Abstract

We analyze the existence, stability, and propagation of dissipative discrete localized modes in one- and two-dimensional nonlinear lattices composed of weakly coupled split-ring resonators (SRRs) excited by an external electromagnetic field. We employ the near-field interaction approach for describing quasi-static electric and magnetic interaction between the resonators, and demonstrate the crucial importance of the electric coupling, which can completely reverse the sign of the overall interaction between the resonators. We derive the effective nonlinear model and analyze the properties of nonlinear localized modes excited in one-and two-dimensional lattices. In particular, we study nonlinear magnetic domain walls (the so-called switching waves) separating two different states of nonlinear magnetization, and reveal the bistable dependence of the domain wall velocity on the external field. Then, we study two-dimensional localized modes in nonlinear lattices of SRRs and demonstrate that larger domains may experience modulational instability and splitting.

© 2011 OSA

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  2. D. A. Powell, M. Lapine, M. V. Gorkunov, I. V. Shadrivov, and Yu. S. Kivshar, “Metamaterial tuning by manipulation of near-field interaction,” Phys. Rev. B 82, 155128 (2010).
    [CrossRef]
  3. M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B 28, 263–269 (2002).
    [CrossRef]
  4. A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011).
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    [CrossRef]
  7. J. S. Hong and J. Lancaster, “Couplings of microstrip square open-loop resonators for cross-coupled planar microwave filters,” IEEE Trans. Microw. Theory Tech. 44, 2099–2109 (1996).
    [CrossRef]
  8. E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magneto-inductive waveguide,” Electron. Lett. 38, 371–373 (2002).
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  9. M. Beruete, F. Falcone, M. J. Freire, R. Marqus, and J. D. Baena, “Electroinductive Waves in Chain of Complementary Metamaterials Elements,” Appl. Phys. Lett. 88, 083503 (2006).
    [CrossRef]
  10. N. Lazarides, M. Eleftheriou, and G. P. Tsironis, “Discrete breathers in nonlinear magnetic metamaterials,” Phys. Rev. Lett. 97, 157406 (2006).
    [CrossRef] [PubMed]
  11. M. Eleftheriou, N. Lazarides, and G. P. Tsironis, “Magnetoinductive breathers in metamaterials,” Phys. Rev. E 77, 036608 (2008).
    [CrossRef]
  12. N. Lazarides, G. P. Tsironis, and Yu. S. Kivshar, “Surface breathers in discrete magnetic metamaterials,” Phys. Rev. E 77, 065601(R) (2008).
    [CrossRef]
  13. M. Molina, N. Lazarides, and G. P. Tsironis, “Bulk and surface magnetoinductive breathers in binary metamaterials,” Phys. Rev. E 80, 046605 (2009).
    [CrossRef]
  14. W. Cui, Y. Zhu, H. Li, and S. Liu, “Soliton excitations in a one-dimensional nonlinear diatomic chain of split-ring resonators,” Phys. Rev. E 81, 016604 (2010).
    [CrossRef]
  15. D. A. Powell, K. Hannam, I. V. Shadrivov, and Yu. S. Kivshar, “Near-field interaction of twisted split-ring resonators,” Phys. Rev. B 83, 235420 (2011).
    [CrossRef]
  16. F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B 244, 1170–1175 (2007).
    [CrossRef]
  17. M. D. Turner, Md M. Hossain, and M. Gu, “The effects of metallic nanostructures,” New J. Phys. 12, 083062 (2010).
    [CrossRef]
  18. N. N. Rosanov, N. V. Vysotina, A. N. Shatsev, I. V. Shadrivov, and Yu.S. Kivshar, “Hysteresis of switching waves and dissipative solitons in nonlinear magnetic metamaterials,” JETP Lett. 93, 743–746 (2011).
    [CrossRef]
  19. N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, Berlin, 2002).
  20. Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 (2007).
    [CrossRef] [PubMed]

2011

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011).
[CrossRef]

D. A. Powell, K. Hannam, I. V. Shadrivov, and Yu. S. Kivshar, “Near-field interaction of twisted split-ring resonators,” Phys. Rev. B 83, 235420 (2011).
[CrossRef]

N. N. Rosanov, N. V. Vysotina, A. N. Shatsev, I. V. Shadrivov, and Yu.S. Kivshar, “Hysteresis of switching waves and dissipative solitons in nonlinear magnetic metamaterials,” JETP Lett. 93, 743–746 (2011).
[CrossRef]

