Abstract

We carry out a modulation instability (MI) analysis in nonlinear complex parity-time (PT) symmetric periodic structures. All three regimes defined by the PT-symmetry breaking point or threshold, namely, below threshold, at threshold, and above threshold, are discussed. It is found that MI exists even beyond the PT-symmetry threshold, indicating the possible existence of solitons or solitary waves, in conformity with some recent reports. We find that MI does not exist at the PT-symmetry breaking point in the case of normal dispersion below a certain nonlinear threshold. However, in the case of the anomalous dispersion regime, MI does exist even at the PT-symmetry breaking point.

© 2014 Optical Society of America

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    [Crossref]
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  3. C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
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    [Crossref]
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    [Crossref]
  6. C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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  14. T. Kottos, “Optical physics: broken symmetry makes light work,” Nat. Phys. 6, 166–167 (2010).
    [Crossref]
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    [Crossref]
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    [Crossref]
  18. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
    [Crossref]
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  27. Y. Xiang, X. Dai, S. Wen, and D. Fan, “Modulation instability in metamaterials with saturable nonlinearity,” J. Opt. Soc. Am. B 28, 908–916 (2011).
    [Crossref]
  28. A. K. Sarma and P. Kumar, “Modulation instability of ultrashort pulses in quadratic nonlinear media beyond the slowly varying envelope approximation,” Appl. Phys. B 106, 289–293 (2012).
    [Crossref]
  29. N. Akhmediev and A. Ankiewicz, “Modulation instability, Fermi-Pasta-Ulam recurrence, rogue waves, nonlinear phase shift, and exact solutions of the Ablowitz-Ladik equation,” Phys. Rev. E 83, 046603 (2011).
    [Crossref]
  30. E. Kengne, S. T. Chui, and W. M. Liu, “Modulational instability criteria for coupled nonlinear transmission lines with dispersive elements,” Phys. Rev. E 74, 036614 (2006).
    [Crossref]
  31. Z. Xu, L. Li, Z. Li, and G. Zhou, “Modulation instability and solitons on a cw background in an optical fiber with higher-order effects,” Phys. Rev. E 67, 026603 (2003).
    [Crossref]
  32. M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901 (2011).
    [Crossref]
  33. C. M. de Sterke, “Theory of modulational instability in fiber Bragg gratings,” J. Opt. Soc. Am. B 15, 2660–2667 (1998).
    [Crossref]
  34. R. Ganapthy, K. Senthilnathan, and K. Porsezian, “Modulational instability in a fibre and a fibre Bragg grating,” J. Opt. B 6, S436–S452 (2004).
    [Crossref]
  35. G. P. Agrawal, Applications of Nonlinear Fiber Optics, 2nd ed. (Academic, 2007).
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    [Crossref]
  37. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
    [Crossref]
  38. D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
    [Crossref]
  39. W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Symmetric and asymmetric solitons in linearly coupled Bragg gratings,” Phys. Rev. E 69, 066610 (2004).
    [Crossref]
  40. N. M. Litchinitser, C. J. McKinstrie, C. M. de Sterke, and G. P. Agrawal, “Spatiotemporal instabilities in nonlinear bulk media with Bragg gratings,” J. Opt. Soc. Am. B 18, 45–54 (2001).
    [Crossref]

2014 (1)

S. K. Gupta and A. K. Sarma, “Solitary waves in parity-time (PT)–symmetric Bragg grating structure and the existence of optical rogue waves,” Europhys. Lett. 105, 44001 (2014).
[Crossref]

2013 (1)

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

2012 (5)

M. A. Miri, P. L. Wa, and D. N. Christodoulides, “Large area single-mode parity–time-symmetric laser amplifiers,” Opt. Lett. 37, 764–766 (2012).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

M. Nazari, F. Nazari, and M. K. Moravvej-Farshi, “Dynamic behavior of spatial solitons propagating along Scarf II parity–time symmetric cells,” J. Opt. Soc. Am. B 29, 3057–3062 (2012).
[Crossref]

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

A. K. Sarma and P. Kumar, “Modulation instability of ultrashort pulses in quadratic nonlinear media beyond the slowly varying envelope approximation,” Appl. Phys. B 106, 289–293 (2012).
[Crossref]

2011 (9)

N. Akhmediev and A. Ankiewicz, “Modulation instability, Fermi-Pasta-Ulam recurrence, rogue waves, nonlinear phase shift, and exact solutions of the Ablowitz-Ladik equation,” Phys. Rev. E 83, 046603 (2011).
[Crossref]

