Abstract

We analyze the modulation stability of spatiotemporal solitary and traveling wave solutions to the multidimensional nonlinear Schrödinger equation and the Gross-Pitaevskii equation with variable coefficients that were obtained using Jacobi elliptic functions. For all the solutions we obtain either unconditional stability, or a conditional stability that can be furnished through the use of dispersion management.

© 2015 Optical Society of America

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  1. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).
  2. L. Ostrovskii, “Propagation of wave packets and space-time self focusing in a nonlinear medium,” Sov. Phys. JETP 24, 797–800 (1969).
  3. Y. S. Kivshar and M. Peyrard, “Modulational instabilities in discrete lattices,” Phys. Rev. A 46, 3198–3205 (1992).
    [Crossref] [PubMed]
  4. N. Akhmediev and A. Ankiewicz, Solitons (Chapman and Hall, 1997).
  5. Y. S. Kivshar and G. Agrawal, Optical Solitons, from Fibers to Photonic Crystals (Academic, 2003).
  6. A. Hasegava and M. Matsumoto, Optical Solitons in Fibers (Springer, 2003).
    [Crossref]
  7. B. Malomed, Soliton Management in Periodic Systems (SpringerUS, 2006).
  8. Z. Rapti, P. G. Kevrekidis, A. Smerzi, and A. R. Bishop, “Variational approach to the modulational instability,” Phys. Rev. E 69, 017601 (2004).
    [Crossref]
  9. Y. S. Kivshar and M. Salerno, “Modulational instabilities in the discrete deformable nonlinear Schrodinger equation,” Phys. Rev. E 49, 3543–3546 (1994).
    [Crossref]
  10. P. Shukla and J. Rasmussen, “Modulational instability of short pulses in long optical fibers,” Opt. Lett. 11, 171–173 (1986).
    [Crossref] [PubMed]
  11. M. Saha and A. Sarma, “Modulation instability in nonlinear metamaterials induced by cubicquintic nonlinearities and higher order dispersive effects,” Opt. Commun. 291, 321–325 (2013).
    [Crossref]
  12. H. Sakaguchi and Y. Kageyama, “Modulational instability and breathing motion in the two-dimensional nonlinear Schrodinger equation with a one-dimensional harmonic potential,” Phys. Rev. E 88, 053203 (2013).
    [Crossref]
  13. M. Potasek, “Modulation instability in an extended nonlinear Schrdinger equation,” Opt. Lett. 12, 921–923 (1987).
    [Crossref] [PubMed]
  14. M. Centurion, M. A. Porter, Y. Pu, P. G. Kevrekidis, D. J. Frantzeskakis, and D. Psaltis, “Modulational instability in a layered Kerr medium: theory and experiment,” Phys. Rev. Lett. 97, 234101 (2006).
    [Crossref]
  15. S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
    [Crossref]
  16. S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self-steepening effect in modulation instability in negative-index material,” Opt. Express 14, 1568–1575 (2006).
    [Crossref] [PubMed]
  17. Y. Xiang, S. Wen, X. Dai, Z. Tang, W. Su, and D. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063 (2007).
    [Crossref]
  18. 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]
  19. X. Dai, Y. Xiang, S. Wen, and D. Fan, “Modulation instability of copropagating light beams in nonlinear metamaterials,” J. Opt. Soc. Am. B 26, 564–571 (2009).
    [Crossref]
  20. A. 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]
  21. 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]
  22. I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005).
    [Crossref]
  23. I. Kourakis, N. Lazarides, and G. P. Tsironis, “Self-focusing and envelope pulse generation in nonlinear magnetic metamaterials,” Phys. Rev. E 75, 067601 (2007).
    [Crossref]
  24. W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, G. Chen, and L. Yi, “Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrdinger equation with distributed coefficients,” Phys. Rev. A 78, 023821 (2008).
    [Crossref]
  25. M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (3+ 1)-dimensional nonlinear Schrdinger equation,” Phys. Rev. Lett. 101, 123904 (2008).
    [Crossref]
  26. N. Petrović, M. Belić, W. Zhong, R. Xie, and G. Chen, “Exact spatiotemporal wave and soliton solutions to the generalized (3+ 1)-dimensional Schrdinger equation for both normal and anomalous dispersion,” Opt. Lett. 34, 1609–1611 (2009).
    [Crossref]
  27. N. Z. Petrović, M. Belić, and W. P. Zhong, “Spatiotemporal wave and soliton solutions to the generalized (3+1)-dimensional Gross-Pitaevskii equation,” Phys. Rev. E 81, 016610 (2010).
    [Crossref]
  28. N. Z. Petrović, N. B. Aleksić, A. A. Bastami, and M. R. Belić, “Analytical traveling-wave and solitary solutions to the generalized Gross-Pitaevskii equation with sinusoidal time-varying diffraction and potential,” Phys. Rev. E 83, 036609 (2011).
    [Crossref]
  29. Y. B. Zhou, M. L. Wang, and Y. M. Wang, “Periodic wave solutions to a coupled KdV equations with variable coefficients,” Phys. Lett. A 308, 31–36 (2003).
    [Crossref]
  30. Y. B. Zhou, M. L. Wang, and T. D. Miao, “The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations,” Phys. Lett. A 323, 77–88 (2004).
    [Crossref]
  31. G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
    [Crossref] [PubMed]
  32. I. Towers and B. A. Malomed, “Stable (2+ 1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity,” J. Opt. Soc. Am. 19, 537–543 (2002).
    [Crossref]
  33. S. A. Ponomarenko and G. P. Agrawal, “Interactions of chirped and chirp-free similaritons in optical fiber amplifiers,” Opt. Express 15, 2963–2973 (2007).
    [Crossref] [PubMed]
  34. C. Dai, Y. Wang, and J. Zhang, “Analytical spatiotemporal localizations for the generalized (3+1)-dimensional nonlinear Schrödinger equation,” Opt. Lett. 35, 1437–1439 (2010).
    [Crossref] [PubMed]
  35. F. Ndzana, A. Mohamadou, and T. C. Kofane, “Modulational instability in the cubicquintic nonlinear Schrödinger equation through the variational approach,” Opt. Commun. 275, 421–428 (2007).
    [Crossref]
  36. E. Wamba, A. Mohamadou, and T. C. Kofane, “A variational approach to the modulational instability of a Bose-Einstein condensate in a parabolic trap,” J. Phys. B: At. Mol. Opt. Phys. 41, 225403 (2008).
    [Crossref]
  37. N. Petrović, H. Zahreddine, and M. Belić, “Exact spatiotemporal wave and soliton solutions to the generalized (3+ 1)-dimensional nonlinear Schrödinger equation with linear potential,” Phys. Scripta 83, 065001 (2011).
    [Crossref]
  38. Y. Castin, Coherent Atomic Matter Waves, R. Kaiser, C. Westbrook, and F. David, eds. (Les Houches Session LXXII, Springer, 2001).