2010

W. Cui, Y. Zhu, H. Li, and S. Liu, “Soliton excitations in a one-dimensional nonlinear diatomic chain of split-ring resonators,” Phys. Rev. E 81, 016604 (2010).
[CrossRef]

M. D. Turner, Md M. Hossain, and M. Gu, “The effects of metallic nanostructures,” New J. Phys. 12, 083062 (2010).
[CrossRef]

D. A. Powell, M. Lapine, M. V. Gorkunov, I. V. Shadrivov, and Yu. S. Kivshar, “Metamaterial tuning by manipulation of near-field interaction,” Phys. Rev. B 82, 155128 (2010).
[CrossRef]

2009

M. Molina, N. Lazarides, and G. P. Tsironis, “Bulk and surface magnetoinductive breathers in binary metamaterials,” Phys. Rev. E 80, 046605 (2009).
[CrossRef]

2008

M. Eleftheriou, N. Lazarides, and G. P. Tsironis, “Magnetoinductive breathers in metamaterials,” Phys. Rev. E 77, 036608 (2008).
[CrossRef]

N. Lazarides, G. P. Tsironis, and Yu. S. Kivshar, “Surface breathers in discrete magnetic metamaterials,” Phys. Rev. E 77, 065601(R) (2008).
[CrossRef]

2007

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B 244, 1170–1175 (2007).
[CrossRef]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

2006

M. Beruete, F. Falcone, M. J. Freire, R. Marqus, and J. D. Baena, “Electroinductive Waves in Chain of Complementary Metamaterials Elements,” Appl. Phys. Lett. 88, 083503 (2006).
[CrossRef]

N. Lazarides, M. Eleftheriou, and G. P. Tsironis, “Discrete breathers in nonlinear magnetic metamaterials,” Phys. Rev. Lett. 97, 157406 (2006).
[CrossRef] [PubMed]

I. V. Shadrivov, A. A. Zharov, N. A. Zharova, and Yu. S. Kivshar, “Nonlinear magnetoinductive waves and domain walls in composite metamaterials,” Photonics Nanostruct. Fundam. Appl. 4, 69–74 (2006).
[CrossRef]

2002

M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B 28, 263–269 (2002).
[CrossRef]

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magneto-inductive waveguide,” Electron. Lett. 38, 371–373 (2002).
[CrossRef]

1996

J. S. Hong and J. Lancaster, “Couplings of microstrip square open-loop resonators for cross-coupled planar microwave filters,” IEEE Trans. Microw. Theory Tech. 44, 2099–2109 (1996).
[CrossRef]

Baena, J. D.

M. Beruete, F. Falcone, M. J. Freire, R. Marqus, and J. D. Baena, “Electroinductive Waves in Chain of Complementary Metamaterials Elements,” Appl. Phys. Lett. 88, 083503 (2006).
[CrossRef]

Bartal, G.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

Belov, P. A.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011).
[CrossRef]

Beruete, M.

M. Beruete, F. Falcone, M. J. Freire, R. Marqus, and J. D. Baena, “Electroinductive Waves in Chain of Complementary Metamaterials Elements,” Appl. Phys. Lett. 88, 083503 (2006).
[CrossRef]

Cui, W.

W. Cui, Y. Zhu, H. Li, and S. Liu, “Soliton excitations in a one-dimensional nonlinear diatomic chain of split-ring resonators,” Phys. Rev. E 81, 016604 (2010).
[CrossRef]

Edwards, D. J.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B 244, 1170–1175 (2007).
[CrossRef]

Eleftheriou, M.

M. Eleftheriou, N. Lazarides, and G. P. Tsironis, “Magnetoinductive breathers in metamaterials,” Phys. Rev. E 77, 036608 (2008).
[CrossRef]

N. Lazarides, M. Eleftheriou, and G. P. Tsironis, “Discrete breathers in nonlinear magnetic metamaterials,” Phys. Rev. Lett. 97, 157406 (2006).
[CrossRef] [PubMed]

Falcone, F.

M. Beruete, F. Falcone, M. J. Freire, R. Marqus, and J. D. Baena, “Electroinductive Waves in Chain of Complementary Metamaterials Elements,” Appl. Phys. Lett. 88, 083503 (2006).
[CrossRef]

Faulkner, G.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B 244, 1170–1175 (2007).
[CrossRef]

Freire, M. J.

M. Beruete, F. Falcone, M. J. Freire, R. Marqus, and J. D. Baena, “Electroinductive Waves in Chain of Complementary Metamaterials Elements,” Appl. Phys. Lett. 88, 083503 (2006).
[CrossRef]

Genov, D. A.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

Gorkunov, M.

