Integrable (2 + 1)-Dimensional Spin Models with Self-Consistent Potentials
Abstract
:1. Introduction
2. A Brief Review on Integrable Spin Systems in 2 + 1 Dimensions
2.1. The Ishimori Equation
2.2. The Myrzakulov-I Equation
2.3. The Myrzakulov–Lakshmanan I Equation
2.4. The (2 + 1)-Dimensional Heisenberg Ferromagnet Equation
3. The Myrzakulov–Lakshmanan II Equation
3.1. Reductions
3.2. Lax Representation
3.3. Gauge Equivalent Counterpart of the ML-II Equation
3.4. Integral of Motion
4. The Myrzakulov–Lakshmanan III Equation
4.1. Lax Representation
4.2. Reductions
4.2.1. Case I:
4.2.2. Case II:
4.2.3. Case III:
4.3. Equivalent Counterpart of the ML-III Equation
5. The Myrzakulov–Lakshmanan IV Equation
5.1. Lax Representation
5.2. Reductions
5.2.1. Case I:
5.2.2. Case II:
5.2.3. Case III:
5.2.4. Case IV:
5.2.5. Case V:
5.3. Equivalent Counterpart of the ML-IV Equation
6. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
Appendix A: Gauge Equivalence
Appendix B: Nonisospectral Problem
References
- Lakshmanan, M. The Fascinating World of Landau-Lifshitz-Gilbert Equation: An Overview. Phil. Trans. R. Soc. A 2011, 369, 1280–1300. [Google Scholar]
- Hillebrands, B.; Ounadjela, K. Spin Dynamics in Confined Magnetic Structures; Springer-Verlag: Berlin, Germany, 2002; Volumes I and II. [Google Scholar]
- Bertotti, G.; Mayergoyz, I.; Serpico, C. Nonlinear Magnetization Dynamics in Nanosystems; Elsevier: Amsterdam, The Netherlands, 2009. [Google Scholar]
- Stiles, M.D.; Miltat, J. Spin Transfer Torque and Dynamics. Top. Appl. Phys. 2006, 101, 225–308. [Google Scholar]
- Slonczewski, J.C. Current-driven excitation of magnetic multilayers. J. Magn. Magn. Mater. 1996, 159, L261–L268. [Google Scholar]
- Lakshmanan, M. Continuum spin system as an exactly solvable dynamical system. Phys. Lett. A 1977, 61, 53–54. [Google Scholar]
- Takhtajan, L.A. Integration of the continuous Heisenberg spin chain through the inverse scattering method. Phys. Lett. A 1977, 64, 235–238. [Google Scholar]
- Senthilkumar, C.; Lakshmanan, M.; Grammaticos, B.; Ramani, A. Nonintegrability of (2+1) dimensional continuum isotropic Heisenberg spin system: Painleve analysis. Phys. Lett. A 2006, 356, 339–345. [Google Scholar]
- Ishimori, Y. Multi-vortex solutions of a two-dimensional nonlinear wave equation. Prog. Theor. Phys. 1984, 72, 33–37. [Google Scholar]
- Myrzakulov, R.; Vijayalakshmi, S.; Nugmanova, G.; Lakshmanan, M. A (2+ 1)-dimensional integrable spin model: Geometrical and gauge equivalent counterpart, solitons and localized coherent structures. Phys. Lett. A 1997, 233, 391–396. [Google Scholar]
- Ablowitz, M.J.; Clarkson, P.A. Solitons, Nonlinear Evolution Equations Inverse Scattering; Cambridge University Press: New York, NY, USA, 1991. [Google Scholar]
- Myrzakulov, R.; Vijayalakshmi, S.; Syzdykova, R.; Lakshmanan, M. On the simplest (2+1) dimensional integrable spin systems and their equivalent nonlinear Schrodinger equations. J. Math. Phys. 1998, 39, 2122–2140. [Google Scholar]