2013 (2)

M. Saha and A. Sarma, “Modulation instability in nonlinear metamaterials induced by cubicquintic nonlinearities and higher order dispersive effects,” Opt. Commun. 291, 321–325 (2013).
[Crossref]

H. Sakaguchi and Y. Kageyama, “Modulational instability and breathing motion in the two-dimensional nonlinear Schrodinger equation with a one-dimensional harmonic potential,” Phys. Rev. E 88, 053203 (2013).
[Crossref]

2011 (4)

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]

A. 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]

N. Z. Petrović, N. B. Aleksić, A. A. Bastami, and M. R. Belić, “Analytical traveling-wave and solitary solutions to the generalized Gross-Pitaevskii equation with sinusoidal time-varying diffraction and potential,” Phys. Rev. E 83, 036609 (2011).
[Crossref]

N. Petrović, H. Zahreddine, and M. Belić, “Exact spatiotemporal wave and soliton solutions to the generalized (3+ 1)-dimensional nonlinear Schrödinger equation with linear potential,” Phys. Scripta 83, 065001 (2011).
[Crossref]

2010 (2)

C. Dai, Y. Wang, and J. Zhang, “Analytical spatiotemporal localizations for the generalized (3+1)-dimensional nonlinear Schrödinger equation,” Opt. Lett. 35, 1437–1439 (2010).
[Crossref] [PubMed]

