Hamiltonian flows on stationary manifolds (Q1381218)

It is known that ordinary differential equations (ODEs), which are obtained as stationary versions of partial differential equations (PDEs) that admit a representation in the form of a Lax pair and, hence, are integrable by means of the inverse scattering transform, are also integrable, and they admit a Hamiltonian representation. However, it is not obvious how to derive the Hamiltonian and symplectic structure (Poisson brackets) for these ODEs from the underlying PDE Lax pair. This is the subject of the present work. The authors demonstrate that the ODE Hamiltonian structure can be derived by means of reformulating the Lax representation in terms of the reversed variables \(x\) and \(t\). This allows one to construct a Hamiltonian structure of the PDE in which \(x\) is the evolutional variable, and, finally, the reduction to the stationary manifold directly leads to the ODE Hamiltonian structure sought for. In many cases, a bi-Hamiltonian or tri-Hamiltonian structure can be obtained for the integrable flows on the stationary manifolds of the integrable PDEs. For illustration, the Korteweg-de Vries hierarchy of integrable PDEs and their stationary reductions are used. Coupled systems of integrable equations are considered too.