On the equivalence of the discrete nonlinear Schr\"odinger equation and the discrete isotropic Heisenberg magnet

Abstract

The equivalence of the discrete isotropic Heisenberg magnet (IHM) model and the discrete nonlinear Schr\"odinger equation (NLSE) given by Ablowitz and Ladik is shown. This is used to derive the equivalence of their discretization with the one by Izergin and Korepin. Moreover a doubly discrete IHM is presented that is equivalent to Ablowitz' and Ladiks doubly discrete NLSE.