N. Z. Petrović, M. Belić, and W. P. Zhong, “Spatiotemporal wave and soliton solutions to the generalized (3+1)-dimensional Gross-Pitaevskii equation,” Phys. Rev. E 81, 016610 (2010).
[Crossref]

2009 (2)

2008 (3)

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, G. Chen, and L. Yi, “Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrdinger equation with distributed coefficients,” Phys. Rev. A 78, 023821 (2008).
[Crossref]

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (3+ 1)-dimensional nonlinear Schrdinger equation,” Phys. Rev. Lett. 101, 123904 (2008).
[Crossref]

E. Wamba, A. Mohamadou, and T. C. Kofane, “A variational approach to the modulational instability of a Bose-Einstein condensate in a parabolic trap,” J. Phys. B: At. Mol. Opt. Phys. 41, 225403 (2008).
[Crossref]

2007 (4)

F. Ndzana, A. Mohamadou, and T. C. Kofane, “Modulational instability in the cubicquintic nonlinear Schrödinger equation through the variational approach,” Opt. Commun. 275, 421–428 (2007).
[Crossref]

S. A. Ponomarenko and G. P. Agrawal, “Interactions of chirped and chirp-free similaritons in optical fiber amplifiers,” Opt. Express 15, 2963–2973 (2007).
[Crossref] [PubMed]

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

I. Kourakis, N. Lazarides, and G. P. Tsironis, “Self-focusing and envelope pulse generation in nonlinear magnetic metamaterials,” Phys. Rev. E 75, 067601 (2007).
[Crossref]

2006 (3)

M. Centurion, M. A. Porter, Y. Pu, P. G. Kevrekidis, D. J. Frantzeskakis, and D. Psaltis, “Modulational instability in a layered Kerr medium: theory and experiment,” Phys. Rev. Lett. 97, 234101 (2006).
[Crossref]

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
[Crossref]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self-steepening effect in modulation instability in negative-index material,” Opt. Express 14, 1568–1575 (2006).
[Crossref] [PubMed]

2005 (2)

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]

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

2004 (2)

Z. Rapti, P. G. Kevrekidis, A. Smerzi, and A. R. Bishop, “Variational approach to the modulational instability,” Phys. Rev. E 69, 017601 (2004).
[Crossref]

Y. B. Zhou, M. L. Wang, and T. D. Miao, “The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations,” Phys. Lett. A 323, 77–88 (2004).
[Crossref]

2003 (1)

Y. B. Zhou, M. L. Wang, and Y. M. Wang, “Periodic wave solutions to a coupled KdV equations with variable coefficients,” Phys. Lett. A 308, 31–36 (2003).
[Crossref]

2002 (1)

I. Towers and B. A. Malomed, “Stable (2+ 1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity,” J. Opt. Soc. Am. 19, 537–543 (2002).
[Crossref]

1994 (1)

Y. S. Kivshar and M. Salerno, “Modulational instabilities in the discrete deformable nonlinear Schrodinger equation,” Phys. Rev. E 49, 3543–3546 (1994).
[Crossref]

1992 (1)

Y. S. Kivshar and M. Peyrard, “Modulational instabilities in discrete lattices,” Phys. Rev. A 46, 3198–3205 (1992).
[Crossref] [PubMed]

1987 (2)

M. Potasek, “Modulation instability in an extended nonlinear Schrdinger equation,” Opt. Lett. 12, 921–923 (1987).
[Crossref] [PubMed]

G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[Crossref] [PubMed]

1986 (1)

1969 (1)

L. Ostrovskii, “Propagation of wave packets and space-time self focusing in a nonlinear medium,” Sov. Phys. JETP 24, 797–800 (1969).

Agrawal, G.

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

Agrawal, G. P.

S. A. Ponomarenko and G. P. Agrawal, “Interactions of chirped and chirp-free similaritons in optical fiber amplifiers,” Opt. Express 15, 2963–2973 (2007).
[Crossref] [PubMed]

G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[Crossref] [PubMed]

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

Akhmediev, N.

