Abstract
It is known that the Schrödinger flow on a complex Grassmann manifold is equivalent to the matrix non-linear Schrödinger equation and the Ferapontov flow on a principal Adjoint U(n)-orbit is equivalent to the n-wave equation. In this paper, we give a systematic method to construct integrable geometric curve flows on Adjoint U-orbits from flows in the soliton hierarchy associated to a compact Lie group U. There are natural geometric bi-Hamiltonian structures on the space of curves on Adjoint orbits, and they correspond to the order two and three Hamiltonian structures on soliton equations under our construction. We study the Hamiltonian theory of these geometric curve flows and also give several explicit examples.
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Dedicated to Professor S. S. Chern on his 90th Birthday
Research supported in part by NSF Grant DMS 9972172 and Humboldt Senior Scientist Award.
Research supported in part by the Deutsche Forschungsgemeinschaft.
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Terng, CL., Thorbergsson, G. Completely integrable curve flows on Adjoint orbits. Results. Math. 40, 286–309 (2001). https://doi.org/10.1007/BF03322713
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DOI: https://doi.org/10.1007/BF03322713