In the previous chapters we have discussed in detail the various manifestations of the Yang-Baxter equation for both discrete and continuous integrable systems. Discrete integrable systems have a close analogy with vertex models of statistical mechanics in two dimensions, which were studied extensively by Baxter [64], Onsager, Mattis, Lieb [32] and others. Our focus here is on the applications of the Yang Baxter equation in the ananlysis of specific nonlinear models. It should be borne in mind that the motivation of solution in classical and quantum problems is quite different. While in classical IST one is interested in the form of the solutions, that is, in the structure of the solitary wave, in the quantum case one searches for the excitation spectrum, the Bethe eigenmomenta equation and the energy eigenvalues, besides being interested in the structure of the string states. Since there are certain subtle problems associated with the continuous Lax operators we begin with discrete models. A model that has received quite a lot of attention in recent years and is structurally simple is the discrete self-trapping (DST) model, which we analyse below.