Integrability of invariant metrics on the diffeomorphism group of the circle

Adrian Constantin (1) , Boris Kolev (2)

1 Department of Mathematics [Lund University]
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités

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 X₀ and X₁ are bi-Hamiltonian relatively to a modified Lie-Poisson structure.

Mots clés: Bi-Hamiltonian formalism; Diffeomorphisms group of the circle; Camassa-Holm equation

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]

Adrian Constantin, Boris Kolev. Integrability of invariant metrics on the diffeomorphism group of the circle. Journal of Nonlinear Science, 2006, 16, pp.109-122. ⟨10.1007/s00332-005-0707-4⟩. ⟨hal-00095916⟩