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 SIGMA, 2011, Volume 7, 073, 14 pp. (Mi sigma631)

Singularities of Type-Q ABS Equations

James Atkinson

School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia

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.

Keywords: singularities; integrable systems; difference equations; multidimensional consistency

DOI: https://doi.org/10.3842/SIGMA.2011.073

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Bibliographic databases:

ArXiv: 1102.2675
MSC: 35Q58
Received: February 14, 2011; in final form July 13, 2011; Published online July 20, 2011
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Citation: James Atkinson, “Singularities of Type-Q ABS Equations”, SIGMA, 7 (2011), 073, 14 pp.

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Atkinson J., Joshi N., “Singular-Boundary Reductions of Type-Q Abs Equations”, Int. Math. Res. Notices, 2013, no. 7, 1451–1481
2. Ormerod Ch.M., “Tropical Geometric Interpretation of Ultradiscrete Singularity Confinement”, J. Phys. A-Math. Theor., 46:30 (2013), 305204
3. van der Kamp P.H., “Initial Value Problems For Quad Equations”, J. Phys. A-Math. Theor., 48:6 (2015), 065204
4. Atkinson J., Howes Ph., Joshi N., Nakazono N., “Geometry of an elliptic difference equation related to Q4”, J. Lond. Math. Soc.-Second Ser., 93:3 (2016), 763–784
5. Zhang D., Zhang D.-j., “On Decomposition of the Abs Lattice Equations and Related Backlund Transformations”, J. Nonlinear Math. Phys., 25:1 (2018), 34–53
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