Classification problems for Hamiltonian evolutionary equations and their discretizations

In the present Thesis, we deal mainly with the subclass of Hamiltonian evolutionary PDEs and their Poisson brackets (PBs). We address the problem of classifying discrete differential-geometric PBs (dDGPBs) of any fixed order on target space of dimension 1 and describing their compatible pencils. Furthermore, we explain a new criterium about the existence of tri-Hamiltonian structures for nonlinear wave systems.