Advertisement

The continuous classical Heisenberg ferromagnet equation with in-plane asymptotic conditions. II. IST and closed-form soliton solutions

  • F. Demontis
  • G. Ortenzi
  • M. Sommacal
  • C. van der Mee
Article

Abstract

A new, general, closed-form soliton solution formula for the classical Heisenberg ferromagnet equation with in-plane asymptotic conditions is obtained by means of the inverse scattering transform technique and the matrix triplet method. This formula encompasses the soliton solutions already known in the literature as well as a new class of soliton solutions (the so-called multipole solutions), allowing their classification and description. Examples from all classes are provided and discussed.

Keywords

Classical Heisenberg ferromagnet equation Soliton solutions Inverse scattering transform Magnetic droplet Ferromagnetic materials 

Mathematics Subject Classification

35C08 35G25 35P25 35Q40 35Q51 

Notes

Acknowledgements

The results contained in the present paper have been partially presented in Wascom 2017. The research leading to this article was supported in part by INdAM-GNFM and the London Mathematical Society Scheme 4 (Research in Pairs) Grant on “Propagating, localised waves in ferromagnetic nanowires” (Ref No. 41622). We also wish to thank an anonymous referee for his/her valuable comments.

References

  1. 1.
    Lakshmanan, M.: Continuum spin system as an exactly solvable dynamical system. Phys. Lett. A 61, 53–54 (1977)CrossRefGoogle Scholar
  2. 2.
    Takhtajan, L.A.: Integration of the continuous Heisenberg spin chain through the inverse scattering method. Phys. Lett. A 64, 235–237 (1977)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Zakharov, V.E., Takhtajan, L.A.: Equivalence of the nonlinear Schrödinger equation and the equation of a Heisenberg ferromagnet. Theor. Math. Phys. 38, 17–23 (1979)CrossRefGoogle Scholar
  4. 4.
    Fogedby, H.C.: Solitons and magnons in the classical Heisenberg chain. J. Phys. A: Math. Gen. 13, 1467–1499 (1980)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Demontis, F., Ortenzi, G., Sommacal, M., van der Mee, C.: The continuous classical Heisenberg ferromagnet equation with in-plane asymptotic conditions. I. Direct and inverse scattering theory. Ricerche Mat. (2018).  https://doi.org/10.1007/s11587-018-0394-8
  6. 6.
    Mohseni, S.M., Sani, S.R., Persson, J., Anh Nguyen, T.N., Chung, S., Pogoryelov, Y., Muduli, P.K., Iacocca, E., Eklund, A., Dumas, R.K., Bonetti, S., Deac, A., Hoefer, M.A., Åkerman, J.: Spin torque-generated magnetic droplet solitons. Science 339, 1295–1298 (2013)CrossRefGoogle Scholar
  7. 7.
    Macià, F., Backes, D., Kent, A.D.: Stable magnetic droplet solitons in spin-transfer nanocontacts. Nat. Nanotechnol. 9, 992–996 (2014)CrossRefGoogle Scholar
  8. 8.
    Mohseni, S.M., Sani, S.R., Dumas, R.K., Persson, J., Anh Nguyen, T.N., Chung, S., Pogoryelov, Ye, Muduli, P.K., Iacocca, E., Eklund, A., Åkerman, J.: Magnetic droplet solitons in orthogonal nano-contact spin torque oscillators. Phys. B 435, 84–87 (2014)CrossRefGoogle Scholar
  9. 9.
    Chung, S., Mohseni, S.M., Sani, S.R., Iacocca, E., Dumas, R.K., AnhNguyen, T.N., Pogoryelov, Ye, Muduli, P.K., Eklund, A., Hoefer, M., Åkerman, J.: Spin transfer torque generated magnetic droplet solitons. J. App. Phys. 115, 172612 (2014)CrossRefGoogle Scholar
