Abstract

A simple model based on the 1D nonlinear Schrödinger equation is studied, which contains both spatially and temporally dispersive terms. Parametric instabilities for plane waves are analyzed in detail, and solitary waves (both bright and dark) are found. The model presented here is able to describe the non-trivial unstable dynamics of intense, nonlinear light propagation near a material resonance in presence of negative spatial dispersion. We provide as a practical example the light propagation near the tail of an exciton-polariton resonance in a specially designed semiconductor superlattice.

© 2008 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, San Diego, 2001).
  2. B. Benjamin and K. Hasselmann, "Instability of Periodic Wavetrains in Nonlinear Dispersive Systems [and Discussion]," Proc. R. Soc. A 299, 59 (1967)
    [CrossRef]
  3. T. Taniuti and H. Washimi, "Self-Trapping and Instability of Hydromagnetic Waves Along the Magnetic Field in a Cold Plasma," Phys. Rev. Lett. 21, 209 (1968).
    [CrossRef]
  4. A. Hasegawa, Plasma Instabilities and Nonlinear Effects (Springer-Verlag, Heidelberg, 1975).
    [CrossRef]
  5. G. Millot, E. Seve, S. Wabnitz and M. Haelterman, "Observation of induced modulational polarization instabilities and pulse-train generation in the normal-dispersion regime of a birefringent optical fiber," J. Opt. Soc. Am. B, 1266 (1998).
  6. J. Klinger, H. Martin and Z. Chen, "Experiments on induced modulational instability of an incoherent optical beam," Opt. Lett. 26, 271 (2001).
    [CrossRef]
  7. E. Fermi, J. Pasta, H. C. Ulam and M. Tsingou, "Studies of Nonlinear Problems," in Collected Papers of Enrico Fermi, edited by E. Segre??? (University of Chicago, 1965); G. Van Symaeys, Ph. Emplit and M. Haelterman, "Experimental Demonstration of the Fermi-Pasta-Ulam Recurrence in a Modulationally Unstable Optical Wave," Phys. Rev. Lett. 87, 033902 (2001).
    [CrossRef]
  8. N. J. Zabusky and M. D. Kruskal, "Interaction of ???Solitons??? in a Collisionless Plasma and the Recurrence of Initial States," Phys. Rev. Lett. 15, 240 (1965).
    [CrossRef]
  9. Y. S. Kivshar and G. P. Agrawal, Optical Solitons, (Academic Press, San Diego, 2003).
  10. F. Biancalana, D. V. Skryabin and P. St. J. Russell, "Four-wave mixing instabilities in photonic-crystal and tapered fibers," Phys. Rev. E 68, 046603 (2003).
    [CrossRef]
  11. F. Bassani and G. Pastori Parravicini, Electronic States and Optical Transitions in Solids, (Pergamon Press, Oxford, 1976).
  12. P. Chamorro-Posada, G. S. McDonald, G. H. C. New, "Non-Paraxial Solitons," J. Mod. Opt. 45, 1111 (1998).
    [CrossRef]
  13. J. M. Christian, G. S. McDonald and P. Chamorro-Posada, "Helmholtz Bright and Boundary Solitons," J. Phys. A: Math. and Theor. 40, 1545 (2007); J. M. Christian, G. S. McDonald, P. Chamorro-Posada, "Bistable Helmholtz Solitons in Cubic-Quintic Materials," Phys. Rev. A 76, 033833 (2007).
    [CrossRef]
  14. T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer and F. Lederer, "Anomalous Refraction and Diffraction in Discrete Optical Systems," Phys. Rev. Lett. 88, 093901 (2002).
    [CrossRef] [PubMed]
  15. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, (World Scientific, Singapore, 2004).
  16. S. I. Pekar, "Supplementary light waves in crystals and exciton absorption," Sov. Phys. Uspekhi 5, 515 (1962).
    [CrossRef]
  17. V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons, (Springer, Berlin, 1984).
  18. S. V. Chernikov and P. V. Mamyshev, "Femtosecond soliton propagation in fibers with slowly decreasing dispersion," J. Opt. Soc. Am. B 8, 1633 (1991).
    [CrossRef]
  19. A. S. Davydov, Theory of Molecular Excitons, (Plenum Press, New York-London, 1971).
  20. V. M. Agranovich, Y. R. Shen, R. H. Baughman and A. A. Zakhidov, "Optical bulk and surface waves with negative refraction," J. Lumin. 110, 167 (2004).
    [CrossRef]
  21. V. M. Agranovich and Y. N. Gartstein, "Spatial dispersion and negative refraction of light," Phys. Usp. 49, 1029 (2006).
    [CrossRef]
  22. A. Je. Semjonow and U.W. Pohl, "Strain-dependent negative excitonic masses in ZnCdSe/Zn(S)Se superlattices," J. Cryst. Growth 214/215, 646 (2000); A. Je. Semjonow, U. W. Pohl and R. Engelhardt, "Negative exciton mass in ZnCdSe/SnSSe superlattices observed by excitonic polariton Raman scattering," J. Phys. Condens. Matter 11, 1735 (1999).
    [CrossRef]
  23. A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young and E. W. Van Stryland, "Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe and ZnTe," J. Opt. Soc. Am. B 9, 405 (1992).
    [CrossRef]

