Lattice equations, hierarchies and Hamiltonian structures

We investigate the continuum limit properties of an integrable lattice version of the Kadomtsev-Petviashvili (KP) equation. By applying continuum limits with vertex operators involving an infinite number of continous variables to the lattice KP we obtain hierarchies of integrable equations for fields depending on one continous variable at the sites of a two-dimensional lattice. The direct linearization of the hierarchies is obtained applying the same limit to the free-wave function in the integral equation for lattice KP.