On the local systems hamiltonian in the weakly nonlocal Poisson brackets .

On the local systems hamiltonian in the weakly nonlocal Poisson brackets. Abstract We consider nonlocal field-theoretical Poisson brackets containing the operator of integration in the nonlocal part. The main attention is given to the nonlocal brackets of Hydrodynamic Type for which we introduce the Physical and Canonical forms. We use the Canonical form of these brackets for the investigation of a Poisson structure on the loop spaces defined by the corresponding pseudo-differential expression. Introduction. We study in this work the important class of nonlocal Poisson brackets which we call weakly nonlocal. They appeared recently in some investigations in the Soliton Theory. However there was no theory of such brackets except very special first order case. Even in this case the theory was not developed enough. In particular, we introduce the Physical forms and find all set of Casimirs, Momentum and Canonical forms for the most important