Remarks on KdV-type Flows on Star-Shaped Curves
arXiv:0808.3593 · doi:10.1016/j.physd.2009.01.007
Abstract
We study the relation between the centro-affine geometry of star-shaped planar curves and the projective geometry of parametrized maps into $\RP^1$. We show that projectivization induces a map between differential invariants and a bi-Poisson map between Hamiltonian structures. We also show that a Hamiltonian evolution equation for closed star-shaped planar curves, discovered by Pinkall, has the Schwarzian KdV equation as its projectivization. (For both flows, the curvature evolves by the KdV equation.) Using algebro-geometric methods and the relation of group-based moving frames to AKNS-type representations, we construct examples of closed solutions of Pinkall's flow associated with periodic finite-gap KdV potentials.
21 pages, 5 figures; revised for publication
References in corpus (1)
Cited by in corpus (7)
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- Hamiltonian flows on null curves
- Integrable Flows for Starlike Curves in Centroaffine Space
- Motions of Curves in the Projective Plane Inducing the Kaup-Kupershmidt Hierarchy
- Multi-Hamiltonian Structures on Spaces of Closed Equicentroaffine Plane Curves Associated to Higher KdV Flows
- Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems
- mKdV-Related Flows for Legendrian Curves in the Pseudohermitian 3-Sphere