2007 (1)

J. M. Christian, G. S. McDonald and P. Chamorro-Posada, "Helmholtz Bright and Boundary Solitons," J. Phys. A: Math. and Theor. 40, 1545 (2007); J. M. Christian, G. S. McDonald, P. Chamorro-Posada, "Bistable Helmholtz Solitons in Cubic-Quintic Materials," Phys. Rev. A 76, 033833 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald and P. Chamorro-Posada, "Helmholtz Bright and Boundary Solitons," J. Phys. A: Math. and Theor. 40, 1545 (2007); J. M. Christian, G. S. McDonald, P. Chamorro-Posada, "Bistable Helmholtz Solitons in Cubic-Quintic Materials," Phys. Rev. A 76, 033833 (2007).
[CrossRef]

2006 (1)

V. M. Agranovich and Y. N. Gartstein, "Spatial dispersion and negative refraction of light," Phys. Usp. 49, 1029 (2006).
[CrossRef]

2004 (1)

V. M. Agranovich, Y. R. Shen, R. H. Baughman and A. A. Zakhidov, "Optical bulk and surface waves with negative refraction," J. Lumin. 110, 167 (2004).
[CrossRef]

2003 (1)

F. Biancalana, D. V. Skryabin and P. St. J. Russell, "Four-wave mixing instabilities in photonic-crystal and tapered fibers," Phys. Rev. E 68, 046603 (2003).
[CrossRef]

2002 (1)

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer and F. Lederer, "Anomalous Refraction and Diffraction in Discrete Optical Systems," Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

2001 (1)

2000 (1)

A. Je. Semjonow and U.W. Pohl, "Strain-dependent negative excitonic masses in ZnCdSe/Zn(S)Se superlattices," J. Cryst. Growth 214/215, 646 (2000); A. Je. Semjonow, U. W. Pohl and R. Engelhardt, "Negative exciton mass in ZnCdSe/SnSSe superlattices observed by excitonic polariton Raman scattering," J. Phys. Condens. Matter 11, 1735 (1999).
[CrossRef]

A. Je. Semjonow and U.W. Pohl, "Strain-dependent negative excitonic masses in ZnCdSe/Zn(S)Se superlattices," J. Cryst. Growth 214/215, 646 (2000); A. Je. Semjonow, U. W. Pohl and R. Engelhardt, "Negative exciton mass in ZnCdSe/SnSSe superlattices observed by excitonic polariton Raman scattering," J. Phys. Condens. Matter 11, 1735 (1999).
[CrossRef]

1998 (2)

P. Chamorro-Posada, G. S. McDonald, G. H. C. New, "Non-Paraxial Solitons," J. Mod. Opt. 45, 1111 (1998).
[CrossRef]

G. Millot, E. Seve, S. Wabnitz and M. Haelterman, "Observation of induced modulational polarization instabilities and pulse-train generation in the normal-dispersion regime of a birefringent optical fiber," J. Opt. Soc. Am. B, 1266 (1998).

