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

SIGMA 7 (2011), 061, 14 pages      arXiv:1105.4413
Contribution to the Proceedings of the Conference “Symmetries and Integrability of Difference Equations (SIDE-9)”

Soliton Taxonomy for a Modification of the Lattice Boussinesq Equation

Jarmo Hietarinta a, b and Da-jun Zhang c
a) Department of Physics and Astronomy, University of Turku, FIN-20014 Turku, Finland
b) LPTHE / CNRS / UPMC, 4 place Jussieu 75252 Paris CEDEX 05, France
c) Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China

Received May 24, 2011, in final form July 01, 2011; Published online July 06, 2011; Figure 3 changed July 20, 2011

Integrable multi-component lattice equations of the Boussinesq family have been known for some time. Recently some new equations of this type were found using the Consistency-Around-the-Cube approach. Here we investigate one of these models, B-2, and in particular the consequences of a nonzero deformation parameter b0>0, which allows special kinds of solitons in the parameter range −b0/3<k<b0.

Key words: lattice Boussinesq equation; integrable lattice equations; solitons; kinks.

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  1. Tongas A., Nijhoff F., The Boussinesq integrable system: compatible lattice and continuum structures, Glasg. Math. J. 47 (2005), no. A, 205-219, nlin.SI/0402053.
  2. Bobenko A.I., Suris Y.B., Integrable systems on quad-graphs, Int. Math. Res. Not. 2002 (2002), no. 11, 573-611, nlin.SI/0110004.
  3. Nijhoff F.W., Walker A.J., The discrete and continuous Painlevé VI hierarchy and the Garnier systems, Glasg. Math. J. 43 (2001), no. A, 109-123, nlin.SI/0001054.
  4. Nijhoff F.W., Lax pair for the Adler (lattice Krichever-Novikov) system, Phys. Lett. A 297 (2002), 49-58, nlin.SI/0110027.
  5. Hietarinta J., Boussinesq-like multi-component lattice equations and multi-dimensional consistency, J. Phys. A: Math. Theor. 44 (2011), 165204, 22 pages, arXiv:1011.1978.
  6. Hietarinta J., Zhang D.J., Multisoliton solutions to the lattice Boussinesq equation, J. Math. Phys. 51 (2010), 033505, 12 pages, arXiv:0906.3955.
  7. 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.
  8. 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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