Department of Mathematics, Northwest University
Xi'an, 710069, P. R. China
e-mail: qu_changzheng@hotmail.com

Integrable equations arise from the motion of curves in Klein Geometry

Abstract:
The motion of curves in Klein geometry is studied. It is shown that many integrable equations such as KdV, Harry Dym, Sawada-Kotera, Burgers hierarchies and the Kaup-Kupershmidt, Boussinesq, Tzitzeica, Hirota-Satsuma equations naturally arise from the motions of plane or space curves in Centro-affine, special affine, similarity geometries. These local and nonlocal dynamics conserve global geometric quantities of curves such as perimetet and area. Curves and their motions corresponding to the taveling waves of the KdV, Burgers and Sawada-Kotera equations respectively in centro-affine, similarity and special affine geometries are discussed in detail.