Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 073, 14 pages      arXiv:1102.2675      http://dx.doi.org/10.3842/SIGMA.2011.073
Contribution to the Proceedings of the Conference “Symmetries and Integrability of Difference Equations (SIDE-9)”

Singularities of Type-Q ABS Equations

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

Received February 14, 2011, in final form July 13, 2011; Published online July 20, 2011

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.

Key words: singularities; integrable systems; difference equations; multidimensional consistency.

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References

  1. Adler V.E., Bobenko A.I., Suris Yu.B., Classification of integrable equations on quad-graphs. The consistency approach, Comm. Math. Phys. 233 (2003), 513-543, nlin.SI/0202024.
  2. Nijhoff F.W., Walker A.J., The discrete and continuous Painlevé VI hierarchy and the Garnier systems, Glasg. Math. J. 43A (2001), 109-123, nlin.SI/0001054.
  3. Bobenko A.I., Suris Yu.B., Integrable systems on quad-graphs, Int. Math. Res. Not. 2002 (2002), no. 11, 573-611, nlin.SI/0110004.
  4. Atkinson J., Hietarinta J., Nijhoff F.W., Seed and soliton solutions for Adler's lattice equation, J. Phys. A: Math. Theor. 40 (2007), F1-F8, nlin.SI/0609044.
  5. Adler V.E., Bobenko A.I., Suris Yu.B., Discrete nonlinear hyperbolic equations: classification of integrable cases, Funct. Anal. Appl. 43 (2009), 3-17, arXiv:0705.1663.
  6. Adler V.E., Bäcklund transformation for the Krichever-Novikov equation, Int. Math. Res. Not. 1998 (1998), no. 1, 1-4, solv-int/9707015.
  7. Adler V.E., Veselov A.P., Cauchy problem for integrable discrete equations on quad-graphs, Acta Appl. Math. 84 (2004), 237-262, math-ph/0211054.
  8. Dolbilin N.P., Sedrakyan A.G., Shtan'ko M.A., Shtogrin M.I., Topology of a family of parametrizations of two-dimensional cycles arising in the three-dimensional Ising model, Dokl. Akad. Nauk SSSR 295 (1987), no. 1, 19-23 (English transl.: Soviet Math. Dokl. 36 (1988), no. 1, 11-15).
  9. Boll R., Classification of 3D consistent quad-equations, arXiv:1009.4007.
  10. Adler V.E., Suris Yu.B., Q4: integrable master equation related to an elliptic curve, Int. Math. Res. Not. 2004 (2004), no. 47, 2523-2553, nlin.SI/0309030.

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