Formal diagonalisation of Darboux transformation and conservation laws of integrable PDEs, PD$\Delta $Es and P$\Delta $Es

Formal diagonalisation of Lax operators leads to formal diagonalisation of the corresponding Darboux transformations. The latter enables us to find recurrent relations for generating conservation laws and establish natural relations between the canonical series of local conservation laws for partial differential equations (PDEs), differential-difference equations ( PD $\Delta $Es) and partial difference equations (P$\Delta $Es), PD$\Delta $Es and P$\Delta $Es. In particular we show that the canonical densities of conservation laws for the symmetries of P$\Delta $Es are canonical densities of the partial difference equations themselves.