Higher-Dimensional Integrable Systems Induced by Motions of Curves in Affine Geometries

We discuss the motions of curves by introducing an extra spatial variable or equivalently, moving surfaces in affine geometries. It is shown that the 2+1-dimensional breaking soliton equation and a 2+1-dimensional nonlinear evolution equation regarded as a generalization to the 1+1-dimensional KdV equation arise from such motions.