Skip to main content
SpringerLink
Log in

Intrinsic Localized Spin Wave Modes and Modulational Instability in a Two-Dimensional Heisenberg Ferromagnet

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

An analytical study on the properties of intrinsic localized modes and modulational instability in a quantum two-dimensional ferromagnet with single-ion uniaxial anisotropy is completed in the semiclassical limit. By making use of the semidiscrete multiple-scale method, we obtain a line localized solution and a radially symmetric localized solution, and analyze their existence conditions. Taking into account that the existence of bright localized solutions is closely linked to the modulational instability of plane waves, we analytically study the discrete modulational instability of plane spin waves. The result of the modulational instability analysis show that the uniaxial anisotropy plays a key role in the appearance of our intrinsic localized spin wave modes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Zhang, H., et al.: Graphene mode locked, wavelengthtunable, dissipative soliton fiber laser. Appl. Phys. Lett. 96, 111112 (2010)

    Article  ADS  Google Scholar 

  2. Zhao, C., Zou, Y., Chen, Y., Wang, Z., Lu, S., Zhang, H., Wen, S., Tang, D.: Wavelength-tunable picosecond soliton fiber laser with topological insulator: Bi2Se3 as a mode locker. Opt. Express 20, 27888–27895 (2012)

    Article  ADS  Google Scholar 

  3. Lü, X., Chen, S.T., Ma, W.X.: Constructing lump solutions to a generalized Kadomtsev-Petviashvili-Boussinesq equation. Nonlinear Dyn. 86, 523–534 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Lü, X., Ma, W.X., Chen, S.T., Khalique, C.M.: A note on rational solutions to a Hirota-Satsuma-like equation. Appl. Math. Lett. 58, 13–18 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lü, X., Ling, L.: Vector bright solitons associated with positive coherent coupling via Darboux transformation. Chaos 25, 123103 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  6. Lü, X., Ma, W.X.: Envelope bright- and dark-soliton solutions for the Gerdjikov-Ivanov model. Nonlinear Dyn. 85, 1217–1222 (2016)

    Article  Google Scholar 

  7. Lü, X., Ma, W.X., Zhou, Y., Khalique, C.M.: Rational solutions to an extended Kadomtsev-Petviashvili-like equation with symbolic computation. Comput. Math. Appl. 71, 1560–1567 (2016)

    Article  MathSciNet  Google Scholar 

  8. Wong, P., Pang, L., Wu, Y., Lei, M., Liu, W.: Novel asymmetric representation method for solving the higher-order Ginzburg-Landau equation. Sci. Rep. 6, 24613 (2016)

    Article  ADS  Google Scholar 

  9. Wang, L., Wang, Z.Q., Sun, W.R., Shi, Y.Y., Li, M.: Dynamics of Peregrine combs and Peregrine walls in an inhomogeneous Hirota and Maxwell-Bloch system. Commun. Nonlinear Sci. Numer. Simulat. 47, 190–199 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  10. Liu, W., Huang, L., Huang, P., Li, Y., Lei, M.: Dark soliton control in inhomogeneous optical fibers. Appl. Math. Lett. 61, 80–87 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Liu, W., Pang, L., Han, H., Tian, W., Chen, H., Lei, M., Yan, P., Wei, Z.: 70-fs mode-locked erbiumdoped fiber laser with topological insulator. Sci. Rep. 5, 19997 (2016)

    Article  ADS  Google Scholar 

  12. Liu, W., Pang, L., Han, H., Liu, M., Lei, M., Fang, S., Teng, H., Wei, Z.: Tungsten disulfide saturable absorbers for 67 fs mode-locked erbium-doped fiber lasers. Opt. Express 25, 2950–2959 (2017)

    Article  ADS  Google Scholar 

  13. Flach, S., Gorbach, A.V.: Discrete breathers – advances in theory and applications. Phys. Rep. 467, 1–116 (2008)

    Article  ADS  MATH  Google Scholar 

  14. Sievers, A.J., Takeno, S.: Intrinsic localized modes in an harmonic crystals. Phys. Rev. Lett. 61, 970–973 (1988)

    Article  ADS  Google Scholar 

  15. Aubry, S.: Discrete breathers: localization and transfer of energy in discrete Hamiltonian nonlinear systems. Physica D 216, 1–30 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Yoshimura, K.: Existence and stability of discrete breathers in diatomic Fermi-Pasta-Ulam type lattices. Nonlinearity 24, 293–317 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Aubry, S., Kopidakis, G., Kadelburg, V.: Variational proof for hard discrete breathers in some classes of Hamiltonian dynamical systems. Discrete Contin. Dynam. Syst. B 1, 271–298 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zolotaryuk, Y., Flach, S., Fleurov, V.: Discrete breathers in classical spin lattices. Phys. Rev. B 63, 214422 (2001)

    Article  ADS  Google Scholar 

  19. Lai, R., Kiselev, S.A., Sievers, A.J.: Intrinsic localized spin-wave modes in antiferromagnetic chains with single-ion easy-axis anisotropy. Phys. Rev. B 54, R12665–R12668 (1996)

    Article  ADS  Google Scholar 

  20. Lai, R., Kiselev, S.A., Sievers, A.J.: Intrinsic localized spin-wave resonances in ferromagnetic chains with nearest- and next-nearest-neighbor exchange interactions. Phys. Rev. B 56, 5345–5354 (1997)

    Article  ADS  Google Scholar 

  21. Nguenang, J.-P., Peyrard, M., Kenfack, A.J., Kofané, T.: On modulational instability of nonlinear waves in 1D ferromagnetic spin chains. J. Phys.: Condens. Matter 17, 3083 (2005)