N. Akhmediev and A. Ankiewicz, Solitons (Chapman and Hall, 1997).

Aleksic, N. B.

N. Z. Petrović, N. B. Aleksić, A. A. Bastami, and M. R. Belić, “Analytical traveling-wave and solitary solutions to the generalized Gross-Pitaevskii equation with sinusoidal time-varying diffraction and potential,” Phys. Rev. E 83, 036609 (2011).
[Crossref]

Ankiewicz, A.

N. Akhmediev and A. Ankiewicz, Solitons (Chapman and Hall, 1997).

Bastami, A. A.

N. Z. Petrović, N. B. Aleksić, A. A. Bastami, and M. R. Belić, “Analytical traveling-wave and solitary solutions to the generalized Gross-Pitaevskii equation with sinusoidal time-varying diffraction and potential,” Phys. Rev. E 83, 036609 (2011).
[Crossref]

Belic, M.

N. Petrović, H. Zahreddine, and M. Belić, “Exact spatiotemporal wave and soliton solutions to the generalized (3+ 1)-dimensional nonlinear Schrödinger equation with linear potential,” Phys. Scripta 83, 065001 (2011).
[Crossref]

N. Z. Petrović, M. Belić, and W. P. Zhong, “Spatiotemporal wave and soliton solutions to the generalized (3+1)-dimensional Gross-Pitaevskii equation,” Phys. Rev. E 81, 016610 (2010).
[Crossref]

N. Petrović, M. Belić, W. Zhong, R. Xie, and G. Chen, “Exact spatiotemporal wave and soliton solutions to the generalized (3+ 1)-dimensional Schrdinger equation for both normal and anomalous dispersion,” Opt. Lett. 34, 1609–1611 (2009).
[Crossref]

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (3+ 1)-dimensional nonlinear Schrdinger equation,” Phys. Rev. Lett. 101, 123904 (2008).
[Crossref]

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, G. Chen, and L. Yi, “Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrdinger equation with distributed coefficients,” Phys. Rev. A 78, 023821 (2008).
[Crossref]

Belic, M. R.

N. Z. Petrović, N. B. Aleksić, A. A. Bastami, and M. R. Belić, “Analytical traveling-wave and solitary solutions to the generalized Gross-Pitaevskii equation with sinusoidal time-varying diffraction and potential,” Phys. Rev. E 83, 036609 (2011).
[Crossref]

Bishop, A. R.

Z. Rapti, P. G. Kevrekidis, A. Smerzi, and A. R. Bishop, “Variational approach to the modulational instability,” Phys. Rev. E 69, 017601 (2004).
[Crossref]

Castin, Y.

Y. Castin, Coherent Atomic Matter Waves, R. Kaiser, C. Westbrook, and F. David, eds. (Les Houches Session LXXII, Springer, 2001).

Centurion, M.

M. Centurion, M. A. Porter, Y. Pu, P. G. Kevrekidis, D. J. Frantzeskakis, and D. Psaltis, “Modulational instability in a layered Kerr medium: theory and experiment,” Phys. Rev. Lett. 97, 234101 (2006).
[Crossref]

Chen, G.

N. Petrović, M. Belić, W. Zhong, R. Xie, and G. Chen, “Exact spatiotemporal wave and soliton solutions to the generalized (3+ 1)-dimensional Schrdinger equation for both normal and anomalous dispersion,” Opt. Lett. 34, 1609–1611 (2009).
[Crossref]

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, G. Chen, and L. Yi, “Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrdinger equation with distributed coefficients,” Phys. Rev. A 78, 023821 (2008).
[Crossref]

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (3+ 1)-dimensional nonlinear Schrdinger equation,” Phys. Rev. Lett. 101, 123904 (2008).
[Crossref]

Dai, C.

Dai, X.

Fan, D.

Frantzeskakis, D. J.

M. Centurion, M. A. Porter, Y. Pu, P. G. Kevrekidis, D. J. Frantzeskakis, and D. Psaltis, “Modulational instability in a layered Kerr medium: theory and experiment,” Phys. Rev. Lett. 97, 234101 (2006).
[Crossref]

Fu, X.

