Lattice Equations, Hierarchies and Hamiltonian Structures

Abstract

By now many integrable systems with soliton solutions are known in two dimensions, for fields u(ⁿ1,ⁿ2) defined at the sites (ⁿlⁿ2) of a two-dimensional (2D) lattice, for time-dependent fields u(n,t) defined at the sites n of a one-dimensional chain, and for fields u(x,t) depending on two continuous variables x and t. The relation between an integrable partial differential equation and its integrable discrete versions can be treated in the framework of the direct linearization method [1, 2] which is based on a linear integral equation with arbitrary measure and contour [3, 4]. In the treatment use is made of Bäcklund transformations (BT’s) which are generated by scalar multiplications of the free-wave function and/or measure in the integral equation [5]. The integrable lattice equations are obtained in the form of Bianchi identities expressing the commutativity of BT’s. The integrable equations with one or more continuous variables and their direct linearizations are obtained applying suitable continuum limits to the lattice equation and the integral equation at the same time.