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


SIGMA 9 (2013), 044, 16 pages      arXiv:1302.0637      http://dx.doi.org/10.3842/SIGMA.2013.044

Two-Dimensional Toda-Heisenberg Lattice

Vadim E. Vekslerchik
Institute for Radiophysics and Electronics of NAS of Ukraine, 12, Proskura Str., Kharkiv, 61085, Ukraine

Received February 06, 2013, in final form June 04, 2013; Published online June 12, 2013

Abstract
We consider a nonlinear model that is a combination of the anisotropic two-dimensional classical Heisenberg and Toda-like lattices. In the framework of the Hirota direct approach, we present the field equations of this model as a bilinear system, which is closely related to the Ablowitz-Ladik hierarchy, and derive its N-soliton solutions.

Key words: classical Heisenberg model; Toda-like lattices; Hirota direct method; Ablowitz-Ladik hierarchy; soliton.

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References

  1. Ablowitz M.J., Ladik J.F., Nonlinear differential-difference equations, J. Math. Phys. 16 (1975), 598-603.
  2. Ablowitz M.J., Ladik J.F., Nonlinear differential-difference equations and Fourier analysis, J. Math. Phys. 17 (1976), 1011-1018.
  3. Baxter R.J., Exactly solved models in statistical mechanics, Academic Press Inc., London, 1982.
  4. Brown H.A., Luttinger J.M., Ferromagnetic and antiferromagnetic Curie temperatures, Phys. Rev. 100 (1955), 685-692.
  5. Bruschi M., Ragnisco O., Lax representation and complete integrability for the periodic relativistic Toda lattice, Phys. Lett. A 134 (1989), 365-370.
  6. Dorizzi B., Grammaticos B., Ramani A., Winternitz P., Are all the equations of the Kadomtsev-Petviashvili hierarchy integrable?, J. Math. Phys. 27 (1986), 2848-2852.
  7. Hietarinta J., A search for bilinear equations passing Hirota's three-soliton condition. I. KdV-type bilinear equations, J. Math. Phys. 28 (1987), 1732-1742.
  8. Hietarinta J., A search for bilinear equations passing Hirota's three-soliton condition. II. mKdV-type bilinear equations, J. Math. Phys. 28 (1987), 2094-2101.
  9. Hietarinta J., A search for bilinear equations passing Hirota's three-soliton condition. III. Sine-Gordon-type bilinear equations, J. Math. Phys. 28 (1987), 2586-2592.
  10. Hietarinta J., A search for bilinear equations passing Hirota's three-soliton condition. IV. Complex bilinear equations, J. Math. Phys. 29 (1988), 628-635.
  11. Hietarinta J., Zhang D.J., Hirota's method and the search for integrable partial difference equations. 1. Equations on a 3×3 stencil, J. Difference Equ. Appl., to appear, arXiv:1210.4708.
  12. Hietarinta J., Zhang D.J., Hirota's method and the search for integrable partial difference equations. 2. Equations on a 2×N stencil, in Report of RIAM Symposium No. 22AO-S8 "Development in Nonlinear Wave: Phenomena and Modeling", Research Institute for Applied Mechanics, Kyushu University, 2011, 30-36.
  13. Hirota R., Nonlinear partial difference equations. I. A difference analogue of the Korteweg-de Vries equation, J. Phys. Soc. Japan 43 (1977), 1424-1433.
  14. Hirota R., Nonlinear partial difference equations. II. Discrete-time Toda equation, J. Phys. Soc. Japan 43 (1977), 2074-2078.
  15. Hirota R., The direct method in soliton theory, Cambridge Tracts in Mathematics, Vol. 155, Cambridge University Press, Cambridge, 2004.
  16. Ishimori Y., An integrable classical spin chain, J. Phys. Soc. Japan 51 (1982), 3417-3418.
  17. Leznov A.N., Saveliev M.V., Smirnov V.G., Explicit solutions to two-dimensionalized Volterra equations, Lett. Math. Phys. 4 (1980), 445-449.
  18. Mattis D.C., The theory of magnetism. I. Statics and dynamics, Springer Series in Solid-State Sciences, Vol. 17, Springer-Verlag, Berlin, 1981.
  19. Newell A.C., Yunbo Z., The Hirota conditions, J. Math. Phys. 27 (1986), 2016-2021.
  20. Papageorgiou V., Grammaticos B., Ramani A., Orthogonal polynomial approach to discrete Lax pairs for initial-boundary value problems of the QD algorithm, Lett. Math. Phys. 34 (1995), 91-101.
  21. Pritula G.M., Vekslerchik V.E., Toda-Heisenberg chain: interacting σ-fields in two dimensions, J. Nonlinear Math. Phys. 18 (2011), 443-459, arXiv:1108.5937.
  22. Ruijsenaars S.N.M., Relativistic Toda systems, Comm. Math. Phys. 133 (1990), 217-247.
  23. Vekslerchik V.E., Explicit solutions for a (2+1)-dimensional Toda-like chain, J. Phys. A: Math. Theor. 46 (2013), 055202, 22 pages, arXiv:1301.0414.
  24. Vekslerchik V.E., Functional representation of the Ablowitz-Ladik hierarchy. II, J. Nonlinear Math. Phys. 9 (2002), 157-180, solv-int/9812020.
  25. Vekslerchik V.E., The 2D Toda lattice and the Ablowitz-Ladik hierarchy, Inverse Problems 11 (1995), 463-479.

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