Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2008-08-26
Physica D, vol. 238 #8 (2009), pp.788-797
Nonlinear Sciences
Exactly Solvable and Integrable Systems
21 pages, 5 figures; revised for publication
Scientific paper
10.1016/j.physd.2009.01.007
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.
Beffa Gloria Mari
Calini Annalisa
Ivey Thomas
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