The Korteweg–de Vries, modified Korteweg–de Vries, and Harry Dym hierarchies of integrable systems are shown to be equivalent to a hierarchy of chiral shape dynamics of closed curves in the plane. These purely local dynamics conserve an infinite number of global geometric properties of the curves, such as perimeter and enclosed area. Several techniques used to study these integrable systems are shown to have simple differential-geometric interpretations. Connections with incompressible, inviscid fluid flow in two dimensions are suggested.

• Received 9 September 1991

DOI:https://doi.org/10.1103/PhysRevLett.67.3203