- Lakshmanan, M.; Myrzakulov, R.; Vijayalakshmi, S.; Danlybaeva, A. Motion of curves and surfaces and nonlinear evolution equations in (2+1) dimensions. J. Math. Phys. 1998, 39, 3765–3771. [Google Scholar]
- Konopelchenko, B.G.; Matkarimov, B.T. Inverse spectral transform for the Ishimori equation: I. Initial value problem. J. Math. Phys. 1990, 31, 2737–2746. [Google Scholar]
- Fokas, A.S.; Santini, P.M. Dromions and a boundary value problem for the Davey-Stewartson 1 equation. Phys. D Nonlinear Phenom. 1990, 44, 99–130. [Google Scholar]
- Konopelchenko, B.G. Solitons in Multidimensions: Inverse Spectral Transform; World Scientific: Singapore, 1993. [Google Scholar]
- Chen, C.; Zhou, Z.-X. Darboux Transformation and Exact Solutions of the Myrzakulov–I Equation. Chin. Phys. Lett. 2009, 26. [Google Scholar] [CrossRef]
- Chen, H.; Zhou, Z.-X. Darboux Transformation with a Double Spectral Parameter for the Myrzakulov–I Equation. Chin. Phys. Lett. 2014, 31. [Google Scholar] [CrossRef]
- Zakharov, V.E. The Inverse Scattering Method. In Solitons; Bullough, R.K., Caudrey, P.J., Eds.; Springer: Berlin, Germany, 1980. [Google Scholar]
- Strachan, I.A.B. Some integrable hierarchies in (2+1) dimensions and their twistor description. J. Math. Phys. 1993, 34, 243–259. [Google Scholar]
- Strachan, I.A.B. Wave solutions of a (2+1)–dimensional generalization of the nonlinear Schrodinger equation. Inverse Problems 1992, 8. [Google Scholar] [CrossRef]
- Calogero, F.; Degasperis, A. Extension of the Spectral Transform Method for Solving Nonlinear Evolution Equations. Lett. Nuovo Cimento 1978, 22, 131–137. [Google Scholar]
- Calogero, F.; Degasperis, A. Extension of the Spectral Transform Method for Solving Nonlinear Evolution Equations. II. Lett. Nuovo Cimento 1978, 22, 263–269. [Google Scholar]
- Lakshmanan, M.; Bullough, R.K. Geometry of generalised nonlinear Schrodinger and Heisenberg ferromagnetic spin equations with linearly x-dependent coefficients. Phys. Lett. A 1980, 80, 287–292. [Google Scholar]
- Balakrishnan, R. Inverse spectral transform analysis of a nonlinear Schrodinger equation with x–dependent coefficients. Phys. D Nonlinear Phenom. 1985, 16, 405–413. [Google Scholar]
- Lakshmanan, M.; Ganesan, S. Geometrical and gauge equivalence of the generalized hirota, Heisenberg and wkis equations with linear inhomogeneities. Phys. A Stat. Mech. Appl. 1985, 132, 117–142. [Google Scholar]
- Balakrishnan, R. Dynamics of a generalised classical Heisenberg chain. Phys. Lett. A 1982, 92, 243–246. [Google Scholar]
- Blumenfeld, R.; Balakrishnan, R. Exact multi–twist solutions to the Belavin–Polyakov equation and applications to magnetic systems. J. Phys. A Math. Gen. 2000, 33, 2459–2468. [Google Scholar]
- Esmakhanova, K.R.; Nugmanova, G.N.; Zhao, W.-Z.; Wu, K. Integrable Inhomogeneous Lakshmanan-Myrzakulov Equation. ArXiv E-Prints 2006. arXiv:nlin/0604034. [Google Scholar]