1992 (1)

1991 (1)

1968 (1)

T. Taniuti and H. Washimi, "Self-Trapping and Instability of Hydromagnetic Waves Along the Magnetic Field in a Cold Plasma," Phys. Rev. Lett. 21, 209 (1968).
[CrossRef]

1967 (1)

B. Benjamin and K. Hasselmann, "Instability of Periodic Wavetrains in Nonlinear Dispersive Systems [and Discussion]," Proc. R. Soc. A 299, 59 (1967)
[CrossRef]

B. Benjamin and K. Hasselmann, "Instability of Periodic Wavetrains in Nonlinear Dispersive Systems [and Discussion]," Proc. R. Soc. A 299, 59 (1967)
[CrossRef]

1965 (1)

N. J. Zabusky and M. D. Kruskal, "Interaction of ???Solitons??? in a Collisionless Plasma and the Recurrence of Initial States," Phys. Rev. Lett. 15, 240 (1965).
[CrossRef]

1962 (1)

S. I. Pekar, "Supplementary light waves in crystals and exciton absorption," Sov. Phys. Uspekhi 5, 515 (1962).
[CrossRef]

Agranovich, V. M.

V. M. Agranovich and Y. N. Gartstein, "Spatial dispersion and negative refraction of light," Phys. Usp. 49, 1029 (2006).
[CrossRef]

V. M. Agranovich, Y. R. Shen, R. H. Baughman and A. A. Zakhidov, "Optical bulk and surface waves with negative refraction," J. Lumin. 110, 167 (2004).
[CrossRef]

Baughman, R. H.

V. M. Agranovich, Y. R. Shen, R. H. Baughman and A. A. Zakhidov, "Optical bulk and surface waves with negative refraction," J. Lumin. 110, 167 (2004).
[CrossRef]

Benjamin, T. B.

B. Benjamin and K. Hasselmann, "Instability of Periodic Wavetrains in Nonlinear Dispersive Systems [and Discussion]," Proc. R. Soc. A 299, 59 (1967)
[CrossRef]

B. Benjamin and K. Hasselmann, "Instability of Periodic Wavetrains in Nonlinear Dispersive Systems [and Discussion]," Proc. R. Soc. A 299, 59 (1967)
[CrossRef]

Biancalana, F.

F. Biancalana, D. V. Skryabin and P. St. J. Russell, "Four-wave mixing instabilities in photonic-crystal and tapered fibers," Phys. Rev. E 68, 046603 (2003).
[CrossRef]

Brauer, A.

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer and F. Lederer, "Anomalous Refraction and Diffraction in Discrete Optical Systems," Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

Chamorro-Posada, P.

J. M. Christian, G. S. McDonald and P. Chamorro-Posada, "Helmholtz Bright and Boundary Solitons," J. Phys. A: Math. and Theor. 40, 1545 (2007); J. M. Christian, G. S. McDonald, P. Chamorro-Posada, "Bistable Helmholtz Solitons in Cubic-Quintic Materials," Phys. Rev. A 76, 033833 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald and P. Chamorro-Posada, "Helmholtz Bright and Boundary Solitons," J. Phys. A: Math. and Theor. 40, 1545 (2007); J. M. Christian, G. S. McDonald, P. Chamorro-Posada, "Bistable Helmholtz Solitons in Cubic-Quintic Materials," Phys. Rev. A 76, 033833 (2007).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, G. H. C. New, "Non-Paraxial Solitons," J. Mod. Opt. 45, 1111 (1998).
[CrossRef]

Chen, Z.

Chernikov, S. V.

Christian, J. M.

J. M. Christian, G. S. McDonald and P. Chamorro-Posada, "Helmholtz Bright and Boundary Solitons," J. Phys. A: Math. and Theor. 40, 1545 (2007); J. M. Christian, G. S. McDonald, P. Chamorro-Posada, "Bistable Helmholtz Solitons in Cubic-Quintic Materials," Phys. Rev. A 76, 033833 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald and P. Chamorro-Posada, "Helmholtz Bright and Boundary Solitons," J. Phys. A: Math. and Theor. 40, 1545 (2007); J. M. Christian, G. S. McDonald, P. Chamorro-Posada, "Bistable Helmholtz Solitons in Cubic-Quintic Materials," Phys. Rev. A 76, 033833 (2007).
[CrossRef]

Feir, J. E.

