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 centroaffine geometry. Since then, a number of authors interpreted various properties of KdV and its generalizations in terms of centroaffine 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.

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 [5,14].…”

“…5 , s 7 , s 3 ), v 2 s 1 s 3 − s 5 s 7 s 1 s 5 − s 3 s 7 , v 4 s 1 s 7 − s 3 s 5 s 1 s 3 − s 5 s 7 .Then after the second recutting we have(s 1 , s 7 , s 3 , s 5 ), v 2 s 1 s 3 − s 5 s 7 s 1 s 7 − s 3 s 5 , v 4 s 1 s 7 − s 3 s 5 s 1 s 5 − s 3 s 7 ,In this section we consider the moduli space X 5,S . This material is parallel to the one in[4, Section 7.1.3].The space X 5,S is 2-dimensional: the variables v 2i satisfy the Ptolemy-Plücker relation v 2 v 4 + v 8 s 3 = s 1 s 5 and its cyclic permutations.…”