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
[Crossref]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self-steepening effect in modulation instability in negative-index material,” Opt. Express 14, 1568–1575 (2006).
[Crossref] [PubMed]

Hasegava, A.

A. Hasegava and M. Matsumoto, Optical Solitons in Fibers (Springer, 2003).
[Crossref]

Hu, Y.

Kageyama, Y.

H. Sakaguchi and Y. Kageyama, “Modulational instability and breathing motion in the two-dimensional nonlinear Schrodinger equation with a one-dimensional harmonic potential,” Phys. Rev. E 88, 053203 (2013).
[Crossref]

Kevrekidis, P. G.

M. Centurion, M. A. Porter, Y. Pu, P. G. Kevrekidis, D. J. Frantzeskakis, and D. Psaltis, “Modulational instability in a layered Kerr medium: theory and experiment,” Phys. Rev. Lett. 97, 234101 (2006).
[Crossref]

Z. Rapti, P. G. Kevrekidis, A. Smerzi, and A. R. Bishop, “Variational approach to the modulational instability,” Phys. Rev. E 69, 017601 (2004).
[Crossref]

Kivshar, Y. S.

Y. S. Kivshar and M. Salerno, “Modulational instabilities in the discrete deformable nonlinear Schrodinger equation,” Phys. Rev. E 49, 3543–3546 (1994).
[Crossref]

Y. S. Kivshar and M. Peyrard, “Modulational instabilities in discrete lattices,” Phys. Rev. A 46, 3198–3205 (1992).
[Crossref] [PubMed]

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

Kofane, T. C.

E. Wamba, A. Mohamadou, and T. C. Kofane, “A variational approach to the modulational instability of a Bose-Einstein condensate in a parabolic trap,” J. Phys. B: At. Mol. Opt. Phys. 41, 225403 (2008).
[Crossref]

F. Ndzana, A. Mohamadou, and T. C. Kofane, “Modulational instability in the cubicquintic nonlinear Schrödinger equation through the variational approach,” Opt. Commun. 275, 421–428 (2007).
[Crossref]

Kourakis, I.

I. Kourakis, N. Lazarides, and G. P. Tsironis, “Self-focusing and envelope pulse generation in nonlinear magnetic metamaterials,” Phys. Rev. E 75, 067601 (2007).
[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]

Lazarides, N.

I. Kourakis, N. Lazarides, and G. P. Tsironis, “Self-focusing and envelope pulse generation in nonlinear magnetic metamaterials,” Phys. Rev. E 75, 067601 (2007).
[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]

Malomed, B.

B. Malomed, Soliton Management in Periodic Systems (SpringerUS, 2006).

Malomed, B. A.

I. Towers and B. A. Malomed, “Stable (2+ 1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity,” J. Opt. Soc. Am. 19, 537–543 (2002).
[Crossref]

Matsumoto, M.

A. Hasegava and M. Matsumoto, Optical Solitons in Fibers (Springer, 2003).
[Crossref]

Miao, T. D.

Y. B. Zhou, M. L. Wang, and T. D. Miao, “The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations,” Phys. Lett. A 323, 77–88 (2004).
[Crossref]

Mohamadou, A.

E. Wamba, A. Mohamadou, and T. C. Kofane, “A variational approach to the modulational instability of a Bose-Einstein condensate in a parabolic trap,” J. Phys. B: At. Mol. Opt. Phys. 41, 225403 (2008).
[Crossref]

F. Ndzana, A. Mohamadou, and T. C. Kofane, “Modulational instability in the cubicquintic nonlinear Schrödinger equation through the variational approach,” Opt. Commun. 275, 421–428 (2007).
[Crossref]

Ndzana, F.

F. Ndzana, A. Mohamadou, and T. C. Kofane, “Modulational instability in the cubicquintic nonlinear Schrödinger equation through the variational approach,” Opt. Commun. 275, 421–428 (2007).
[Crossref]

Ostrovskii, L.

L. Ostrovskii, “Propagation of wave packets and space-time self focusing in a nonlinear medium,” Sov. Phys. JETP 24, 797–800 (1969).

Petrovic, N.