M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B 28, 263–269 (2002).
[CrossRef]

Gorkunov, M. V.

D. A. Powell, M. Lapine, M. V. Gorkunov, I. V. Shadrivov, and Yu. S. Kivshar, “Metamaterial tuning by manipulation of near-field interaction,” Phys. Rev. B 82, 155128 (2010).
[CrossRef]

Gu, M.

M. D. Turner, Md M. Hossain, and M. Gu, “The effects of metallic nanostructures,” New J. Phys. 12, 083062 (2010).
[CrossRef]

Hannam, K.

D. A. Powell, K. Hannam, I. V. Shadrivov, and Yu. S. Kivshar, “Near-field interaction of twisted split-ring resonators,” Phys. Rev. B 83, 235420 (2011).
[CrossRef]

Hao, T.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B 244, 1170–1175 (2007).
[CrossRef]

Hesmer, F.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B 244, 1170–1175 (2007).
[CrossRef]

Hong, J. S.

J. S. Hong and J. Lancaster, “Couplings of microstrip square open-loop resonators for cross-coupled planar microwave filters,” IEEE Trans. Microw. Theory Tech. 44, 2099–2109 (1996).
[CrossRef]

Hossain, Md M.

M. D. Turner, Md M. Hossain, and M. Gu, “The effects of metallic nanostructures,” New J. Phys. 12, 083062 (2010).
[CrossRef]

Kalinin, V. A.

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magneto-inductive waveguide,” Electron. Lett. 38, 371–373 (2002).
[CrossRef]

Kivshar, Yu. S.

D. A. Powell, K. Hannam, I. V. Shadrivov, and Yu. S. Kivshar, “Near-field interaction of twisted split-ring resonators,” Phys. Rev. B 83, 235420 (2011).
[CrossRef]

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011).
[CrossRef]

D. A. Powell, M. Lapine, M. V. Gorkunov, I. V. Shadrivov, and Yu. S. Kivshar, “Metamaterial tuning by manipulation of near-field interaction,” Phys. Rev. B 82, 155128 (2010).
[CrossRef]

N. Lazarides, G. P. Tsironis, and Yu. S. Kivshar, “Surface breathers in discrete magnetic metamaterials,” Phys. Rev. E 77, 065601(R) (2008).
[CrossRef]

I. V. Shadrivov, A. A. Zharov, N. A. Zharova, and Yu. S. Kivshar, “Nonlinear magnetoinductive waves and domain walls in composite metamaterials,” Photonics Nanostruct. Fundam. Appl. 4, 69–74 (2006).
[CrossRef]

Kivshar, Yu.S.

N. N. Rosanov, N. V. Vysotina, A. N. Shatsev, I. V. Shadrivov, and Yu.S. Kivshar, “Hysteresis of switching waves and dissipative solitons in nonlinear magnetic metamaterials,” JETP Lett. 93, 743–746 (2011).
[CrossRef]

Lancaster, J.

J. S. Hong and J. Lancaster, “Couplings of microstrip square open-loop resonators for cross-coupled planar microwave filters,” IEEE Trans. Microw. Theory Tech. 44, 2099–2109 (1996).
[CrossRef]

Lapine, M.

D. A. Powell, M. Lapine, M. V. Gorkunov, I. V. Shadrivov, and Yu. S. Kivshar, “Metamaterial tuning by manipulation of near-field interaction,” Phys. Rev. B 82, 155128 (2010).
[CrossRef]

M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B 28, 263–269 (2002).
[CrossRef]

Lazarides, N.

M. Molina, N. Lazarides, and G. P. Tsironis, “Bulk and surface magnetoinductive breathers in binary metamaterials,” Phys. Rev. E 80, 046605 (2009).
[CrossRef]

M. Eleftheriou, N. Lazarides, and G. P. Tsironis, “Magnetoinductive breathers in metamaterials,” Phys. Rev. E 77, 036608 (2008).
[CrossRef]

N. Lazarides, G. P. Tsironis, and Yu. S. Kivshar, “Surface breathers in discrete magnetic metamaterials,” Phys. Rev. E 77, 065601(R) (2008).
[CrossRef]

N. Lazarides, M. Eleftheriou, and G. P. Tsironis, “Discrete breathers in nonlinear magnetic metamaterials,” Phys. Rev. Lett. 97, 157406 (2006).
[CrossRef] [PubMed]

Li, H.

