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

SIGMA 15 (2019), 028, 17 pages      arXiv:1812.06465

Explicit Solutions for a Nonlinear Vector Model on the Triangular Lattice

V.E. Vekslerchik
Usikov Institute for Radiophysics and Electronics, 12 Proskura Str., Kharkiv, 61085, Ukraine

Received December 28, 2018, in final form April 04, 2019; Published online April 13, 2019

We present a family of explicit solutions for a nonlinear classical vector model with anisotropic Heisenberg-like interaction on the triangular lattice.

Key words: classical Heisenberg-type models; triangular lattice; bilinear approach; explicit solutions; solitons.

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  1. Ablowitz M.J., Ladik J.F., Nonlinear differential-difference equations, J. Math. Phys. 16 (1975), 598-603.
  2. Adler V.E., Legendre transforms on a triangular lattice, Funct. Anal. Appl. 34 (2000), 1-9, arXiv:solv-int/9808016.
  3. Adler V.E., Discrete equations on planar graphs, J. Phys. A: Math. Gen. 34 (2001), 10453-10460.
  4. Adler V.E., Suris Yu.B., ${\rm Q}_4$: integrable master equation related to an elliptic curve, Int. Math. Res. Not. 2004 (2004), 2523-2553.
  5. Bobenko A.I., Hoffmann T., Hexagonal circle patterns and integrable systems: patterns with constant angles, Duke Math. J. 116 (2003), 525-566, arXiv:math.CV/0109018.
  6. Bobenko A.I., Hoffmann T., Suris Yu.B., Hexagonal circle patterns and integrable systems: patterns with the multi-ratio property and Lax equations on the regular triangular lattice, Int. Math. Res. Not. 2002 (2002), 111-164, arXiv:math.CV/0104244.
  7. Bobenko A.I., Suris Yu.B., Integrable systems on quad-graphs, Int. Math. Res. Not. 2002 (2002), 573-611, arXiv:nlin.SI/0110004.
  8. Boll R., Suris Yu.B., Non-symmetric discrete Toda systems from quad-graphs, Appl. Anal. 89 (2010), 547-569, arXiv:0908.2822.
  9. Date E., Jinbo M., Miwa T., Method for generating discrete soliton equations. I, J. Phys. Soc. Japan 51 (1982), 4116-4124.
  10. Date E., Jinbo M., Miwa T., Method for generating discrete soliton equations. II, J. Phys. Soc. Japan 51 (1982), 4125-4131.
  11. Doliwa A., Nieszporski M., Santini P.M., Integrable lattices and their sublattices. II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices, J. Math. Phys. 48 (2007), 113506, 17 pages, arXiv:0705.0573.
  12. Haldane F.D.M., Excitation spectrum of a generalised Heisenberg ferromagnetic spin chain with arbitrary spin, J. Phys. C 15 (1982), L1309-L1313.
  13. Hirota R., Discrete analogue of a generalized Toda equation, J. Phys. Soc. Japan 50 (1981), 3785-3791.
  14. Ishimori Y., An integrable classical spin chain, J. Phys. Soc. Japan 51 (1982), 3417-3418.
  15. Nimmo J.J.C., Darboux transformations and the discrete KP equation, J. Phys. A: Math. Gen. 30 (1997), 8693-8704, arXiv:solv-int/9410001.
  16. Papanicolaou N., Complete integrability for a discrete Heisenberg chain, J. Phys. A: Math. Gen. 20 (1987), 3637-3652.
  17. Tokihiro T., Satsuma J., Willox R., On special function solutions to nonlinear integrable equations, Phys. Lett. A 236 (1997), 23-29.
  18. Vekslerchik V.E., Functional representation of the Ablowitz-Ladik hierarchy. II, J. Nonlinear Math. Phys. 9 (2002), 157-180, arXiv:solv-int/9812020.
  19. Vekslerchik V.E., Soliton Fay identities: I. Dark soliton case, J. Phys. A: Math. Theor. 47 (2014), 415202, 19 pages, arXiv:1409.0406.
  20. Vekslerchik V.E., Soliton Fay identities: II. Bright soliton case, J. Phys. A: Math. Theor. 48 (2015), 445204, 18 pages, arXiv:1510.00908.
  21. Vekslerchik V.E., Explicit solutions for a nonlinear model on the honeycomb and triangular lattices, J. Nonlinear Math. Phys. 23 (2016), 399-422, arXiv:1606.06470.
  22. Vekslerchik V.E., Solitons of a vector model on the honeycomb lattice, J. Phys. A: Math. Theor. 49 (2016), 455202, 16 pages, arXiv:1610.03242.
  23. Willox R., Tokihiro T., Satsuma J., Darboux and binary Darboux transformations for the nonautonomous discrete KP equation, J. Math. Phys. 38 (1997), 6455-6469.

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