N. Petrović, H. Zahreddine, and M. Belić, “Exact spatiotemporal wave and soliton solutions to the generalized (3+ 1)-dimensional nonlinear Schrödinger equation with linear potential,” Phys. Scripta 83, 065001 (2011).
[Crossref]

N. Petrović, M. Belić, W. Zhong, R. Xie, and G. Chen, “Exact spatiotemporal wave and soliton solutions to the generalized (3+ 1)-dimensional Schrdinger equation for both normal and anomalous dispersion,” Opt. Lett. 34, 1609–1611 (2009).
[Crossref]

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, G. Chen, and L. Yi, “Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrdinger equation with distributed coefficients,” Phys. Rev. A 78, 023821 (2008).
[Crossref]

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (3+ 1)-dimensional nonlinear Schrdinger equation,” Phys. Rev. Lett. 101, 123904 (2008).
[Crossref]

Petrovic, N. Z.

N. Z. Petrović, N. B. Aleksić, A. A. Bastami, and M. R. Belić, “Analytical traveling-wave and solitary solutions to the generalized Gross-Pitaevskii equation with sinusoidal time-varying diffraction and potential,” Phys. Rev. E 83, 036609 (2011).
[Crossref]

N. Z. Petrović, M. Belić, and W. P. Zhong, “Spatiotemporal wave and soliton solutions to the generalized (3+1)-dimensional Gross-Pitaevskii equation,” Phys. Rev. E 81, 016610 (2010).
[Crossref]

Peyrard, M.

Y. S. Kivshar and M. Peyrard, “Modulational instabilities in discrete lattices,” Phys. Rev. A 46, 3198–3205 (1992).
[Crossref] [PubMed]

Ponomarenko, S. A.

Porter, M. A.

M. Centurion, M. A. Porter, Y. Pu, P. G. Kevrekidis, D. J. Frantzeskakis, and D. Psaltis, “Modulational instability in a layered Kerr medium: theory and experiment,” Phys. Rev. Lett. 97, 234101 (2006).
[Crossref]

Potasek, M.

Psaltis, D.

M. Centurion, M. A. Porter, Y. Pu, P. G. Kevrekidis, D. J. Frantzeskakis, and D. Psaltis, “Modulational instability in a layered Kerr medium: theory and experiment,” Phys. Rev. Lett. 97, 234101 (2006).
[Crossref]

Pu, Y.

M. Centurion, M. A. Porter, Y. Pu, P. G. Kevrekidis, D. J. Frantzeskakis, and D. Psaltis, “Modulational instability in a layered Kerr medium: theory and experiment,” Phys. Rev. Lett. 97, 234101 (2006).
[Crossref]

Rapti, Z.

Z. Rapti, P. G. Kevrekidis, A. Smerzi, and A. R. Bishop, “Variational approach to the modulational instability,” Phys. Rev. E 69, 017601 (2004).
[Crossref]

Rasmussen, J.

Saha, M.

M. Saha and A. Sarma, “Modulation instability in nonlinear metamaterials induced by cubicquintic nonlinearities and higher order dispersive effects,” Opt. Commun. 291, 321–325 (2013).
[Crossref]

A. 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]

Sakaguchi, H.

H. Sakaguchi and Y. Kageyama, “Modulational instability and breathing motion in the two-dimensional nonlinear Schrodinger equation with a one-dimensional harmonic potential,” Phys. Rev. E 88, 053203 (2013).
[Crossref]

Salerno, M.

Y. S. Kivshar and M. Salerno, “Modulational instabilities in the discrete deformable nonlinear Schrodinger equation,” Phys. Rev. E 49, 3543–3546 (1994).
[Crossref]

Sarma, A.

M. Saha and A. Sarma, “Modulation instability in nonlinear metamaterials induced by cubicquintic nonlinearities and higher order dispersive effects,” Opt. Commun. 291, 321–325 (2013).
[Crossref]

A. 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]

Shukla, P.

Shukla, P. K.

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

Smerzi, A.

Z. Rapti, P. G. Kevrekidis, A. Smerzi, and A. R. Bishop, “Variational approach to the modulational instability,” Phys. Rev. E 69, 017601 (2004).
[Crossref]

Su, W.

Tang, Z.

Towers, I.