  10. 10.
    Maiden, M.D., Bookman, L.D., Hoefer, M.A.: Attraction, merger, reflection, and annihilation in magnetic droplet soliton scattering. Phys. Rev. B 89(18), 180409 (2014)CrossRefGoogle Scholar
  11. 11.
    Chung, S., Mohseni, S.M., Eklund, A., Dürrenfeld, P., Ranjbar, M., Sani, S.R., Anh Nguyen, T.N., Dumas, R.K., åkerman, J.: Magnetic droplet solitons in orthogonal spin valves. Low Temp. Phys. 41, 833 (2015)CrossRefGoogle Scholar
  12. 12.
    Bookman, L.D., Hoefer, M.A.: Perturbation theory for propagating magnetic droplet solitons. Proc. R. Soc. A 471(2179), 20150042 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chung, S., Eklund, A., Iacocca, E., Mohseni, S.M., Sani, S.R., Bookman, L., Hoefer, M.A., Dumas, R.K., Åkerman, J.: Magnetic droplet nucleation boundary in orthogonal spin-torque nano-oscillators. Nat. Commun. 7, 11209 (2016)CrossRefGoogle Scholar
  14. 14.
    Wang, C., Xiao, D., Liu, Y.: Merging magnetic droplets by a magnetic field pulse. AIP Adv. 8, 056021 (2018)CrossRefGoogle Scholar
  15. 15.
    Ivanov, B., Kosevich, A.: Bound-states of a large number of magnons in a ferromagnet with one-ion anisotropy. Zh. Eksp. Teor. Fiz. 72(5), 2000–2015 (1977)Google Scholar
  16. 16.
    Kosevich, A., Ivanov, B., Kovalev, A.: Magnetic solitons. Phys. Rep. 194(3–4), 117–238 (1990)CrossRefGoogle Scholar
  17. 17.
    Hoefer, M.A., Silva, T., Keller, M.: Theory for a dissipative droplet soliton excited by a spin torque nanocontact. Phys. Rev. B 82(5), 054432 (2010)CrossRefGoogle Scholar
  18. 18.
    Bonetti, S., Tiberkevich, V., Consolo, G., Finocchio, G., Muduli, P., Mancoff, F., Slavin, A., Åkerman, J.: Experimental evidence of self-localized and propagating spin wave modes in obliquely magnetized current-driven nanocontacts. Phys. Rev. Lett. 105, 217204 (2010)CrossRefGoogle Scholar
  19. 19.
    Ivanov, B.A., Stephanovich, V.A.: Two-dimensional soliton dynamics in ferromagnets. Phys. Lett. A 141(1), 89–94 (1989)CrossRefGoogle Scholar
  20. 20.
    Piette, B., Zakrzewski, W.J.: Localized solutions in a two-dimensional Landau–Lifshitz model. Physica D 119(3), 314–326 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ivanov, B.A., Zaspel, C.E., Yastremsky, I.A.: Small-amplitude mobile solitons in the two-dimensional ferromagnet. Phys. Rev. B 63(13), 134413 (2001)CrossRefGoogle Scholar
  22. 22.
    Hoefer, M.A., Sommacal, M.: Propagating two-dimensional magnetic droplets. Physica D 241, 890–901 (2012)CrossRefzbMATHGoogle Scholar
  23. 23.
    Hoefer, M.A., Sommacal, M., Silva, T.: Propagation and control of nano-scale, magnetic droplet solitons. Phys. Rev. B 85(21), 214433 (2012)CrossRefGoogle Scholar
  24. 24.
    Iacocca, E., Dumas, R.K., Bookman, L., Mohseni, M., Chung, S., Hoefer, M.A., Åkerman, J.: Confined dissipative droplet solitons in spin-valve nanowires with perpendicular magnetic anisotropy. Phys. Rev. Lett. 112, 047201 (2014)CrossRefGoogle Scholar
  25. 25.
    Demontis, F., Lombardo, S., Sommacal, M., van der Mee, C., Vargiu, F.: Effective generation of closed-form soliton solutions of the continuous classical Heisenberg ferromagnet equation (submitted)Google Scholar
  26. 26.
    Chen, A.-H., Wang, F.-F.: Darboux transformation and exact solutions of the continuous Heisenberg spin chain equation. Z. Nat. Teil A 69, 9–16 (2014)Google Scholar
