Klein Curvilinear Motion Of The Integrable Geometry

As well-known, the motions of curve have a wide range of application. Many interesting integrable equations naturally arise from the motions of curves. In this paper, motions of non-stretching curves in some Klein geometries, determined by the transformation groups acting on S~1×R, are studied.In Chapter 1, we provide some background material and notions of invariant curve flow and integrable equations. A classification of Lie groups or Lie algebra of vector fields acting on S~1×Ris listed. The Arc-length and Curvature of a curve in every Klein geometry are also given.In Chapter 2, we discuss motion of inextensible curves in these Klein geometries. It is shown that several 1+1-dimensional integrable equations including the KdV, mKdV, defocusing mKdV, Sawada-Kotera, Burgers equations and their hierarchies arise naturally from motions of non-stretching curves in such geometries.