Stationary flows: Hamiltonian structures and canonical transformations

Abstract

Abstract We consider the restriction of isospectral flows, Miura maps and Darboux transformations to stationary manifolds. Specifically, we present: 1. (i) A systematic construction of Hamiltonian structures on stationary manifolds. 2. (ii) The reduction of Darboux transformations to stationary manifolds. Using the relationship between the factorisation of the Schrodinger operator and the MKdV hierarchy, we derive interesting canonical transformations which preserve both Hamiltonian structures and all commuting Hamiltonians for the stationary flows. 3. (iii) Some interesting connections between stationary flows of integrable nonlinear evolution equations and integrable Hamiltonian systems of natural type. The corresponding Miura maps give rise to canonical transformations between various of these integrable Hamiltonian systems.