Lattice equations, hierarchies and Hamiltonian structures

Abstract

By applying a suitable continuum limit to a lattice version of the Korteweg-de Vries (KdV) equation we obtain an infinite hierarchy of integrable partial difference equations for time-dependent fields, defined on the sites of a one-dimensional chain, together with the associated Hamiltonian structure. A second continuum limit applied to any member of the hierarchy yields the well-known hierarchy of higher-order KdV equations with the recursion operator, as well as the Hamiltonian structure. A similar procedure is worked out for the hierarchies related to the modified Korteweg-de Vries (mKdV) equation, starting from a lattice version of the mKdV. For all the equations a direct linearization is given of an integral equation with arbitrary measure and contour.