Pre-Hamiltonian structures for integrable nonlinear systems

Abstract:
We consider pre-Hamiltonian operators in total derivatives; they are defined by the property that their images in the Lie algebras of evolutionary vector fields are closed with respect to the commutation. These operators generalize the Hamiltonian formalism for PDE and are close to the anchors of Lie algebroids over infinite jet spaces. We assign a class of pre-Hamiltonian operators and the Lie-type brackets in their inverse images to integrable KdV-type hierarchies of symmetry flows on hyperbolic Euler-Lagrange systems of the Liouville type (e.g., the 2D Toda lattices associated with the semi-simple Lie algebras).

Joint work with Johan van de Leur (Utrecht University).