Singular-boundary reductions of type-Q ABS equations

TL;DR

This paper investigates singular boundary problems for discrete elliptic integrable models, introducing a novel solution technique based on multidimensional consistency and singularity analysis, with applications to Toda systems and periodic solutions.

Contribution

It develops a new method for solving singular boundary problems in type-Q ABS equations using multidimensional consistency and tau-functions, advancing the understanding of their singularity structure.

Findings

Constructed exact solutions on singularity-bounded strips.

Linked solutions to Toda-type systems with open boundary conditions.

Provided insights into the singularity structure in multidimensions.

Abstract

We study the fully discrete elliptic integrable model Q4 and its immediate trigonometric 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 multidimensional 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 associated Toda-type systems, and have interesting application to the construction of periodic solutions.