W. Cui, Y. Zhu, H. Li, and S. Liu, “Soliton excitations in a one-dimensional nonlinear diatomic chain of split-ring resonators,” Phys. Rev. E 81, 016604 (2010).
[CrossRef]

Liu, S.

W. Cui, Y. Zhu, H. Li, and S. Liu, “Soliton excitations in a one-dimensional nonlinear diatomic chain of split-ring resonators,” Phys. Rev. E 81, 016604 (2010).
[CrossRef]

Liu, Y.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

Marqus, R.

M. Beruete, F. Falcone, M. J. Freire, R. Marqus, and J. D. Baena, “Electroinductive Waves in Chain of Complementary Metamaterials Elements,” Appl. Phys. Lett. 88, 083503 (2006).
[CrossRef]

Molina, M.

M. Molina, N. Lazarides, and G. P. Tsironis, “Bulk and surface magnetoinductive breathers in binary metamaterials,” Phys. Rev. E 80, 046605 (2009).
[CrossRef]

Orlov, A. A.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011).
[CrossRef]

Powell, D. A.

D. A. Powell, K. Hannam, I. V. Shadrivov, and Yu. S. Kivshar, “Near-field interaction of twisted split-ring resonators,” Phys. Rev. B 83, 235420 (2011).
[CrossRef]

D. A. Powell, M. Lapine, M. V. Gorkunov, I. V. Shadrivov, and Yu. S. Kivshar, “Metamaterial tuning by manipulation of near-field interaction,” Phys. Rev. B 82, 155128 (2010).
[CrossRef]

Radkovskaya, A. A.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B 244, 1170–1175 (2007).
[CrossRef]

Ringhofer, K. H.

M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B 28, 263–269 (2002).
[CrossRef]

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magneto-inductive waveguide,” Electron. Lett. 38, 371–373 (2002).
[CrossRef]

Rosanov, N. N.

N. N. Rosanov, N. V. Vysotina, A. N. Shatsev, I. V. Shadrivov, and Yu.S. Kivshar, “Hysteresis of switching waves and dissipative solitons in nonlinear magnetic metamaterials,” JETP Lett. 93, 743–746 (2011).
[CrossRef]

N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, Berlin, 2002).

Shadrivov, I. V.

D. A. Powell, K. Hannam, I. V. Shadrivov, and Yu. S. Kivshar, “Near-field interaction of twisted split-ring resonators,” Phys. Rev. B 83, 235420 (2011).
[CrossRef]

N. N. Rosanov, N. V. Vysotina, A. N. Shatsev, I. V. Shadrivov, and Yu.S. Kivshar, “Hysteresis of switching waves and dissipative solitons in nonlinear magnetic metamaterials,” JETP Lett. 93, 743–746 (2011).
[CrossRef]

D. A. Powell, M. Lapine, M. V. Gorkunov, I. V. Shadrivov, and Yu. S. Kivshar, “Metamaterial tuning by manipulation of near-field interaction,” Phys. Rev. B 82, 155128 (2010).
[CrossRef]

I. V. Shadrivov, A. A. Zharov, N. A. Zharova, and Yu. S. Kivshar, “Nonlinear magnetoinductive waves and domain walls in composite metamaterials,” Photonics Nanostruct. Fundam. Appl. 4, 69–74 (2006).
[CrossRef]

Shamonin, M.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B 244, 1170–1175 (2007).
[CrossRef]

Shamonina, E.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B 244, 1170–1175 (2007).
[CrossRef]

M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B 28, 263–269 (2002).
[CrossRef]

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magneto-inductive waveguide,” Electron. Lett. 38, 371–373 (2002).
[CrossRef]

Shatsev, A. N.

N. N. Rosanov, N. V. Vysotina, A. N. Shatsev, I. V. Shadrivov, and Yu.S. Kivshar, “Hysteresis of switching waves and dissipative solitons in nonlinear magnetic metamaterials,” JETP Lett. 93, 743–746 (2011).
[CrossRef]

Solymar, L.

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magneto-inductive waveguide,” Electron. Lett. 38, 371–373 (2002).
[CrossRef]

Stevens, C. J.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B 244, 1170–1175 (2007).
[CrossRef]

Tatartschuk, E.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B 244, 1170–1175 (2007).
[CrossRef]

Tsironis, G. P.

