Abstract

We consider the generalized nonlinear Schrödinger equation describing the propagation of ultrashort pulses in nonlinear metamaterials and find exact combined solitary wave solutions describing the simultaneous propagation of bright and dark solitary waves in a combined form. Based on the Drude model, we discuss the existence conditions and the properties of the combined solitary waves. Furthermore, we analytically and numerically investigate the stability of the combined solitary waves under small perturbations. The results show that bright and dark solitary waves may combine together under certain conditions and propagate stably with a proper form in metamaterials.

© 2011 Optical Society of America

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  1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
    [CrossRef]
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    [CrossRef] [PubMed]
  3. C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
    [CrossRef] [PubMed]
  4. C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refraction index at optical wavelengths,” Science 315, 47–49 (2007).
    [CrossRef] [PubMed]
  5. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photon. 1, 41–48 (2007).
    [CrossRef]
  6. J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
    [CrossRef] [PubMed]
  7. A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
    [CrossRef] [PubMed]
  8. M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
    [CrossRef]
  9. N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 036614 (2005).
    [CrossRef]
  10. M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
    [CrossRef] [PubMed]
  11. S. C. Wen, Y. J. Xiang, X. Y. Dai, Z. X. TangW. H. Su, and D. Y. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
    [CrossRef]
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    [CrossRef]
  13. I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005).
    [CrossRef]
  14. Y. J. Xiang, S. C. Wen, X. Y. Dai, Z. X. Tang, W. H. Su, and D. Y. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063(2007).
    [CrossRef]
  15. G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Bright and dark gap solitons in a negative index Fabry–Perot etalon,” Phys. Rev. Lett. 93, 213902 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  21. Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, “New types of solitary wave solutions for the higher order nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
    [CrossRef] [PubMed]
  22. R. C. Yang, L. Li, R. Y. Hao, Z. H. Li, and G. S. Zhou, “Combined solitary wave solutions for the inhomogeneous higher-order nonlinear Schrödinger equation,” Phys. Rev. E 71, 036616 (2005).
    [CrossRef]
  23. R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
    [CrossRef]
  24. A. K. Popov and V. M. Shalaev, “Compensating losses in negative-index metamaterials by optical parametric amplification,” Opt. Lett. 31, 2169–2171 (2006).
    [CrossRef] [PubMed]
  25. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006).
    [CrossRef] [PubMed]

2009 (1)

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
[CrossRef]

2008 (1)

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

2007 (4)

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refraction index at optical wavelengths,” Science 315, 47–49 (2007).
[CrossRef] [PubMed]

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photon. 1, 41–48 (2007).
[CrossRef]

S. C. Wen, Y. J. Xiang, X. Y. Dai, Z. X. TangW. H. Su, and D. Y. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Y. J. Xiang, S. C. Wen, X. Y. Dai, Z. X. Tang, W. H. Su, and D. Y. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063(2007).
[CrossRef]

2006 (4)

2005 (6)

R. C. Yang, L. Li, R. Y. Hao, Z. H. Li, and G. S. Zhou, “Combined solitary wave solutions for the inhomogeneous higher-order nonlinear Schrödinger equation,” Phys. Rev. E 71, 036616 (2005).
[CrossRef]

N. A. Zharova, I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Nonlinear transmission and spatiotemporal solitons in metamaterials with negative refraction,” Opt. Express 13, 1291–1298 (2005).
[CrossRef] [PubMed]

A. D. Boardman, L. Velasco, and N. King, “Ultra-narrow bright spatial solitons interacting with left-handed surfaces,” J. Opt. Soc. Am. B 22, 1443–1452 (2005).
[CrossRef]

I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005).
[CrossRef]

N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 036614 (2005).
[CrossRef]

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

2004 (1)

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Bright and dark gap solitons in a negative index Fabry–Perot etalon,” Phys. Rev. Lett. 93, 213902 (2004).
[CrossRef] [PubMed]

2003 (3)

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[CrossRef]

2001 (1)

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

2000 (2)

Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, “New types of solitary wave solutions for the higher order nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Akozbek, N.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Banerjee, P. P.

