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


SIGMA 7 (2011), 046, 11 pages      arXiv:1105.1583      http://dx.doi.org/10.3842/SIGMA.2011.046
Contribution to the Proceedings of the Conference “Integrable Systems and Geometry”

Rational Solutions of the H3 and Q1 Models in the ABS Lattice List

Ying Shi and Da-jun Zhang
Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China

Received January 31, 2011, in final form May 04, 2011; Published online May 09, 2011

Abstract
In the paper we present rational solutions for the H3 and Q1 models in the Adler-Bobenko-Suris lattice list. These solutions are in Casoratian form and are generated by considering difference equation sets satisfied by the basic Casoratian column vector.

Key words: Casoratian; bilinear; rational solutions; H3; Q1.

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References

  1. Nijhoff F.W., Walker A.J., The discrete and continuous Painlevé VI hierarchy and the Garnier systems, Glasgow Math. J. 43A (2001), 109-123, nlin.SI/0001054.
  2. 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.
  3. Atkinson J., Hietarinta J., Nijhoff F., Seed and soliton solutions of Adler's lattice equation, J. Phys. A: Math. Theor. 40 (2007), F1-F8, nlin.SI/0609044.
  4. Atkinson J., Hietarinta J., Nijhoff F., Soliton solutions for Q3, J. Phys. A: Math. Theor. 41 (2008), 142001, 11 pages, arXiv:0801.0806.
  5. Nijhoff F., Atkinson J., Hietarinta J., Soliton solutions for ABS lattice equations. I. Cauchy matrix approach, J. Phys. A: Math. Theor. 42 (2009), 404005, 34 pages, arXiv:0902.4873.
  6. Hietarinta J., Zhang D.J., Soliton solutions for ABS lattice equations. II. Casoratians and bilinearization, J. Phys. A: Math. Theor. 42 (2009), 404006, 30 pages, arXiv:0903.1717.
  7. Hietarinta J., Zhang D.J., Multisoliton solutions to the lattice Boussinesq equation, J. Math. Phys. 51 (2010), 033505, 12 pages, arXiv:0906.3955.
  8. Atkinson J., Nijhoff F., A constructive approach to the soliton solutions of integrable quadrilateral lattice equations, Comm. Math. Phys. 299 (2010), 283-304, arXiv:0911.0458.
  9. Nijhoff F., Atkinson J., Elliptic N-soliton solutions of ABS lattice equations, Int. Math. Res. Not. 2010 (2010), no. 20, 3837-3895, arXiv:0911.0461.
  10. Ablowitz M.J., Satsuma J., Solitons and rational solutions of nonlinear evolution equations, J. Math. Phys. 19 (1978), 2180-2186.
  11. Zhang D.J., Notes on solutions in Wronskian form to soliton equations: KdV-type, nlin.SI/0603008.
  12. Zhang D.J., Hietarinta J., Generalized solutions for the H1 model in ABS list of lattice equations, in Nonlinear and Modern Mathematical Physics (July 15-21, 2009, Beijing), Editors W.X. Ma, X.B. Hu and Q.P. Liu, AIP Conf. Proc., Vol. 1212, Amer. Inst. Phys., Melville, NY, 2010, 154-161.
  13. Freeman N.C., Nimmo J.J.C., Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili equations: the Wronskian technique, Phys. Lett. A 95 (1983), 1-3.

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