B. Benjamin and K. Hasselmann, "Instability of Periodic Wavetrains in Nonlinear Dispersive Systems [and Discussion]," Proc. R. Soc. A 299, 59 (1967)
[CrossRef]

Gartstein, Y. N.

V. M. Agranovich and Y. N. Gartstein, "Spatial dispersion and negative refraction of light," Phys. Usp. 49, 1029 (2006).
[CrossRef]

Haelterman, M.

G. Millot, E. Seve, S. Wabnitz and M. Haelterman, "Observation of induced modulational polarization instabilities and pulse-train generation in the normal-dispersion regime of a birefringent optical fiber," J. Opt. Soc. Am. B, 1266 (1998).

Hagan, D. J.

Hasselmann, K.

B. Benjamin and K. Hasselmann, "Instability of Periodic Wavetrains in Nonlinear Dispersive Systems [and Discussion]," Proc. R. Soc. A 299, 59 (1967)
[CrossRef]

Je, A.

A. Je. Semjonow and U.W. Pohl, "Strain-dependent negative excitonic masses in ZnCdSe/Zn(S)Se superlattices," J. Cryst. Growth 214/215, 646 (2000); A. Je. Semjonow, U. W. Pohl and R. Engelhardt, "Negative exciton mass in ZnCdSe/SnSSe superlattices observed by excitonic polariton Raman scattering," J. Phys. Condens. Matter 11, 1735 (1999).
[CrossRef]

A. Je. Semjonow and U.W. Pohl, "Strain-dependent negative excitonic masses in ZnCdSe/Zn(S)Se superlattices," J. Cryst. Growth 214/215, 646 (2000); A. Je. Semjonow, U. W. Pohl and R. Engelhardt, "Negative exciton mass in ZnCdSe/SnSSe superlattices observed by excitonic polariton Raman scattering," J. Phys. Condens. Matter 11, 1735 (1999).
[CrossRef]

Klinger, J.

Kruskal, M. D.

N. J. Zabusky and M. D. Kruskal, "Interaction of ???Solitons??? in a Collisionless Plasma and the Recurrence of Initial States," Phys. Rev. Lett. 15, 240 (1965).
[CrossRef]

Lederer, F.

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer and F. Lederer, "Anomalous Refraction and Diffraction in Discrete Optical Systems," Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

Mamyshev, P. V.

Martin, H.

McDonald, G. S.

J. M. Christian, G. S. McDonald and P. Chamorro-Posada, "Helmholtz Bright and Boundary Solitons," J. Phys. A: Math. and Theor. 40, 1545 (2007); J. M. Christian, G. S. McDonald, P. Chamorro-Posada, "Bistable Helmholtz Solitons in Cubic-Quintic Materials," Phys. Rev. A 76, 033833 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald and P. Chamorro-Posada, "Helmholtz Bright and Boundary Solitons," J. Phys. A: Math. and Theor. 40, 1545 (2007); J. M. Christian, G. S. McDonald, P. Chamorro-Posada, "Bistable Helmholtz Solitons in Cubic-Quintic Materials," Phys. Rev. A 76, 033833 (2007).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, G. H. C. New, "Non-Paraxial Solitons," J. Mod. Opt. 45, 1111 (1998).
[CrossRef]

Millot, G.

G. Millot, E. Seve, S. Wabnitz and M. Haelterman, "Observation of induced modulational polarization instabilities and pulse-train generation in the normal-dispersion regime of a birefringent optical fiber," J. Opt. Soc. Am. B, 1266 (1998).

New, G. H. C.

P. Chamorro-Posada, G. S. McDonald, G. H. C. New, "Non-Paraxial Solitons," J. Mod. Opt. 45, 1111 (1998).
[CrossRef]

Pekar, S. I.

