interest in centroaffine differential geometry, namely its close relation with the Korteweg-de Vries equation, discovered by U. Pinkall [9] and studied by a number of authors since then. When the exponent a equals 2, the extremal curves are periodic solutions to Lamé's equation thoroughly studied in this context in a recent paper [4].…”

A family of integrable transformations of centroaffine polygons: geometrical aspects

“…The c-relation is a discretization of a relation on centroaffine curves, which is a geometrical realization of the Bäcklund transformation of the KdV equation studied in [5,16]. Two consecutive pairs of vertices of c-related polygons form a quadrilateral satisfying…”