Singularities of Type-Q ABS Equations

TL;DR

This paper investigates the nature of singularities in type-Q ABS equations, revealing geometric criteria for their occurrence and showing that singularities can disrupt the global integrability despite local consistency.

Contribution

It introduces geometric criteria for singularities in type-Q ABS equations and demonstrates how they can cause breakdowns in global consistency, extending previous local analyses.

Findings

Singularities are characterized by simple geometric criteria.

Presence of singularities can lead to failure of global consistency.

Monodromy quantifies the breakdown of global integrability.

Abstract

The type-Q equations lie on the top level of the hierarchy introduced by Adler, Bobenko and Suris (ABS) in their classification of discrete counterparts of KdV-type integrable partial differential equations. We ask what singularities are possible in the solutions of these equations, and examine the relationship between the singularities and the principal integrability feature of multidimensional consistency. These questions are considered in the global setting and therefore extend previous considerations of singularities which have been local. What emerges are some simple geometric criteria that determine the allowed singularities, and the interesting discovery that generically the presence of singularities leads to a breakdown in the global consistency of such systems despite their local consistency property. This failure to be globally consistent is quantified by introducing a natural notion of monodromy for isolated singularities.