S. I. Pekar, "Supplementary light waves in crystals and exciton absorption," Sov. Phys. Uspekhi 5, 515 (1962).
[CrossRef]

Pertsch, T.

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer and F. Lederer, "Anomalous Refraction and Diffraction in Discrete Optical Systems," Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

Peschel, U.

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer and F. Lederer, "Anomalous Refraction and Diffraction in Discrete Optical Systems," Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

Russell, P. St. J.

F. Biancalana, D. V. Skryabin and P. St. J. Russell, "Four-wave mixing instabilities in photonic-crystal and tapered fibers," Phys. Rev. E 68, 046603 (2003).
[CrossRef]

Said, A. A.

Seve, E.

G. Millot, E. Seve, S. Wabnitz and M. Haelterman, "Observation of induced modulational polarization instabilities and pulse-train generation in the normal-dispersion regime of a birefringent optical fiber," J. Opt. Soc. Am. B, 1266 (1998).

Sheik-Bahae, M.

Shen, Y. R.

V. M. Agranovich, Y. R. Shen, R. H. Baughman and A. A. Zakhidov, "Optical bulk and surface waves with negative refraction," J. Lumin. 110, 167 (2004).
[CrossRef]

Skryabin, D. V.

F. Biancalana, D. V. Skryabin and P. St. J. Russell, "Four-wave mixing instabilities in photonic-crystal and tapered fibers," Phys. Rev. E 68, 046603 (2003).
[CrossRef]

Taniuti, T.

T. Taniuti and H. Washimi, "Self-Trapping and Instability of Hydromagnetic Waves Along the Magnetic Field in a Cold Plasma," Phys. Rev. Lett. 21, 209 (1968).
[CrossRef]

Van Stryland, E. W.

Wabnitz, S.

G. Millot, E. Seve, S. Wabnitz and M. Haelterman, "Observation of induced modulational polarization instabilities and pulse-train generation in the normal-dispersion regime of a birefringent optical fiber," J. Opt. Soc. Am. B, 1266 (1998).

Wang, J.

Washimi, H.

T. Taniuti and H. Washimi, "Self-Trapping and Instability of Hydromagnetic Waves Along the Magnetic Field in a Cold Plasma," Phys. Rev. Lett. 21, 209 (1968).
[CrossRef]

Wei, T. H.

Young, J.

Zabusky, N. J.

N. J. Zabusky and M. D. Kruskal, "Interaction of ???Solitons??? in a Collisionless Plasma and the Recurrence of Initial States," Phys. Rev. Lett. 15, 240 (1965).
[CrossRef]

Zakhidov, A. A.

V. M. Agranovich, Y. R. Shen, R. H. Baughman and A. A. Zakhidov, "Optical bulk and surface waves with negative refraction," J. Lumin. 110, 167 (2004).
[CrossRef]

Zentgraf, T.

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer and F. Lederer, "Anomalous Refraction and Diffraction in Discrete Optical Systems," Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

J. Cryst. Growth (1)

A. Je. Semjonow and U.W. Pohl, "Strain-dependent negative excitonic masses in ZnCdSe/Zn(S)Se superlattices," J. Cryst. Growth 214/215, 646 (2000); A. Je. Semjonow, U. W. Pohl and R. Engelhardt, "Negative exciton mass in ZnCdSe/SnSSe superlattices observed by excitonic polariton Raman scattering," J. Phys. Condens. Matter 11, 1735 (1999).
[CrossRef]

J. Lumin. (1)

V. M. Agranovich, Y. R. Shen, R. H. Baughman and A. A. Zakhidov, "Optical bulk and surface waves with negative refraction," J. Lumin. 110, 167 (2004).
[CrossRef]

J. Mod. Opt. (1)

P. Chamorro-Posada, G. S. McDonald, G. H. C. New, "Non-Paraxial Solitons," J. Mod. Opt. 45, 1111 (1998).
[CrossRef]

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

J. Phys. A: Math. and Theor. (1)

J. M. Christian, G. S. McDonald and P. Chamorro-Posada, "Helmholtz Bright and Boundary Solitons," J. Phys. A: Math. and Theor. 40, 1545 (2007); J. M. Christian, G. S. McDonald, P. Chamorro-Posada, "Bistable Helmholtz Solitons in Cubic-Quintic Materials," Phys. Rev. A 76, 033833 (2007).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. E (1)

