Abstract
It has been realized in recent times that the elementary excitations in Heisenberg ferromagnetic systems can be characterized by spatially compact solitons in addition to the linear spin waves (magnons).1–4 Hence rigorous attempts were made after the discovery of the concepts of soliton and inverse scattering transform to identify soliton possessing integrable spin models. It was successful in the case of simple spin models like one-dimensional classical isotropic and anisotropic spin chains.1, 5 At the same time Lakshmanan1 approached the spin chain problem through an understanding of the underlying geometry of the system. This enabled Lakshmanan and his coworkers2, 3 to investigate a large class of integrable spin chain models by including different kinds of magnetic interactions. Some of the systems in this list include bilinear, biquadratic, deformed, site dependent, and radially symmetric Heisenberg ferromagnetic spin chains. Similarly gauge equivalence formalism introduced by Zakharov and Takhtajan connects the integrable nonlinear Schrödinger (NLS) family of equations with integrable spin models. The main advantage of the above equivalence method is that individual treatment of each one of them would become unnecessary and full information about only a few basic equations would be sufficient to solve the rest generated from them.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
M. Lakshmanan, Continuum spin system as an exactly solvable dynamical system, Phys. Lett. A61:53(1977).
M. Daniel, “Nonlinear excitations in the Heisenberg ferromagnetic spin chain,” Ph. D. thesis, University of Madras (1983).
K. Porsezian, “On the nonlinear dynamics of the discrete and continuum spin systems,” Ph. D. thesis, Bharathidasan University (1990).
V. E. Zakharov and L. A. Takhtajan, Equivalent of the nonlinear Schrödinger equation and the equation of a Heisenberg ferromagnet, Theor. Math. Phys. 38:17(1979).
A. F. Borovik, Exact integration of the nonlinear Landau-Lifshitz equation, Solid State Commun. 34:721(1980).
E. K. Sklyanin, The complete integrability of the Landau-Lifshitz equation, Lomi preprint (1979).
K. Porsezian, M. Lakshmanan and K. M. Tamizhmani, Geometrical equivalence of a deformed Heisenberg spin equation and the generalized nonlinear Schrödinger equation, Phys. Lett. A124:159(1987).
K. Porsezian and M. Lakshmanan, On the dynamics of the radially symmetric Heisenberg ferromagnetic spin system, J. Math. Phys. 32:2923(1991).
K. Porsezian, M. Daniel and M. Lakshmanan, On the integrability aspects of the one-dimensional classical continuum isotropic biquadratic Heisenberg spin chain, J. Math. Phys. 33:(1992) (in press)
Y. Ishimori, A relationship between the AKNS and WKI schemes of the inverse scattering method, J. Phys. Soc. Japan 51:394(1983).
M. Lakshmanan and S. Ganesan, Geometrical and gauage equivalence of the generalized Hirota, Heisenberg and WKIS equation with linear inhomogeneties, Physica A132:117(1985).
B. G. Konopelchenko and B. J. Matkarimov, On the inverse scattering transform for the Ishimori equation, Phys. Lett. A135:183(1989).
Y. Ishimori, An integrable spin chain, J. Phys. Soc. Japan 51:3417(1989).
V. S. Gerdjikov, M. I. Ivanov and Y. S. Vaklev, Gauge transformations and gererating operators for the discrete Akharov-Shabat system, Inverse Prob. 2:413(1986).
K. Porsezian and M. Lakshmanan, Discretized Hirota equation equivalent spin system and Bäcklund transformations, Inverse Prob. 5:L15(1988).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer Science+Business Media New York
About this chapter
Cite this chapter
Porsezian, K. (1993). On the Discrete and Continuum Integrable Heisenberg Spin Chain Models. In: Christiansen, P.L., Eilbeck, J.C., Parmentier, R.D. (eds) Future Directions of Nonlinear Dynamics in Physical and Biological Systems. NATO ASI Series, vol 312. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1609-9_42
Download citation
DOI: https://doi.org/10.1007/978-1-4899-1609-9_42
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4899-1611-2
Online ISBN: 978-1-4899-1609-9
eBook Packages: Springer Book Archive