  27. 27.
    Yersultanova, Z.S., Zhassybayeva, M., Yesmakhanova, K., Nugmanova, G., Myrzakulov, R.: Darboux transformation and exact solutions of the integrable Heisenberg ferromagnetic equation with self-consistent potentials. Int. J. Geom. Methods Mod. Phys. 13, 1550134 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia (1981)CrossRefzbMATHGoogle Scholar
  29. 29.
    Calogero, F., Degasperis, A.: Spectral Transforms and Solitons. North-Holland, Amsterdam (1982)zbMATHGoogle Scholar
  30. 30.
    Faddeev, L.D., Takhtajan, L.A.: Hamiltonian Methods in the Theory of Solitons. Springer, Berlin (1987)CrossRefzbMATHGoogle Scholar
  31. 31.
    Aktosun, T., Demontis, F., van der Mee, C.: Exact solutions to the focusing nonlinear Schrödinger equation. Inverse Prob. 23, 2171–2195 (2007)CrossRefzbMATHGoogle Scholar
  32. 32.
    Demontis, F.: Exact solutions to the modified Korteweg–de Vries equation. Theor. Math. Phys. 168, 886–897 (2011)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Aktosun, T., Demontis, F., van der Mee, C.: Exact solutions to the Sine–Gordon equation. J. Math. Phys. 51, 123521 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Demontis, F., van der Mee, C.: Closed form solutions to the integrable discrete nonlinear Schrödinger equation. J. Nonlinear Math. Phys. 19(2), 1250010 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Demontis, F., Ortenzi, G., van der Mee, C.: Exact solutions of the Hirota equation and vortex filaments motion. Physica D 313, 61–80 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Demontis, F.: Matrix Zakharov–Shabat system and inverse scattering transform (2012), also: Direct and inverse scattering of the matrix Zakharov–Shabat system. Ph.D. Thesis, University of Cagliari, Italy (2007)Google Scholar
  37. 37.
    Dym, H.: Linear Algebra in Action, Vol. 78, Graduate Studies in Mathematics. American Mathematical Society, Providence (2007)Google Scholar
  38. 38.
    van der Mee, C.: Nonlinear evolution models of integrable type, vol. 11, e-Lectures Notes, SIMAI (2013)Google Scholar
  39. 39.
    Egorov, R.F., Bostrem, I.G., Ovchinnikov, A.S.: The variational symmetries and conservation laws in classical theory of Heisenberg (anti)ferromagnet. Phys. Lett. A 292, 325–334 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Zhao, Z., Han, B.: Lie symmetry analysis of the Heisenberg equation. Commun. Nonlinear Sci. Numer. Simul. 45, 220–234 (2017)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Olmedilla, E.: Multiple pole solutions of the non-linear Schrödinger equation. Physica D 25, 330–346 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Schiebold, C.: Asymptotics for the multiple pole solutions of the nonlinear Schrödinger equation. Mid Sweden University, Reports of the Department of Mathematics 1, 51 pages (2014)Google Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2018

Authors and Affiliations

  • F. Demontis
    • 1
  • G. Ortenzi
    • 2
  • M. Sommacal
    • 3
  • C. van der Mee
    • 1
  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi di CagliariCagliariItaly
  2. 2.Dipartimento di Matematica e ApplicazioniUniversità degli Studi di Milano BicoccaMilanItaly
  3. 3.Department of Mathematics, Physics and Electrical EngineeringUniversity of Northumbria at NewcastleNewcastle upon TyneUK

Personalised recommendations