Motions of curves in the complex hyperbola and the Burgers hierarchy

Abstract

We study theory of curves in the complex hyperbola and show that special motions of curves are linked with the Burgers hierarchy, which also leads to a Hamiltonian formulation of the hierarchy and to the difference Burgers equation via their discretization. 1. Introduction It is widely recognized that a lot of differential equations in soliton theory arise from differential geometry especially theory of curves or surfaces ([1, 2, 3]). For example surfaces in the Euclidean 3-space with constant negative Gaussian or non-zero mean curvature are described by the sine or sinh Gordon equation respectively and proper affine spheres are described by the Tzitzeica equatio n. If we refer to theory of curves, the curvature of curves in the Euclidean 2-space evolves according to the mKdV equation under special motions. Pinkall ([4]) showed that the space of closed centroaffine curves in the centroaffine plane possesses a nat ural symplectic structure and the centroaffine curvature evolves according to the KdV e quation when the flow is generated by a Hamiltonian given by the total centroaffine curvature. Chou and Qu ([5, 6]) showed that many soliton equations arise from special motions of plane or space curves. Moreover, Hoffmann and Kutz ([7]) showed that special motions of curves in the complex 1 or 2-space or the complex projective line whose curvature evolves according to the mKdV or KdV equation can be discretized by use of cross ratios. In this paper we study theory of curves in the complex hyperbola which are determined by a certain curvature up to some symmetry. We shall show that special motions of curves are linked with the Burgers hierarchy, which can be formulated as a Hamiltonian system and also leads to the difference Burgers equation ([8]) via their discretization.