    ADS  Google Scholar 

  22. Kavitha, L., Parasuraman, E., Gopi, D., Prabhu, A., Vicencio, R.A.: Nonlinear nano-scale localized breather modes in a discrete weak ferromagnetic spin lattice. J. Magn. Magn. Mater. 401, 394–405 (2016)

    Article  ADS  Google Scholar 

  23. Kavitha, L., Mohamadou, A., Parasuraman, E., Gopi, D., Akila, N., Prabhu, A.: Modulational instability and nano-scale energy localization in ferromagnetic spin chain with higher order dispersive interactions. J. Magn. Magn. Mater. 404, 91–118 (2016)

    Article  Google Scholar 

  24. Tang, B., Li, G., Fu, M.: Modulational instability and localized modes in Heisenberg ferromagnetic chains with single-ion easy-axis anisotropy. J. Magn. Magn. Mater. 426, 429–434 (2017)

    Article  ADS  Google Scholar 

  25. Kevrekidis, P.G., Rasmussen, K.Ø., Bishop, A.R.: Comparison of one-dimensional and two-dimensional discrete breathers. Math. Comput. Simulat. 55, 449–462 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Flach, S., Gorbach, A.: Discrete breathers in Fermi-Pasta-Ulam lattices. Chaos 15, 015112 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Tang, B., Li, D., Tang, Y.: Spin discrete breathers in two-dimensional square anisotropic ferromagnets. Phys. Scr. 89, 095208 (2014)

    Article  ADS  Google Scholar 

  28. Dyson, F.J.: General theory of spin-wave interactions. Phys. Rev. 102, 1217–1230 (1956)

    Article  ADS  MathSciNet  Google Scholar 

  29. Dyson, F.J.: Thermodynamic behavior of an ideal ferromagnet. Phys. Rev. 102, 1230–1244 (1956)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Glauber, R.J.: Coherent and incoherent states of the radiation field. Phys. Rev. 131, 2766–2788 (1963)

    Article  ADS  MathSciNet  Google Scholar 

  31. Smith, H.: Introduction to quantum mechanics. In: Singapore, Proceedings World Scientific, p. 108 (1991)

  32. Remoissenet, M.: Low-amplitude breather and envelope solitons in quasi-one-dimensional physical models. Phys. Rev. B 33, 2386–2392 (1986)

    Article  ADS  Google Scholar 

  33. Butt, I.A., Wattis, J.A.D.: Discrete breathers in a two-dimensional Fermi-Pasta-Ulam lattice. J. Phys. A: Math. Gen. 39, 1239–1264 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang, L., Jiang, D.Y., Qi, F.H., Shi, Y.Y., Zhao, Y.C.: Dynamics of the higher-order rogue waves for a generalized mixed nonlinear schrödinger model. Commun. Nonlinear Sci. Numer. Simulat. 42, 502–519 (2017)

    Article  ADS  Google Scholar 

  35. Wang, L., Li, X., Qi, F.H., Zhang, L.L.: Breather interactions and higher-order nonautonomous rogue waves for the inhomogeneous nonlinear Schrödinger Maxwell-Bloch equations. Ann. Phys. 359, 97–114 (2017)

    Article  MATH  Google Scholar 

  36. Remoissenet, M.: Waves Called Solitons. Concepts and Experiments, 2nd edn., pp. 238–239. Springer (1996)

  37. Wang, L., Zhang, J.-H., Wang, Z.-Q., Liu, C., Li, M., Qi, F.-H., Guo, R.: Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear Schrödinger equation. Phys. Rev. E 93, 012214 (2016)

    Article  ADS  Google Scholar 

  38. Wang, L., Zhu, Y.-J., Qi, F.-H., Li, M., Guo, R.: Modulational instability, higher-order localized wave structures, and nonlinear wave interactions for a nonautonomous Lenells-Fokas equation in inhomogeneous fibers. Chaos 25, 063111 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  39. Zhang, J.H., Wang, L., Liu, C.: Superregular breathers, characteristics of nonlinear stage of modulation instability induced by higher-order effects. Proc. R. Soc. A 473, 20160681 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  40. Abdullaev, F.K.H., Bouketir, A., Messikh, A., Umarov, B.A.: Modulational instability and discrete breathers in the discrete cubic–quintic nonlinear Schrödinger equation. Physica D 232, 54–61 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Wang, L., Zhang, J.H., Liu, C., Li, M., Qi, F.H.: Breather transition dynamics, Peregrine combs and walls, and modulation instability in a variable-coefficient nonlinear Schrödinger equation with higher-order effects. Phys. Rev. E 93, 062217 (2016)

    Article  ADS  Google Scholar 

  42. Daumonty, I., Dauxoisz, T., Peyrard, M.: Modulational instability: first step towards energy localization in nonlinear lattices. Nonlinearity 10, 617–630 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  43. Tang, B.: Quantum breathers and two-breathers in the β-fermi-pasta-ulam chain with the second-neighbor coupling. Commun. Nonlinear Sci. Numer. Simulat. 48, 361–375 (2017)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant No. 11604121, the Scientific Research Fund of Hunan Provincial Education Department under Grant No. 16B210, and the Natural Science Fund Project of Hunan Province under Grant No. 2017JJ3255.

Author information

Authors and Affiliations

  1. College of Physics, Mechanical and Electrical Engineering, Jishou University, Jishou, 416000, Hunan, China

    Tianle Wu, Jiayu Xie, Xia Chang & Bing Tang

Authors
  1. Tianle Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiayu Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xia Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Bing Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bing Tang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, T., Xie, J., Chang, X. et al. Intrinsic Localized Spin Wave Modes and Modulational Instability in a Two-Dimensional Heisenberg Ferromagnet. Int J Theor Phys 56, 3005–3018 (2017). https://doi.org/10.1007/s10773-017-3467-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-017-3467-7

Keywords

Search

Navigation