Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations

Boris Kolev (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités

This paper is a survey article on bi-Hamiltonian systems on the dual of the Lie algebra of vector fields on the circle. We investigate the special case where one of the structures is the canonical Lie-Poisson structure and the second one is constant. These structures called affine or modified Lie-Poisson structures are involved in the integrability of certain Euler equations that arise as models of shallow water waves.

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

Domaines: Physique mathématique [math-ph]

Boris Kolev. Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations. Philosophical Transactions. Series A, Mathematical, Physical and Engineering Sciences, 2007, 365 (1858), pp. 2333-2357. ⟨10.1098/rsta.2007.2012⟩. ⟨hal-00102390⟩