Motions of curves in similarity geometries and Burgers-mKdV hierarchies

Integrable systems satisfied by the curvatures of curves under inextensible motions in similarity geometries are identified. It is shown that motions of curves in two-, three- and n-dimensional (n>3) similarity geometries are described respectively by the Burgers hierarchy, Burgers-mKdV hierarchy and a multi-component generalizations of these hierarchies.