F. Biancalana, D. V. Skryabin and P. St. J. Russell, "Four-wave mixing instabilities in photonic-crystal and tapered fibers," Phys. Rev. E 68, 046603 (2003).
[CrossRef]

Phys. Rev. Lett. (3)

N. J. Zabusky and M. D. Kruskal, "Interaction of ???Solitons??? in a Collisionless Plasma and the Recurrence of Initial States," Phys. Rev. Lett. 15, 240 (1965).
[CrossRef]

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer and F. Lederer, "Anomalous Refraction and Diffraction in Discrete Optical Systems," Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

T. Taniuti and H. Washimi, "Self-Trapping and Instability of Hydromagnetic Waves Along the Magnetic Field in a Cold Plasma," Phys. Rev. Lett. 21, 209 (1968).
[CrossRef]

Phys. Usp. (1)

V. M. Agranovich and Y. N. Gartstein, "Spatial dispersion and negative refraction of light," Phys. Usp. 49, 1029 (2006).
[CrossRef]

Proc. R. Soc. A (1)

B. Benjamin and K. Hasselmann, "Instability of Periodic Wavetrains in Nonlinear Dispersive Systems [and Discussion]," Proc. R. Soc. A 299, 59 (1967)
[CrossRef]

Sov. Phys. Uspekhi (1)

S. I. Pekar, "Supplementary light waves in crystals and exciton absorption," Sov. Phys. Uspekhi 5, 515 (1962).
[CrossRef]

Other (8)

V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons, (Springer, Berlin, 1984).

G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, San Diego, 2001).

E. Fermi, J. Pasta, H. C. Ulam and M. Tsingou, "Studies of Nonlinear Problems," in Collected Papers of Enrico Fermi, edited by E. Segre??? (University of Chicago, 1965); G. Van Symaeys, Ph. Emplit and M. Haelterman, "Experimental Demonstration of the Fermi-Pasta-Ulam Recurrence in a Modulationally Unstable Optical Wave," Phys. Rev. Lett. 87, 033902 (2001).
[CrossRef]

A. Hasegawa, Plasma Instabilities and Nonlinear Effects (Springer-Verlag, Heidelberg, 1975).
[CrossRef]

A. S. Davydov, Theory of Molecular Excitons, (Plenum Press, New York-London, 1971).

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, (World Scientific, Singapore, 2004).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons, (Academic Press, San Diego, 2003).

F. Bassani and G. Pastori Parravicini, Electronic States and Optical Transitions in Solids, (Pergamon Press, Oxford, 1976).

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Figures (5)

Fig. 1.
Fig. 1.

Dispersion k ± of the two branches of the carrier wave in the linear excitation case (I = 0) for (a) anomalous dispersion regime (s = +1) and (b) normal dispersion regime (s = -1). D = 1 in both cases. Solid (dashed) lines indicate the real (imaginary) parts of k ±. Blue (red) lines refer to k + (k -). Gray shaded regions in (a) and (b) indicate the k-bandgap and the ω-inverted bandgap, respectively.

Fig. 2.
Fig. 2.

Dispersion k ± of the two branches of the carrier wave in the nonlinear excitation case (I = 0.3 > I t ) for (a) anomalous dispersion regime (s = +1) and (b) normal dispersion regime (s = -1). D = 1 in both cases. Solid (dashed) lines indicate the real (imaginary) parts of k ±. Blue (red) lines refer to k + (k-).

Fig. 3.
Fig. 3.

Parametric instabilities for anomalous dispersion regime, s = +1. (a,c) Nonlinear dispersion branches of the carrier wave, for D = 0.7<D t < D c [(a)], and for D = 0.85 > D t < D c [(c)]. Solid (dashed) lines are real (imaginary) parts of k ±. Blue (red) lines refer to k + (k-). (b,d) Nonlinear instability gain Im{κ j (δ)}<0 for the parameters described in (a) and (c) respectively. The carrier wave intensity is I = 0.22 in both cases. The values of D t and D c are 0.7576 and 1.1364 respectively.