A. K. Sarma and M. Saha, “Modulational instability of coupled nonlinear field equations for pulse propagation in a negative index material embedded into a Kerr medium,” J. Opt. Soc. Am. B 28, 944–948 (2011).
[Crossref]

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901 (2011).
[Crossref]

Y. Xiang, X. Dai, S. Wen, and D. Fan, “Modulation instability in metamaterials with saturable nonlinearity,” J. Opt. Soc. Am. B 28, 908–916 (2011).
[Crossref]

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84, 040101 (2011).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011).
[Crossref]

2010 (4)

T. Kottos, “Optical physics: broken symmetry makes light work,” Nat. Phys. 6, 166–167 (2010).
[Crossref]

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
[Crossref]

A. K. Sarma, “Modulational instability of few-cycle pulses in optical fibers,” Europhys. Lett. 92, 24004 (2010).
[Crossref]

2009 (1)

V. E. Zakharov and L. A. Ostrosvsky, “Modulation instability: the beginning,” Physica D 238, 540–548 (2009).
[Crossref]

2008 (2)

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT-symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

2007 (1)

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 947–1018 (2007).
[Crossref]

2006 (1)

E. Kengne, S. T. Chui, and W. M. Liu, “Modulational instability criteria for coupled nonlinear transmission lines with dispersive elements,” Phys. Rev. E 74, 036614 (2006).
[Crossref]

2004 (2)

R. Ganapthy, K. Senthilnathan, and K. Porsezian, “Modulational instability in a fibre and a fibre Bragg grating,” J. Opt. B 6, S436–S452 (2004).
[Crossref]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Symmetric and asymmetric solitons in linearly coupled Bragg gratings,” Phys. Rev. E 69, 066610 (2004).
[Crossref]

2003 (2)

Z. Xu, L. Li, Z. Li, and G. Zhou, “Modulation instability and solitons on a cw background in an optical fiber with higher-order effects,” Phys. Rev. E 67, 026603 (2003).
[Crossref]

A. Mostafazadeh, “Exact PT-symmetry is equivalent to Hermiticity,” J. Phys. A 36, 7081–7091 (2003).
[Crossref]

2002 (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

2001 (1)

1999 (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[Crossref]

1998 (2)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

C. M. de Sterke, “Theory of modulational instability in fiber Bragg gratings,” J. Opt. Soc. Am. B 15, 2660–2667 (1998).
[Crossref]

1996 (1)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[Crossref]

1989 (1)

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[Crossref]

1987 (1)

1986 (1)

Aceves, A. B.

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

Agrawal, G. P.

N. M. Litchinitser, C. J. McKinstrie, C. M. de Sterke, and G. P. Agrawal, “Spatiotemporal instabilities in nonlinear bulk media with Bragg gratings,” J. Opt. Soc. Am. B 18, 45–54 (2001).
[Crossref]

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

G. P. Agrawal, Applications of Nonlinear Fiber Optics, 2nd ed. (Academic, 2007).

Akhmediev, N.

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901 (2011).
[Crossref]

N. Akhmediev and A. Ankiewicz, “Modulation instability, Fermi-Pasta-Ulam recurrence, rogue waves, nonlinear phase shift, and exact solutions of the Ablowitz-Ladik equation,” Phys. Rev. E 83, 046603 (2011).
[Crossref]

Almeida, V. R.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Ankiewicz, A.

N. Akhmediev and A. Ankiewicz, “Modulation instability, Fermi-Pasta-Ulam recurrence, rogue waves, nonlinear phase shift, and exact solutions of the Ablowitz-Ladik equation,” Phys. Rev. E 83, 046603 (2011).
[Crossref]

Ayache, M.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Bender, C. M.

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 947–1018 (2007).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Bersch, C.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

Boettcher, S.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Brody, D. C.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Chen, Y. F.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Chen, Y.-F.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Chong, Y. D.

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[Crossref]

Christodoulides, D. N.

M. A. Miri, P. L. Wa, and D. N. Christodoulides, “Large area single-mode parity–time-symmetric laser amplifiers,” Opt. Lett. 37, 764–766 (2012).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT-symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[Crossref]

Chu, P. L.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Symmetric and asymmetric solitons in linearly coupled Bragg gratings,” Phys. Rev. E 69, 066610 (2004).
[Crossref]

Chui, S. T.

E. Kengne, S. T. Chui, and W. M. Liu, “Modulational instability criteria for coupled nonlinear transmission lines with dispersive elements,” Phys. Rev. E 74, 036614 (2006).
[Crossref]

Dai, X.

de Sterke, C. M.

