Integrable Lattice Equations

Abstract

Integrable Lattice systems are space-and time-discretizations of integrable partial differential equations (PDE’s). The most convenient way of introducing them is based on the direct linearization method (DLM), which was introduced by Fokas and Ablowitz (1981),[1]. This method employs singular linear integral equations with general integration measure and contour. The integral equation introduced in [1] is of the form $${{u}_{k}} + {{\rho }_{k}}\int_{C} {d\lambda (\ell )\frac{{{{u}_{\ell }}}}{{k + \ell }} = {{\rho }_{k}}} .$$ (1.1) Here u k is a wave function to be solved from the integral equation depending on a complex spectral parameter k and on the coordinates of the system. As we shall note later, these coordinates can be chosen to be discrete as well as continuous, and it is the freedom in this choice that makes integral equations of the type (1.1) a convenient tool to develop discrete integrable systems. Furthermore, in eq. (1.1) C is a contour in the complex k-plane and dλ(k) is a suitably chosen integration measure, whereas ρk is a free-wave function depending in a given way on k and on the coordinates of the system. The contour C and measure dλ (k) need to be chosen to be such that the solution u k of the integral equation for gives ρ k is unique.