Bartal, G.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Bloemer, M. J.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Bright and dark gap solitons in a negative index Fabry–Perot etalon,” Phys. Rev. Lett. 93, 213902 (2004).
[CrossRef] [PubMed]

Boardman, A. D.

D’Aguanno, G.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Bright and dark gap solitons in a negative index Fabry–Perot etalon,” Phys. Rev. Lett. 93, 213902 (2004).
[CrossRef] [PubMed]

Dai, X. Y.

Y. J. Xiang, S. C. Wen, X. Y. Dai, Z. X. Tang, W. H. Su, and D. Y. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063(2007).
[CrossRef]

S. C. Wen, Y. J. Xiang, X. Y. Dai, Z. X. TangW. H. Su, and D. Y. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Dolling, G.

Enkrich, C.

Fan, D. Y.

S. C. Wen, Y. J. Xiang, X. Y. Dai, Z. X. TangW. H. Su, and D. Y. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Y. J. Xiang, S. C. Wen, X. Y. Dai, Z. X. Tang, W. H. Su, and D. Y. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063(2007).
[CrossRef]

Frantzeskakis, D. J.

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
[CrossRef]

Gorkunov, M.

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[CrossRef]

Greegor, R. B.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Hao, R. Y.

R. C. Yang, L. Li, R. Y. Hao, Z. H. Li, and G. S. Zhou, “Combined solitary wave solutions for the inhomogeneous higher-order nonlinear Schrödinger equation,” Phys. Rev. E 71, 036616 (2005).
[CrossRef]

Heyman, E.

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

Kevrekidis, P. G.

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
[CrossRef]

King, N.

Kivshar, Y. S.

Kivshar, Yu. S.

A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

Koltenbah, B. E. C.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Kourakis, I.

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
[CrossRef]

I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005).
[CrossRef]

Lapine, M.

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[CrossRef]

Lazarides, N.

N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 036614 (2005).
[CrossRef]

Li, K.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Li, L.

R. C. Yang, L. Li, R. Y. Hao, Z. H. Li, and G. S. Zhou, “Combined solitary wave solutions for the inhomogeneous higher-order nonlinear Schrödinger equation,” Phys. Rev. E 71, 036616 (2005).
[CrossRef]

Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, “New types of solitary wave solutions for the higher order nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
[CrossRef] [PubMed]

Li, Z. H.

R. C. Yang, L. Li, R. Y. Hao, Z. H. Li, and G. S. Zhou, “Combined solitary wave solutions for the inhomogeneous higher-order nonlinear Schrödinger equation,” Phys. Rev. E 71, 036616 (2005).
[CrossRef]

Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, “New types of solitary wave solutions for the higher order nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
[CrossRef] [PubMed]

Linden, S.

Liu, Y. M.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Liu, Z. W.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Marklund, M.

M. Marklund, P. K. Shukla, and L. Stenflo, “Ultrashort solitons and kinetic effects in nonlinear metamaterials,” Phys. Rev. E 73, 037601 (2006).
[CrossRef]

Mattiucci, N.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Bright and dark gap solitons in a negative index Fabry–Perot etalon,” Phys. Rev. Lett. 93, 213902 (2004).
[CrossRef] [PubMed]

Nehmetallah, G.

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Parazzoli, C. G.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Poliakov, E. Y.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Popov, A. K.

Ringhofer, K. H.

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[CrossRef]

Rompotis, N.

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
[CrossRef]

Scalora, M.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Bright and dark gap solitons in a negative index Fabry–Perot etalon,” Phys. Rev. Lett. 93, 213902 (2004).
[CrossRef] [PubMed]

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Shadrivov, I. V.

Shalaev, V. M.

Shukla, P. K.

M. Marklund, P. K. Shukla, and L. Stenflo, “Ultrashort solitons and kinetic effects in nonlinear metamaterials,” Phys. Rev. E 73, 037601 (2006).
[CrossRef]

I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005).
[CrossRef]

Smith, D. R.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Soukoulis, C. M.

Stacy, A.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Stenflo, L.

M. Marklund, P. K. Shukla, and L. Stenflo, “Ultrashort solitons and kinetic effects in nonlinear metamaterials,” Phys. Rev. E 73, 037601 (2006).
[CrossRef]

Su, W. H.