M. Molina, N. Lazarides, and G. P. Tsironis, “Bulk and surface magnetoinductive breathers in binary metamaterials,” Phys. Rev. E 80, 046605 (2009).
[CrossRef]

M. Eleftheriou, N. Lazarides, and G. P. Tsironis, “Magnetoinductive breathers in metamaterials,” Phys. Rev. E 77, 036608 (2008).
[CrossRef]

N. Lazarides, G. P. Tsironis, and Yu. S. Kivshar, “Surface breathers in discrete magnetic metamaterials,” Phys. Rev. E 77, 065601(R) (2008).
[CrossRef]

N. Lazarides, M. Eleftheriou, and G. P. Tsironis, “Discrete breathers in nonlinear magnetic metamaterials,” Phys. Rev. Lett. 97, 157406 (2006).
[CrossRef] [PubMed]

Turner, M. D.

M. D. Turner, Md M. Hossain, and M. Gu, “The effects of metallic nanostructures,” New J. Phys. 12, 083062 (2010).
[CrossRef]

Voroshilov, P. M.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011).
[CrossRef]

Vysotina, N. V.

N. N. Rosanov, N. V. Vysotina, A. N. Shatsev, I. V. Shadrivov, and Yu.S. Kivshar, “Hysteresis of switching waves and dissipative solitons in nonlinear magnetic metamaterials,” JETP Lett. 93, 743–746 (2011).
[CrossRef]

Zhang, X.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

Zharov, A. A.

I. V. Shadrivov, A. A. Zharov, N. A. Zharova, and Yu. S. Kivshar, “Nonlinear magnetoinductive waves and domain walls in composite metamaterials,” Photonics Nanostruct. Fundam. Appl. 4, 69–74 (2006).
[CrossRef]

Zharova, N. A.

I. V. Shadrivov, A. A. Zharov, N. A. Zharova, and Yu. S. Kivshar, “Nonlinear magnetoinductive waves and domain walls in composite metamaterials,” Photonics Nanostruct. Fundam. Appl. 4, 69–74 (2006).
[CrossRef]

Zhu, Y.

W. Cui, Y. Zhu, H. Li, and S. Liu, “Soliton excitations in a one-dimensional nonlinear diatomic chain of split-ring resonators,” Phys. Rev. E 81, 016604 (2010).
[CrossRef]

Zhuromskyy, O.

F. Hesmer, E. Tatartschuk, O. Zhuromskyy, A. A. Radkovskaya, M. Shamonin, T. Hao, C. J. Stevens, G. Faulkner, D. J. Edwards, and E. Shamonina, “Coupling mechanisms for split ring resonators: Theory and experiment,” Phys. Status Solidi B 244, 1170–1175 (2007).
[CrossRef]

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M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B 28, 263–269 (2002).
[CrossRef]

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[CrossRef]

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N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, Berlin, 2002).

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Figures (5)

Fig. 1
Fig. 1

Electric and magnetic interaction coefficients (solid and dashed curves, respectively) as functions of the spacing between the ring centers for the normalized frequency Ω = −0.2, when the rings are offset in (a) x direction, (b) y direction, and (c) z -direction. Insets show schematically corresponding ring positions.

Fig. 3
Fig. 3

(a) Dependence of the magnetization on the index n for S = 0.024. Solid - standing, and dashed - moving domain wall, corresponding to the top and bottom branches on the panel (c) in this Figure. (b) Dependence of the velocity of the domain wall as a function of the external excitation S. (c) shows close up of the bistable region in panel (b), with arrows indicating directions of the jumps. All results are obtained for Ω = −0.2, Kz = −0.02.

Fig. 2
Fig. 2

Examples of (a,b) arrays and (c,d) two-dimensional lattices of weakly coupled nonlinear split-ring resonators creating the simplest nonlinear magnetic metamaterials.

Fig. 4
Fig. 4

Examples of a nonlinear localized mode in a two-dimensional lattice of SRRs. α = 1, Ω = −0.2, γ = 0.024, S = 0.058. (a) soliton with minimum possible size of excited 1x1 site (b) soliton with the size of 5x1.

Fig. 5
Fig. 5

(a) Initial and (b) final state of the two-dimensional magnetic domain in a SRR lattice in the case of anisotropic modulational instability. The domain decays in two identical modes of a smaller size. Parameters are: α = 1, Ω = −0.2, γ = 0.032, S = 0.063.

Equations (2)

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L d I n , q , m d t + R I n , q , m + Q n , q , m C = n , q , m j ( M j d I j d t + P j Q j ) ,
i d ψ n , q , m d τ ( 2 Ω i γ + α | ψ n , q , m | 2 ) ψ n , q , m = S n , q , m + j K j ψ j ,

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