math.DG/0304245 HAMILTONIAN OPERATORS AND `¤-COVERINGS

Abstract. An e±cient method to construct Hamiltonian structures for nonlinear evolution equations is described. It is based on the notions of variational Schouten bracket and `¤-covering. The latter serves the role of the cotangent bundle in the category of nonlinear evolution PDEs. We ¯rst consider two illustrative examples (the KdV equation and the Boussinesq system) and reconstruct for them the known Hamiltonian structures by our methods. For the coupled KdV-mKdV system, a new Hamiltonian structure is found and its uniqueness (in the class of polynomial (x; t)-independent structures) is proved. We also construct a nonlocal Hamiltonian structure for this system and prove its compatibility with the local one