Y. J. Xiang, S. C. Wen, X. Y. Dai, Z. X. Tang, W. H. Su, and D. Y. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063(2007).
[CrossRef]

S. C. Wen, Y. J. Xiang, X. Y. Dai, Z. X. TangW. H. Su, and D. Y. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Sun, C.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Syrchin, M. S.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Tang, Z. X.

S. C. Wen, Y. J. Xiang, X. Y. Dai, Z. X. TangW. H. Su, and D. Y. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Y. J. Xiang, S. C. Wen, X. Y. Dai, Z. X. Tang, W. H. Su, and D. Y. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063(2007).
[CrossRef]

Tanielian, M.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Tian, H. P.

Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, “New types of solitary wave solutions for the higher order nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
[CrossRef] [PubMed]

Tsironis, G. P.

N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 036614 (2005).
[CrossRef]

Tsitsas, N. L.

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
[CrossRef]

Velasco, L.

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Wang, Y.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Wegener, M.

Wen, S. C.

Y. J. Xiang, S. C. Wen, X. Y. Dai, Z. X. Tang, W. H. Su, and D. Y. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063(2007).
[CrossRef]

S. C. Wen, Y. J. Xiang, X. Y. Dai, Z. X. TangW. H. Su, and D. Y. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Xiang, Y. J.

S. C. Wen, Y. J. Xiang, X. Y. Dai, Z. X. TangW. H. Su, and D. Y. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Y. J. Xiang, S. C. Wen, X. Y. Dai, Z. X. Tang, W. H. Su, and D. Y. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063(2007).
[CrossRef]

Yang, R. C.

R. C. Yang, L. Li, R. Y. Hao, Z. H. Li, and G. S. Zhou, “Combined solitary wave solutions for the inhomogeneous higher-order nonlinear Schrödinger equation,” Phys. Rev. E 71, 036616 (2005).
[CrossRef]

Yao, J.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Zhang, X.

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Zharov, A. A.

Zharova, N. A.

Zheltikov, A. M.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Zhou, G. S.

R. C. Yang, L. Li, R. Y. Hao, Z. H. Li, and G. S. Zhou, “Combined solitary wave solutions for the inhomogeneous higher-order nonlinear Schrödinger equation,” Phys. Rev. E 71, 036616 (2005).
[CrossRef]

Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, “New types of solitary wave solutions for the higher order nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
[CrossRef] [PubMed]

Ziolkowski, R. W.

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

J. Opt. Soc. Am. B (3)

Nat. Photon. (1)

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photon. 1, 41–48 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

S. C. Wen, Y. J. Xiang, X. Y. Dai, Z. X. TangW. H. Su, and D. Y. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Phys. Rev. E (7)

R. C. Yang, L. Li, R. Y. Hao, Z. H. Li, and G. S. Zhou, “Combined solitary wave solutions for the inhomogeneous higher-order nonlinear Schrödinger equation,” Phys. Rev. E 71, 036616 (2005).
[CrossRef]

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
[CrossRef]

M. Marklund, P. K. Shukla, and L. Stenflo, “Ultrashort solitons and kinetic effects in nonlinear metamaterials,” Phys. Rev. E 73, 037601 (2006).
[CrossRef]

I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005).
[CrossRef]

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[CrossRef]

N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 036614 (2005).
[CrossRef]

Phys. Rev. Lett. (6)

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Bright and dark gap solitons in a negative index Fabry–Perot etalon,” Phys. Rev. Lett. 93, 213902 (2004).
[CrossRef] [PubMed]

Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, “New types of solitary wave solutions for the higher order nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
[CrossRef] [PubMed]

Science (2)

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refraction index at optical wavelengths,” Science 315, 47–49 (2007).
[CrossRef] [PubMed]

J. Yao, Z. W. Liu, Y. M. Liu, Y. Wang, C. Sun, G. Bartal, A. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

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Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

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Figures (6)

Fig. 1
Fig. 1

Curves of k 2 , p 5 , and s 1 in Eq. (1) versus ω ˜ for ω m / ω p = 0.8 . Here s 1 in units of χ ( 3 ) , p 5 in units of ( χ ( 3 ) ) 2 .