Fig. 4.
Fig. 4.

Parametric instabilities for normal dispersion regime, s = -1. (a,c) Nonlinear dispersion branches of the carrier wave, for D = 0.7<D t < D c [(a)], and for D = 0.85 > D t < D c [(c)]. Solid (dashed) lines are real (imaginary) parts of k ±. Blue (red) lines refer to k + (k-). (b,d) Nonlinear instability gain Im{κ j (δ)κ<0 for the parameters described in (a) and (c) respectively. The carrier wave intensity is I = 0.22 in both cases. D t and D c are the same as in Fig. 3.

Fig. 5.
Fig. 5.

(a) Region of existence on the parameter space (ν,ω) for bright solitons (s = +1, D = 1), calculated by imposing reality conditions on Eqs. (7)(9), plus the extra condition Eq. (10). (b) Regions of existence on the parameter space (ν,ω) for dark (black + gray) solitons (s = -1, D = 1), calculated by imposing reality conditions on Eqs. (1214), plus the extra condition sνω = ±α|ν||ω|, valid for the black and the gray solitons respectively.

Equations (23)

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i z ψ D z 2 ψ + 1 2 s t 2 ψ + ψ 2 ψ = 0 .
i α [ 1 4 D ( I 2 / 2 ) ] 1 2 z f D z 2 f + 1 2 s t 2 f i s ω t f + I ( f + f ¯ ) = 0 ,
i α σ ̂ z A ̂ z v D z 2 v + M ̂ v = 0 ,
M ̂ [ D ̂ ( i t ) + I , I I , D ̂ ( i t ) + I ] .
I { [ 2 D κ + α ( a a ¯ ) ] κ s δ 2 } + 1 4 [ 2 κ ( a ¯ α D κ ) + s δ ( δ 2 ω ) ] · [ 2 κ ( a α + D κ ) + s δ ( δ + 2 ω ) ] = 0
ψ ( z , t ) = A 0 sech ( [ v t z ] / z 0 ) e ikz i ω t ,
k α = 1 + α 1 4 D ( A 0 2 / 2 s ω 2 / 2 ) 2 D = 1 + α | v | | ω | 2 D ,
A 0 = 1 + s ω 2 ( 2 D s v 2 ) 2 D ,
z 0 = 2 D ( sv 2 2 D ) 1 + 2 ( 2 D sv 2 ) .
svw = α | v | | ω | .
ψ ( z , t ) = A 0 { cos ( θ ) tanh [ ( vt z ) / z 0 ] + i sin ( θ ) } e ikz i ω t ,
k α = 1 + α | v | | ω | 2 D ,
A 0 = 1 + 2 ( 2 D sv 2 ) 4 D ,
z 0 = 4 D ( 2 D sv 2 ) 1 + 2 ( 2 D sv 2 ) sec ( θ ) ,
sin ( θ ) = 4 D ( 2 D sv 2 ) [ 1 + 2 ( 2 D sv 2 ) ] ( α | v | | ω | svω ) ,
ε k , ω = ε b [ 1 Δ ω ω ˜ 0 Γ | k | 2 + i γ ] .
ε k , ω = c 2 | k | 2 ω 2 ,
P k , ω = ( χ k , ω X + χ ω W ) E k , ω ,
[ ε k , ω kc / ω ] E k , ω ( ± ) = 0 ,
{ ε b [ 1 Δ δ ω Γ k 2 + i γ ] 1 / 2 kc / ω } E k , ω ( + ) = 0 ,
{ ε b [ 1 Γ Δ k 2 2 δ ω 2 ] kc / ω } E k , ω ( + ) = 0 .
[ i z + i D ̂ W ( i T ) + ε b Γ Δ ω ˜ 0 2 δ ω 2 c z 2 + γ NL A 2 ] A = 0 ,
D ̂ W ( i T ) j = 1 β j j ! [ i T ] j

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