Hamiltonian Structures in Stationary Manifold Coordinates

We consider the restriction of isospectral flows to stationary manifolds. Specifically, we present a systematic construction of Hamiltonian structures written in stationary manifold coordinates, which demonstrates the close relationship between the Hamiltonian formulations of nonlinear evolution equation (PDE) and its stationary reduction. We illustrate these ideas in the context of the KdV and 5th order KdV equations.We then apply these ideas to the Boussinesq hierarchy, associated with the (trace free) 3rd order Lax operator, together with the Sawada-Kotera and Kaup-Kupershmidt reductions.We use our results to study the integrable cases of the Hénon Heiles equation.KeywordsPoisson BracketNonlinear Evolution EquationHamiltonian StructureMKdV EquationStationary ManifoldThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.