Fig. 2
Fig. 2

Evolution plots of exact (a) brightlike and (b) darklike combined solitary wave solution (6) under strict existence condition with the parameters as follows: λ = 0.4 × 10 5 , ρ = 0.2 × 10 5 , k 2 = 0 . 2066 , ρ 3 = 3.629 × 10 10 , s 1 = 0.5 × 10 10 , corresponding to ω ˜ = 1.155 , ω m / ω p = 1.0 .

Fig. 3
Fig. 3

Initial pulses (solid curve), numerical (dashed curve), and exact (dotted curve) output profiles of brightlike and darklike combined solitary waves (6) under the slight violations of strict condition. The adopted parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

Numerical evolutions of brightlike and darklike combined solitary wave (6) (a), (c) under the perturbation of random noise; (b), (d) the fluctuation of pulse width smaller than the theoretical prediction. The adopted parameters are the same as in Fig. 2.

Fig. 5
Fig. 5

(a) Evolution plots of the solution (12) under the existence condition. (b) Corresponding profiles of copropagating bright and solitary waves. The adopted parameters are ρ = 10 5 , η = 1 , Ω = 1 , k 2 = 0 , p 3 = 1.176 × 10 10 , p 5 = 0.2935 × 10 20 , s 1 = 0.4129 × 10 10 .

Fig. 6
Fig. 6

Numerical evolution of the solution (12) in MMs under the perturbation of 10% random noise. The adopted parameters are the same as in Fig. 5.

Equations (19)

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q ξ = i k 2 2 2 q τ 2 + i p 3 | q | 2 q i p 5 | q | 4 q + s 1 τ ( | q | 2 q ) ,
A ( ξ , τ ) = i λ + ρ tanh [ η ( τ χ ξ ) ] + i β sech [ η ( τ χ ξ ) ] ,
| A ( ξ , τ ) | 2 = ρ 2 + λ 2 + 2 λ β sech [ η ( τ χ ξ ) ] + ( β 2 ρ 2 ) sech 2 [ η ( τ χ ξ ) ] ,
φ ( ξ , τ ) = arctan λ + β sech [ η ( τ χ ξ ) ] ρ tanh [ η ( τ χ ξ ) ] .
k 2 p 5 = 3 s 1 3 ,
A ( ξ , τ ) = i λ + ρ tanh [ η ( τ χ ξ ) ] ± i ρ sech [ η ( τ χ ξ ) ] ,
| A | 2 = ρ 2 + λ 2 ± 2 λ ρ sech [ η ( t χ ξ ) ] ,
η = 2 p 5 λ ρ / s i ,
k = 7 p 5 λ 2 ( λ 2 + 3 ρ 2 ) 3 p 3 ( 2 λ 2 + 3 ρ 2 ) + 31 p 5 ρ 4 / 2 + 3 p 3 2 / 2 p 5 ,
Ω = [ p 3 p 5 ( 2 λ 2 + 3 ρ 2 ) ] / s 1 ,
χ = k 2 p 3 / s 1 9 ( λ 2 + ρ 2 ) s 1 .
A ( ξ , τ ) = ρ tanh [ η ( τ χ ξ ) ] ± i ρ sech [ η ( τ χ ξ ) ] ,
χ = ρ 2 s 1 ,
k = ρ 2 ( p 3 p 5 ρ 2 Ω s 1 ) .
q ( ξ , τ ) = ρ exp { ± i arctan sech [ η ( τ χ ξ ) ] tanh [ η ( τ χ ξ ) ] + i ( k ξ + Ω τ ) } .
q ( ξ , τ ) = ( ρ + a ) exp { ± i arctan sech [ η ( τ χ ξ ) ] tanh [ η ( τ χ ξ ) ] + i ( k ξ + Ω τ ) } ,
α ( ξ , τ ) = a 1 exp [ i ( K a ξ Ω a τ ) ] + a 2 exp [ i ( K a ξ Ω a τ ) ] .
i a ξ = i ρ 2 s 1 ( 2 a τ + a * τ ) + ρ 2 { 2 p 5 ρ 2 p 3 + η sech [ η ( τ χ ξ ) ] } ( a + a * ) ρ 2 Ω s 1 ( a + ρ ) .
K a = ρ 2 s 1 ( Ω + Ω a ) .

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