- Zhunussova, Z.K.; Yesmakhanova, K.R.; Tungushbaeva, D.I.; Mamyrbekova, G.K.; Nugmanova, G.N.; Myrzakulov, R. Integrable Heisenberg Ferromagnet Equations with self-consistent potentials. ArXiv E-Prints 2013. arXiv:1301.1649. [Google Scholar]
- Calogero, F. A Method to Generate Solvable Nonlinear Evolution Equations. Lett. Nuovo Cimento 1975, 14, 443–448. [Google Scholar]
- Calogero, F.; Degasperis, A. Solution by the spectral transform method of a nonlinear evolution equation including as a special case the cylindrical KdV equation. Lett. Nuovo Cimento 1978, 23, 150–154. [Google Scholar]
- Sakhnovich, A. Nonisospectral integrable nonlinear equations with external potentials and their GBDT solutions. J. Phys. A Math. Theor. 2008, 41. [Google Scholar] [CrossRef]
- Li, C.; He, J.; Porsezian, K. Rogue waves of the Hirota and the Maxwell-Bloch equations. Phys. Rev. E 2013, 87. [Google Scholar] [CrossRef]
- Li, C.; He, J. Darboux transformation and positons of the inhomogeneous Hirota and the Maxwell-Bloch equation. Sci. China Phys. Mech. Astron. 2014, 57, 898–907. [Google Scholar]
- Beggs, E.J. Solitons in the chiral equations. Commun. Math. Phys. 1990, 128, 131–139. [Google Scholar]
- Myrzakulov, R.; Mamyrbekova, G.K.; Nugmanova, G.N.; Yesmakhanova, K.R.; Lakshmanan, M. Integrable motion of curves in self-consistent potentials: Relation to spin systems and soliton equations. Phys. Lett. A 2014, 378, 2118–2123. [Google Scholar]
- Porsezian, K.; Nakkeeran, K. Optical Soliton Propagation in an Erbium Doped Nonlinear Light Guide with Higher Order Dispersion. Phys. Rev. Lett. 1995, 74. [Google Scholar] [CrossRef]
- Myrzakulov, R. On Some Integrable and Nonintegrable Soliton Equations of Magnets I-IV; HEPI: Alma-Ata, Kazakhstan, 1987. [Google Scholar]
- Myrzakulov, R.; Danlybaeva, A.K.; Nugmanova, G.N. Geometry and multidimensional soliton equations. Theor. Math. Phys. 1999, 118, 347–356. [Google Scholar]
- Myrzakulov, R.; Nugmanova, G.; Syzdykova, R. Gauge equivalence between (2+1)-dimensional continuous Heisenberg ferromagnetic models and nonlinear Schrodinger-type equations. J. Phys. A Math. Theor. 1998, 31, 9535–9545. [Google Scholar]
- Myrzakulov, R.; Daniel, M.; Amuda, R. Nonlinear spin-phonon excitations in an inhomogeneous compressible biquadratic Heisenberg spin chain. Phys. A 1997, 234, 715–724. [Google Scholar]
- Myrzakulov, R.; Makhankov, V.G.; Pashaev, O. Gauge equivalence SUSY and classical solutions of OSPU(1,1/1)-Heisenberg model and nonlinear Schrodinger equation. Lett. Math. Phys. 1989, 16, 83–92. [Google Scholar]
- Myrzakulov, R.; Makhankov, V.G.; Makhankov, A. General Coherent States and the Continuous Heisenberg XYZ Model with One-Ion Anizotropy. Phys. Scr. 1987, 35, 233–237. [Google Scholar]
- Myrzakulov, R.; Pashaev, O.; Kholmurodov, K. Particle-line excitations in Multicomponent Magnon-Poton System. Phys. Scr. 1986, 33, 378–384. [Google Scholar]
- Anco, S.C.; Myrzakulov, R. Integrable generalizations of Schrodinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces. J. Geom. Phys. 2010, 60, 1576–1603. [Google Scholar]
