Singularities of Type-Q ABS Equations

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.

contribution to the SIDE-9 special issue of SIGMA

References in corpus (3)

• Bäcklund transformation for the Krichever-Novikov equation
• Classification of 3D consistent quad-equations
• Cauchy problem for integrable discrete equations on quad-graphs

Cited by in corpus (5)

• Geometry of an elliptic-difference equation related to Q4
• On the singularity structure of the discrete KdV equation
• On decomposition of the ABS lattice equations and related Bäcklund transformations
• On the singularity structure of a discrete modified-Korteweg-deVries equation
• Classification of integrable boundary equations for integrable quad-graph systems