Dudley, J. M.

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901 (2011).
[Crossref]

Eggleton, B. J.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[Crossref]

Eichelkraut, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

El-Ganainy, R.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011).
[Crossref]

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT-symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

Ellis, F. M.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84, 040101 (2011).
[Crossref]

Erkintalo, M.

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901 (2011).
[Crossref]

Fainman, Y.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Fan, D.

Fegadolli, W. S.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Feng, L.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Finot, C.

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901 (2011).
[Crossref]

Ganapthy, R.

R. Ganapthy, K. Senthilnathan, and K. Porsezian, “Modulational instability in a fibre and a fibre Bragg grating,” J. Opt. B 6, S436–S452 (2004).
[Crossref]

Ge, L.

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[Crossref]

Genty, G.

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901 (2011).
[Crossref]

Gupta, S. K.

S. K. Gupta and A. K. Sarma, “Solitary waves in parity-time (PT)–symmetric Bragg grating structure and the existence of optical rogue waves,” Europhys. Lett. 105, 44001 (2014).
[Crossref]

Hammani, K.

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901 (2011).
[Crossref]

Huang, J.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Jones, H. F.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

Joseph, R. I.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[Crossref]

Kengne, E.

E. Kengne, S. T. Chui, and W. M. Liu, “Modulational instability criteria for coupled nonlinear transmission lines with dispersive elements,” Phys. Rev. E 74, 036614 (2006).
[Crossref]

Kibler, B.

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901 (2011).
[Crossref]

Kip, D.

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Kottos, T.

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84, 040101 (2011).
[Crossref]

T. Kottos, “Optical physics: broken symmetry makes light work,” Nat. Phys. 6, 166–167 (2010).
[Crossref]

Kovanis, V.

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

Krug, P. A.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[Crossref]

Kumar, P.

A. K. Sarma and P. Kumar, “Modulation instability of ultrashort pulses in quadratic nonlinear media beyond the slowly varying envelope approximation,” Appl. Phys. B 106, 289–293 (2012).
[Crossref]

Li, A.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84, 040101 (2011).
[Crossref]

Li, L.

Z. Xu, L. Li, Z. Li, and G. Zhou, “Modulation instability and solitons on a cw background in an optical fiber with higher-order effects,” Phys. Rev. E 67, 026603 (2003).
[Crossref]

Li, Z.

Z. Xu, L. Li, Z. Li, and G. Zhou, “Modulation instability and solitons on a cw background in an optical fiber with higher-order effects,” Phys. Rev. E 67, 026603 (2003).
[Crossref]

Lin, Z.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Litchinitser, N. M.

Liu, W. M.

E. Kengne, S. T. Chui, and W. M. Liu, “Modulational instability criteria for coupled nonlinear transmission lines with dispersive elements,” Phys. Rev. E 74, 036614 (2006).
[Crossref]

Longhi, S.

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
[Crossref]

Lu, M. H.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Lu, M.-H.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Mak, W. C. K.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Symmetric and asymmetric solitons in linearly coupled Bragg gratings,” Phys. Rev. E 69, 066610 (2004).
[Crossref]

Makris, K. G.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011).
[Crossref]

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT-symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

Malomed, B. A.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Symmetric and asymmetric solitons in linearly coupled Bragg gratings,” Phys. Rev. E 69, 066610 (2004).
[Crossref]

McKinstrie, C. J.

Meisinger, P. N.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[Crossref]

Miri, M. A.

M. A. Miri, P. L. Wa, and D. N. Christodoulides, “Large area single-mode parity–time-symmetric laser amplifiers,” Opt. Lett. 37, 764–766 (2012).
[Crossref]

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

Miri, M.-A.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

Moravvej-Farshi, M. K.

Mostafazadeh, A.

A. Mostafazadeh, “Exact PT-symmetry is equivalent to Hermiticity,” J. Phys. A 36, 7081–7091 (2003).
[Crossref]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT-symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

Nazari, F.

Nazari, M.

Oliveira, J. E.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Onishchukov, G.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

Ostrosvsky, L. A.

V. E. Zakharov and L. A. Ostrosvsky, “Modulation instability: the beginning,” Physica D 238, 540–548 (2009).
[Crossref]

Peschel, U.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

Porsezian, K.

R. Ganapthy, K. Senthilnathan, and K. Porsezian, “Modulational instability in a fibre and a fibre Bragg grating,” J. Opt. B 6, S436–S452 (2004).
[Crossref]

Potasek, M. J.

