Abstract

We study the fully discrete elliptic integrable model Q4 and its immediate trigonomet-ric and rational counterparts (Q3, Q2 and Q1). Singular boundary problems for these equations are systematised in the framework of global singularity analysis. We introduce a technique to obtain solutions of such problems, in particular constructing the exact solution on a regular singularity-bounded strip. The solution technique is based on the multidimen-sional consistency and uses new insights into these equations related to the singularity structure in multidimensions and the identification of an associated tau-function. The obtained special solutions can be identified with open boundary problems of the associ-ated Toda-type systems, and have interesting application to the construction of periodic solutions. 1