Hamiltonian flows on the space of star-shaped curves
- 34 Downloads
We provide a geometric interpretation of the KdV equation as an evolution equation on the space of closed curves in the centroaffine plane. There is a natural symplectic structure on this space and the KdV-flow is generated by a Hamiltonian given by the total centroaffine curvature. In this way we obtain another example for a soliton equation coming naturally from a differential geometric problem . Furthermore, we present a group action of the diffeomorphism group of the circle on the space of closed centroaffine curves.
1991 Mathematics Subject Classification35Q53 53A15
Key words and phraseKdV equation affine curves
Unable to display preview. Download preview PDF.
- Bobenko, A., Surfaces in terms of 2 by 2 matrices. Old and new integrable cases, In: Fordy A., Wood J. (eds) “Harmonic Maps and Integrable Systems”, Vieweg (1994)Google Scholar
- Gardner C.S., Greene J.M., Kruskal M.D., Miura R.M., Method for solving the Korteweg — de Vries equation. — Phys. Rev. Lett., 1967, 1095-1097.Google Scholar