Unidirectionalization of Hamiltonian Waves

We show how the restriction of certain bidirectional hamiltonian systems modelling nonlinear, one-dimensional wave propagation to waves moving in a single direction preserves the hamiltonian structure, even though the perturbation expansion of the bidirectional hamiltonian is not correct. A combination of the two approaches of direct hamiltonian perturbation theory and the method of multiple scales helps explain the apperance of integrable bihamiltonian wave models.