Explicit Solutions for a Nonlinear Vector Model on the Triangular Lattice
Symmetry, integrability and geometry: methods and applications, Tome 15 (2019) Cet article a éte moissonné depuis la source Math-Net.Ru

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We present a family of explicit solutions for a nonlinear classical vector model with anisotropic Heisenberg-like interaction on the triangular lattice.
Keywords: classical Heisenberg-type models; triangular lattice; bilinear approach; explicit solutions; solitons.
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     author = {V. E. Vekslerchik},
     title = {Explicit {Solutions} for a {Nonlinear} {Vector} {Model} on the {Triangular} {Lattice}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a27/}
}
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V. E. Vekslerchik. Explicit Solutions for a Nonlinear Vector Model on the Triangular Lattice. Symmetry, integrability and geometry: methods and applications, Tome 15 (2019). http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a27/

[1] Ablowitz M. J., Ladik J. F., “Nonlinear differential-difference equations”, J. Math. Phys., 16 (1975), 598–603 | DOI | MR | Zbl

[2] Adler V. E., “Legendre transforms on a triangular lattice”, Funct. Anal. Appl., 34 (2000), 1–9, arXiv: solv-int/9808016 | DOI | MR | Zbl

[3] Adler V. E., “Discrete equations on planar graphs”, J. Phys. A: Math. Gen., 34 (2001), 10453–10460 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[7] Bobenko A. I., Suris Yu. B., “Integrable systems on quad-graphs”, Int. Math. Res. Not., 2002 (2002), 573–611, arXiv: nlin.SI/0110004 | DOI | MR | Zbl

[8] Boll R., Suris Yu. B., “Non-symmetric discrete Toda systems from quad-graphs”, Appl. Anal., 89 (2010), 547–569, arXiv: 0908.2822 | DOI | MR | Zbl

[9] Date E., Jinbo M., Miwa T., “Method for generating discrete soliton equations. I”, J. Phys. Soc. Japan, 51 (1982), 4116–4124 | DOI | MR

[10] Date E., Jinbo M., Miwa T., “Method for generating discrete soliton equations. II”, J. Phys. Soc. Japan, 51 (1982), 4125–4131 | DOI | MR

[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 pp., arXiv: 0705.0573 | DOI | MR | Zbl

[12] Haldane F. D. M., “Excitation spectrum of a generalised Heisenberg ferromagnetic spin chain with arbitrary spin”, J. Phys. C, 15 (1982), L1309–L1313 | DOI

[13] Hirota R., “Discrete analogue of a generalized Toda equation”, J. Phys. Soc. Japan, 50 (1981), 3785–3791 | DOI | MR

[14] Ishimori Y., “An integrable classical spin chain”, J. Phys. Soc. Japan, 51 (1982), 3417–3418 | DOI | MR

[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 | DOI | MR | Zbl

[16] Papanicolaou N., “Complete integrability for a discrete Heisenberg chain”, J. Phys. A: Math. Gen., 20 (1987), 3637–3652 | DOI | MR | Zbl

[17] Tokihiro T., Satsuma J., Willox R., “On special function solutions to nonlinear integrable equations”, Phys. Lett. A, 236 (1997), 23–29 | DOI | MR | Zbl

[18] Vekslerchik V. E., “Functional representation of the Ablowitz–Ladik hierarchy. II crossref{https://doi.org/10.2991/jnmp.2002.9.2.3}”, J. Nonlinear Math. Phys., 9 (2002), 157–180, arXiv: solv-int/9812020 | DOI | MR | Zbl

[19] Vekslerchik V. E., “Soliton Fay identities: I Dark soliton case”, J. Phys. A: Math. Theor., 47 (2014), 415202, 19 pp., arXiv: 1409.0406 | DOI | MR | Zbl

[20] Vekslerchik V. E., “Soliton Fay identities: II Bright soliton case”, J. Phys. A: Math. Theor., 48 (2015), 445204, 18 pp., arXiv: 1510.00908 | DOI | MR | Zbl

[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 | DOI | MR

[22] Vekslerchik V. E., “Solitons of a vector model on the honeycomb lattice”, J. Phys. A: Math. Theor., 49 (2016), 455202, 16 pp., arXiv: 1610.03242 | DOI | MR | Zbl

[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 | DOI | MR | Zbl