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Rasmussen, J. J.

Regensburger, A.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

Ruter, C. E.

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Saha, M.

Sarma, A. K.

S. K. Gupta and A. K. Sarma, “Solitary waves in parity-time (PT)–symmetric Bragg grating structure and the existence of optical rogue waves,” Europhys. Lett. 105, 44001 (2014).
[Crossref]

A. K. Sarma and P. Kumar, “Modulation instability of ultrashort pulses in quadratic nonlinear media beyond the slowly varying envelope approximation,” Appl. Phys. B 106, 289–293 (2012).
[Crossref]

A. K. Sarma and M. Saha, “Modulational instability of coupled nonlinear field equations for pulse propagation in a negative index material embedded into a Kerr medium,” J. Opt. Soc. Am. B 28, 944–948 (2011).
[Crossref]

A. K. Sarma, “Modulational instability of few-cycle pulses in optical fibers,” Europhys. Lett. 92, 24004 (2010).
[Crossref]

Scherer, A.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Schindler, J.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84, 040101 (2011).
[Crossref]

Segev, M.

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Senthilnathan, K.

R. Ganapthy, K. Senthilnathan, and K. Porsezian, “Modulational instability in a fibre and a fibre Bragg grating,” J. Opt. B 6, S436–S452 (2004).
[Crossref]

Shukla, P. K.

Sipe, J. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[Crossref]

Slusher, R. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[Crossref]

Stone, A. D.

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[Crossref]

Wa, P. L.

Wen, S.

Xiang, Y.

Xu, Y. L.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Xu, Y.-L.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Xu, Z.

Z. Xu, L. Li, Z. Li, and G. Zhou, “Modulation instability and solitons on a cw background in an optical fiber with higher-order effects,” Phys. Rev. E 67, 026603 (2003).
[Crossref]

Zakharov, V. E.

V. E. Zakharov and L. A. Ostrosvsky, “Modulation instability: the beginning,” Physica D 238, 540–548 (2009).
[Crossref]

Zheng, M. C.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84, 040101 (2011).
[Crossref]

Zhou, G.

Z. Xu, L. Li, Z. Li, and G. Zhou, “Modulation instability and solitons on a cw background in an optical fiber with higher-order effects,” Phys. Rev. E 67, 026603 (2003).
[Crossref]

Appl. Phys. B (1)

A. K. Sarma and P. Kumar, “Modulation instability of ultrashort pulses in quadratic nonlinear media beyond the slowly varying envelope approximation,” Appl. Phys. B 106, 289–293 (2012).
[Crossref]

Europhys. Lett. (2)

A. K. Sarma, “Modulational instability of few-cycle pulses in optical fibers,” Europhys. Lett. 92, 24004 (2010).
[Crossref]

S. K. Gupta and A. K. Sarma, “Solitary waves in parity-time (PT)–symmetric Bragg grating structure and the existence of optical rogue waves,” Europhys. Lett. 105, 44001 (2014).
[Crossref]

Int. J. Theor. Phys. (1)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011).
[Crossref]

J. Math. Phys. (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[Crossref]

J. Opt. B (1)

R. Ganapthy, K. Senthilnathan, and K. Porsezian, “Modulational instability in a fibre and a fibre Bragg grating,” J. Opt. B 6, S436–S452 (2004).
[Crossref]

J. Opt. Soc. Am. B (5)

J. Phys. A (1)

A. Mostafazadeh, “Exact PT-symmetry is equivalent to Hermiticity,” J. Phys. A 36, 7081–7091 (2003).
[Crossref]

Nat. Mater. (1)

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nat. Mater. 12, 108–113 (2013).
[Crossref]

Nat. Phys. (2)

T. Kottos, “Optical physics: broken symmetry makes light work,” Nat. Phys. 6, 166–167 (2010).
[Crossref]

C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Nature (1)

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[Crossref]

Opt. Lett. (3)

Phys. Rev. A (3)

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82, 031801 (2010).
[Crossref]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84, 040101 (2011).
[Crossref]

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

Phys. Rev. E (4)

N. Akhmediev and A. Ankiewicz, “Modulation instability, Fermi-Pasta-Ulam recurrence, rogue waves, nonlinear phase shift, and exact solutions of the Ablowitz-Ladik equation,” Phys. Rev. E 83, 046603 (2011).
[Crossref]

E. Kengne, S. T. Chui, and W. M. Liu, “Modulational instability criteria for coupled nonlinear transmission lines with dispersive elements,” Phys. Rev. E 74, 036614 (2006).
[Crossref]

