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

This paper is a survey article on bi-Hamiltonian systems on the dual of the Lie algebra of vector fields on the circle. Here, 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 for shallow water waves.