ELSEVIER Physica D 87 (1995) 20-31 PHYSICA Stationary flows: Hamiltonian structures and canonical transformations Allan R Fordy Department of Applied Mathematical Studies and Centrefor Nonlinear Studies, University of Leeds, Leeds LS2 9JT, UK Abstract We consider the restriction of isospectral flows, Miura maps and Darboux transformations to stationary manifolds. Specifically, we present: (i) A systematic construction of Hamiltonian structures on stationary manifolds. (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. (iii) Some interesting connections between stationary flows of integrable nonfinear 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. 1. Introduction In this paper we consider the restriction of an integrable nonlinear evolution equation to its stationary manifold. We start, in Section 2, by constructing the Poisson brackets and Hamiltonians associated with the isospectral flows of a given spectral problem. By choosing the correct coordinates (and reversing the roles of x and t) these can be adapted to the stationary manifolds. These ideas are illustrated through the stationary KdV equation. The Darboux transformation is related to reversing the factorisation of the Schrfdinger operator. In Section 2 we consider the restriction of the Darboux transformation to the stationary manifolds of the MKdV hierarchy. This gives rise to a remarkable, explicit (and explicitly invertible) transformation, which preserves both Poisson brackets and all the commuting Hamiltonians in the hierarchy. The usual canonical coordinates of the stationary KdV hierarchy are associated with the first Hamiltonian structure B0 = a. If we use the 'squared eigenfunction' representation of the second Hamiltonian structure, then we obtain a different set of canonical coordinates, which in the case of the stationary 5th order KdV equation, transforms the equation onto an integrable case of the Hfnon-Heiles system. This connection (using a reverse procedure) was first presented in [8]. The squared eigenfunction procedure given in Section 4 can be used in conjunction with other non-canonical Poisson brackets to obtain interesting coordinate systems for related 0167-2789/95/$09.50 @ 1995 Elsevier Science B.V. All fights reserved SSDI 0 167 -2789(95)00134-4