On the local systems Hamiltonian in the weakly non-local Poisson brackets

We study in this work the important class of non-local Poisson brackets (PB) which we call weakly non-local. They appeared recently in some investigations in soliton theory. However there was no theory of such brackets except for very special first-order case. Even in this case the theory was not developed enough. In particular, we introduce the physical forms and find Casimirs, momentum and canonical forms for the most important hydrodynamic type PB of that kind and their dependence on the boundary conditions.