- Myrzakulov, R.; Rahimov, F.K.; Myrzakul, K.; Serikbaev, N.S. On the geometry of stationary Heisenberg ferromagnets. In Non-Linear Waves: Classical and Quantum Aspects; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2004; pp. 543–549. [Google Scholar]
- Myrzakulov, R.; Serikbaev, N.S.; Myrzakul, K.; Rahimov, F.K. On continuous limits of some generalized compressible Heisenberg spin chains. J. NATO Sci. Ser. II. Math. Phys. Chem. 2004, 153, 535–542. [Google Scholar]
- Myrzakulov, R.; Martina, L.; Kozhamkulov, T.A.; Myrzakul, K. Integrable Heisenberg ferromagnets and soliton geometry of curves and surfaces. In Nonlinear Physics: Theory and Experiment. II; World Scientific: London, UK, 2003; p. 248. [Google Scholar]
- Myrzakulov, R. Integrability of the Gauss-Codazzi-Mainardi equation in 2 + 1 dimensions. In Mathematical Problems of Nonlinear Dynamics, Proceedings of the International Conference “Progress in Nonlinear sciences”, Nizhny Novgorod, Russia, 2–6 July 2001; 2001; Volume 1, pp. 314–319. [Google Scholar]
- Yan, Z.-W.; Chen, M.-R.; Wu, K.; Zhao, W.-Z. (2+1)-Dimensional Integrable Heisenberg Supermagnet Model. J. Phys. Soc. Jpn. 2012, 81. [Google Scholar] [CrossRef]
- Yan, Z.-W.; Chen, M.-R.; Wu, K.; Zhao, W.-Z. Integrable Deformations of the (2+1)-Dimensional Heisenberg Ferromagnetic Model. Commun. Theor. Phys. 2012, 58. [Google Scholar] [CrossRef]
- Zhang, Z.-H.; Deng, M.; Zhao, W.-Z.; Wu, K. On the Integrable Inhomogeneous Myrzakulov-I Equation. ArXiv E-Prints 2006. arXiv: nlin/0603069. [Google Scholar]
- Martina, L.; Myrzakul, Kur.; Myrzakulov, R.; Soliani, G. Deformation of surfaces, integrable systems, and Chern-Simons theory. J. Math. Phys. 2001, 42. [Google Scholar] [CrossRef]
- Burtsev, S.P.; Gabitov, I.R. Alternative integrable equations of nonlinear optics. Phys. Rev. A 1994, 49, 2065–2070. [Google Scholar]
- Brunelli, J.C. Dispersionless limit of integrable models. Braz. J. Phys. 2000, 30, 455–468. [Google Scholar]
- Whitham, G.B. Linear and Nonlinear Waves; John Wiley & Sons: New York, NY, USA, 1974. [Google Scholar]
© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Myrzakulov, R.; Mamyrbekova, G.; Nugmanova, G.; Lakshmanan, M. Integrable (2 + 1)-Dimensional Spin Models with Self-Consistent Potentials. Symmetry 2015, 7, 1352-1375. https://doi.org/10.3390/sym7031352
Myrzakulov R, Mamyrbekova G, Nugmanova G, Lakshmanan M. Integrable (2 + 1)-Dimensional Spin Models with Self-Consistent Potentials. Symmetry. 2015; 7(3):1352-1375. https://doi.org/10.3390/sym7031352
Chicago/Turabian StyleMyrzakulov, Ratbay, Galya Mamyrbekova, Gulgassyl Nugmanova, and Muthusamy Lakshmanan. 2015. "Integrable (2 + 1)-Dimensional Spin Models with Self-Consistent Potentials" Symmetry 7, no. 3: 1352-1375. https://doi.org/10.3390/sym7031352
APA StyleMyrzakulov, R., Mamyrbekova, G., Nugmanova, G., & Lakshmanan, M. (2015). Integrable (2 + 1)-Dimensional Spin Models with Self-Consistent Potentials. Symmetry, 7(3), 1352-1375. https://doi.org/10.3390/sym7031352