Self-B\"acklund curves in centroaffine geometry and Lam\'e's equation

TL;DR

This paper explores self-Bäcklund curves in centroaffine geometry, linking them to elliptic functions, discretizations, and KdV equation properties, extending classical Euclidean results to a centroaffine setting.

Contribution

It provides a detailed analysis of self-Bäcklund centroaffine curves, including their elliptic function descriptions and discretizations, connecting geometric properties with integrable systems.

Findings

Characterization of self-Bäcklund centroaffine curves using elliptic functions

Discretization of curves relates to KdV and cross-ratio dynamics

Extension of Euclidean geometry results to centroaffine geometry

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 B\"acklund transformation of the Korteweg-de Vries equation can be viewed as a relation between centroaffine curves. Our paper concerns self-B\"acklund 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.