Integrability of invariant metrics on the diffeomorphism group of the circle

Each H^k Sobolev inner product defines a Hamiltonian vector field X_k on the regular dual of the Lie algebra of the diffeomorphism group of the circle. We show that only X_0 and X_1 are bi-Hamiltonian relatively to a modified Lie-Poisson structure

Least action principle for an integrable shallow water equation

For an integrable shallow water equation we describe a geometrical approach showing that any two nearby fluid configurations are successive states of a unique flow minimizing the kinetic energy.Comment: arXiv version is already officia

Poisson structure and Action-Angle variables for the Camassa-Holm equation

The Poisson brackets for the scattering data of the Camassa-Holm equation are computed. Consequently, the action-angle variables are expressed in terms of the scattering data.Comment: 20 pages, LaTeX. The original publication is available at www.springerlink.co