Z. Xu, L. Li, Z. Li, and G. Zhou, “Modulation instability and solitons on a cw background in an optical fiber with higher-order effects,” Phys. Rev. E 67, 026603 (2003).
[Crossref]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Symmetric and asymmetric solitons in linearly coupled Bragg gratings,” Phys. Rev. E 69, 066610 (2004).
[Crossref]

Phys. Rev. Lett. (9)

M. Erkintalo, K. Hammani, B. Kibler, C. Finot, N. Akhmediev, J. M. Dudley, and G. Genty, “Higher-order modulation instability in nonlinear fiber optics,” Phys. Rev. Lett. 107, 253901 (2011).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[Crossref]

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[Crossref]

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106, 093902 (2011).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT-symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Physica D (1)

V. E. Zakharov and L. A. Ostrosvsky, “Modulation instability: the beginning,” Physica D 238, 540–548 (2009).
[Crossref]

Rep. Prog. Phys. (1)

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 947–1018 (2007).
[Crossref]

Science (1)

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Other (2)

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

G. P. Agrawal, Applications of Nonlinear Fiber Optics, 2nd ed. (Academic, 2007).

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Figures (5)

Fig. 1.
Fig. 1.

Dispersion curves traced by q and δ with k=0.5. Upper branch refers to f<0 (anomalous dispersion), and lower branch refers to f>0 (normal dispersion). (a) Case of passive medium with g=0; solid curves represent dispersion curves without nonlinearity, while the dashed ones with nonlinearity, γ=2. (b) Case of PT-symmetric medium with γ=2. Solid curve refers to below threshold (g=0.25), dashed curve refers to at threshold (g=0.5), and dash–dotted one refers to the above threshold case (g=0.75).

Fig. 2.
Fig. 2.

Gain spectrum for different regimes: below PT threshold (g=0.5), at threshold (g=1), and above threshold (g=1.5) with k=1, u0=1, and γ=1. (a) Normal dispersion (f=4). (b) Anomalous dispersion (f=4).

Fig. 3.
Fig. 3.

Contour plot for MI gain with variation of the gain/loss parameter. (a) f=3 (normal dispersion), (b) f=3 (anomalous dispersion), (c) f=1 (top of the photonic bandgap), and (d) f=1 (bottom of the photonic bandgap).

Fig. 4.
Fig. 4.

Contour plot for MI gain with variation of the nonlinear parameter, γ, with k=1, u0=1, and f=2. (a) g=0.5, (b) g=1, and (c) g=1.5.

Fig. 5.
Fig. 5.

Contour plot for MI gain with the variation of the nonlinear parameter, γ, with k=1, u0=1, and f=2. (a) g=0.5, (b) g=1, and (c) g=1.5.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

i(Efz+1vgEft)+δ0Ef+(k0+g0)Eb+γ0(|Ef|2+2|Eb|2)Ef=0,
i(Ebz1vgEbt)δ0Eb(k0g0)Efγ0(2|Ef|2+|Eb|2)Eb=0.
ξ=z/z0,E˜f=Ef/P0,E˜b=Eb/P0,τ=t/T0.
i(E˜fξ+E˜fτ)+δE˜f+(k+g)E˜b+γ(|E˜f|2+2|E˜b|2)E˜f=0,
i(EbξE˜bτ)δE˜b(kg)E˜fγ(2|E˜f|2+|E˜b|2)E˜b=0.
E˜f=AfeiqξandE˜b=Abeiqξ.
δ=k(1+f2)g(1f2)2f3γ2,q=k(1f2)g(1+f2)2fγ2(1f21+f2).
u=(u0+ε1)exp(iqξ)exp(iδξ),
v=(v0+ε2)exp(iqξ)exp(iδξ)
i(ε1ξ+ε1τ)(qδ)ε1+(k+g)ε2+γu02(2ε1+ε1*)+2γu0v0(ε2+ε2*)+2γv02ε1=0,i(ε2ξε2τ)(q+δ)ε1(kg)ε1γv02(2ε2+ε2*)2γu0v0(ε1+ε1*)2γu02ε2=0.
ε1=a1exp[i(KξΩτ)]+a2exp[i(KξΩτ)],
ε2=b1exp[i(KξΩτ)]+b2exp[i(KξΩτ)].
Ω4αΩ2+βΩ+η=0,
η=K4+(BGDEACFH)K2+ACFH+ADFGBCEHBDEG
Ω=±12p±122αp2β/p,

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