Numerical Simulations of Nonlinear Modes in Mica: Past, Present and Future

  • Janis Bajars
  • J. Chris Eilbeck
  • Ben Leimkuhler
Part of the Springer Series in Materials Science book series (SSMATERIALS, volume 221)


We review research on the role of nonlinear coherent phenomena (e.g. breathers and kinks) in the formation of linear decorations in mica crystal. The work is based on a new model for the motion of the mica hexagonal K layer, which allows displacement of the atoms from the unit cell. With a simple piece-wise polynomial inter-particle potential, we verify the existence of localized long-lived breathers in an idealized lattice at 0 K. Moreover, our model allows us to observe long-lived localized kinks. We study the interactions of such localized modes along a lattice direction, and in addition demonstrate fully two dimensional scattering of such pulses for the first time. For large interatomic forces we observe a spreading horseshoe-shaped wave, a type of shock wave but with a breather profile.


ILM Breather Kink 2D crystals 



JB and BJL acknowledge the support of the Engineering and Physical Sciences Research Council which has funded this work as part of the Numerical Algorithms and Intelligent Software Centre under Grant EP/G036136/1.


  1. 1.
    Allen, M.P., Tildesley, D.J.: Computer Simulation of Liquids. Oxford University Press, USA (1989)Google Scholar
  2. 2.
    Bajars, J., Eilbeck, J.C., Leimkuhler, B.: Nonlinear propagating localized modes in a 2D hexagonal crystal lattice. Physica D 301–302, 8–20 (2015)CrossRefGoogle Scholar
  3. 3.
    Cuevas, J., Katerji, C., Archilla, J.F.R., Eilbeck, J.C., Russell, F.M.: Influence of moving breathers on vacancies migration. Phys. Lett. A 315, 364–371 (2003)CrossRefGoogle Scholar
  4. 4.
    Dou, Q., Cuevas, J., Eilbeck, J.C., Russell, F.M.: Breathers and kinks in a simulated crystal experiment. Discret. Contin. Dyn. S. 4, 1107–1118 (2011)CrossRefGoogle Scholar
  5. 5.
    Duncan, D.B., Eilbeck, J.C., Feddersen, H., Wattis, J.A.D.: Solitons on lattices. Physica D 68, 1–11 (1993)CrossRefGoogle Scholar
  6. 6.
    Eilbeck, J.C.: Numerical simulations of the dynamics of polypeptide chains and proteins. In: Kawabata, C., Bishop, A.R. (eds.) Computer Analysis for Life Science, pp. 12–21. Ohmsha, Tokyo (1986)Google Scholar
  7. 7.
    Eilbeck, J.C., Lomdahl, P.S., Scott, A.C.: Soliton structure in crystalline acetanilide. Phys. Rev. B. 30, 4703–4712 (1984)CrossRefGoogle Scholar
  8. 8.
    Eilbeck, J.C., Lomdahl, P.S., Scott, A.C.: The discrete self-trapping equation. Physica D 16, 318–338 (1985)CrossRefGoogle Scholar
  9. 9.
    Feddersen, H.: Solitary wave solutions to the discrete nonlinear Schrödinger equation. In: Remoissenet, M., Peyrard, M. (eds.) Nonlinear Coherent Structures in Physics and Biology, pp. 159–167. Springer (1991)Google Scholar
  10. 10.
    Flach, S.: Discrete breathers in a nutshell. NOLTA 3, 12–26 (2012)CrossRefGoogle Scholar
  11. 11.
    Flach, S., Willis, C.: Discrete breathers. Phys. Rep. 295, 181–264 (1998)CrossRefGoogle Scholar
  12. 12.
    Geim, A.K., Grigorieva, I.V.: Van der Waals heterostructures. Nature 499, 419–425 (2013)CrossRefGoogle Scholar
  13. 13.
    Leimkuhler, B., Reich, S.: Simulating Hamiltonian Dynamics. Cambridge University Press, Cambridge (2005)Google Scholar
  14. 14.
    MacKay, R., Aubry, S.: Proof of existence of breathers for time-reversible or hamiltonian networks of weakly coupled oscillators. Nonlinearity 7, 1623–1643 (1994)CrossRefGoogle Scholar
  15. 15.
    Manton, N., Sutcliffe, P.: Topological Solitons. Cambridge University Press, Cambridge (2004)Google Scholar
  16. 16.
    Marín, J.L., Eilbeck, J.C., Russell, F.M.: Localised moving breathers in a 2-D hexagonal lattice. Phys. Lett. A 248, 225–229 (1998)CrossRefGoogle Scholar
  17. 17.
    Marín, J.L., Eilbeck, J.C., Russell, F.M.: 2-D breathers and applications. In: Christiansen, P.L., Sørensen, M.P., Scott, A.C. (eds.) Nonlinear Science at the Dawn of the 21st century, pp. 293–306. Springer (2000)Google Scholar
  18. 18.
    Marín, J.L., Russell, F.M., Eilbeck, J.C.: Breathers in cuprate superconductor lattices. Phys. Lett. A 281, 225–229 (2001)CrossRefGoogle Scholar
  19. 19.
    Ovchinnikov, A.A.: Localized long-lived vibrational states in molecular crystals. Sov. Phys. JETP 30, 147–150 (1970)Google Scholar
  20. 20.
    Peyrard, M., Kruskal, M.D.: Kink dynamics in the highly discrete sine-gordon system. Physica D 14, 88–102 (1984)CrossRefGoogle Scholar
  21. 21.
    Russell, F.M., Collins, D.R.: Lattice-solitons and non-linear phenomena in track formation. Radiat. Meas. 25, 67–70 (1995)CrossRefGoogle Scholar
  22. 22.
    Russell, F.M., Eilbeck, J.C.: Evidence for moving breathers in a layered crystal insulator at 300k. Phys. Lett. A 78, 10004 (2007)Google Scholar
  23. 23.
    Scott, A.C., MacNeil, L.: Binding energy versus nonlinearity for a small stationary soliton. Phys. Lett. A 98, 87–88 (1983)CrossRefGoogle Scholar
  24. 24.
    Yang, Y., Duan, W.S., Yang, L., Chen, J.M., Lin, M.M.: Rectification and phase locking in overdamped two-dimensional Frenkel-Kontorova model. Europhys. Lett. 93(1), 16001 (2011)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Janis Bajars
    • 1
  • J. Chris Eilbeck
    • 2
  • Ben Leimkuhler
    • 3
  1. 1.College of Arts and Science, School of Science & TechnologyNottingham Trent UniversityNottinghamUK
  2. 2.Maxwell Institute and Department of MathematicsHeriot-Watt UniversityEdinburghUK
  3. 3.Maxwell Institute and School of MathematicsUniversity of EdinburghEdinburghUK

Personalised recommendations