I. Towers and B. A. Malomed, “Stable (2+ 1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity,” J. Opt. Soc. Am. 19, 537–543 (2002).
[Crossref]

Tsironis, G. P.

I. Kourakis, N. Lazarides, and G. P. Tsironis, “Self-focusing and envelope pulse generation in nonlinear magnetic metamaterials,” Phys. Rev. E 75, 067601 (2007).
[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]

Wamba, E.

E. Wamba, A. Mohamadou, and T. C. Kofane, “A variational approach to the modulational instability of a Bose-Einstein condensate in a parabolic trap,” J. Phys. B: At. Mol. Opt. Phys. 41, 225403 (2008).
[Crossref]

Wang, M. L.

Y. B. Zhou, M. L. Wang, and T. D. Miao, “The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations,” Phys. Lett. A 323, 77–88 (2004).
[Crossref]

Y. B. Zhou, M. L. Wang, and Y. M. Wang, “Periodic wave solutions to a coupled KdV equations with variable coefficients,” Phys. Lett. A 308, 31–36 (2003).
[Crossref]

Wang, Y.

C. Dai, Y. Wang, and J. Zhang, “Analytical spatiotemporal localizations for the generalized (3+1)-dimensional nonlinear Schrödinger equation,” Opt. Lett. 35, 1437–1439 (2010).
[Crossref] [PubMed]

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
[Crossref]

Wang, Y. M.

Y. B. Zhou, M. L. Wang, and Y. M. Wang, “Periodic wave solutions to a coupled KdV equations with variable coefficients,” Phys. Lett. A 308, 31–36 (2003).
[Crossref]

Wen, S.

Xiang, Y.

Xie, R.

Xie, R. H.

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, G. Chen, and L. Yi, “Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrdinger equation with distributed coefficients,” Phys. Rev. A 78, 023821 (2008).
[Crossref]

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (3+ 1)-dimensional nonlinear Schrdinger equation,” Phys. Rev. Lett. 101, 123904 (2008).
[Crossref]

Yi, L.

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, G. Chen, and L. Yi, “Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrdinger equation with distributed coefficients,” Phys. Rev. A 78, 023821 (2008).
[Crossref]

Zahreddine, H.

N. Petrović, H. Zahreddine, and M. Belić, “Exact spatiotemporal wave and soliton solutions to the generalized (3+ 1)-dimensional nonlinear Schrödinger equation with linear potential,” Phys. Scripta 83, 065001 (2011).
[Crossref]

Zhang, J.

Zhong, W.

Zhong, W. P.

N. Z. Petrović, M. Belić, and W. P. Zhong, “Spatiotemporal wave and soliton solutions to the generalized (3+1)-dimensional Gross-Pitaevskii equation,” Phys. Rev. E 81, 016610 (2010).
[Crossref]

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (3+ 1)-dimensional nonlinear Schrdinger equation,” Phys. Rev. Lett. 101, 123904 (2008).
[Crossref]

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, G. Chen, and L. Yi, “Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrdinger equation with distributed coefficients,” Phys. Rev. A 78, 023821 (2008).
[Crossref]

Zhou, Y. B.

Y. B. Zhou, M. L. Wang, and T. D. Miao, “The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations,” Phys. Lett. A 323, 77–88 (2004).
[Crossref]

Y. B. Zhou, M. L. Wang, and Y. M. Wang, “Periodic wave solutions to a coupled KdV equations with variable coefficients,” Phys. Lett. A 308, 31–36 (2003).
[Crossref]

J. Opt. Soc. Am. (1)

I. Towers and B. A. Malomed, “Stable (2+ 1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity,” J. Opt. Soc. Am. 19, 537–543 (2002).
[Crossref]

J. Opt. Soc. Am. B (4)

J. Phys. B: At. Mol. Opt. Phys. (1)

E. Wamba, A. Mohamadou, and T. C. Kofane, “A variational approach to the modulational instability of a Bose-Einstein condensate in a parabolic trap,” J. Phys. B: At. Mol. Opt. Phys. 41, 225403 (2008).
[Crossref]

Opt. Commun. (2)

F. Ndzana, A. Mohamadou, and T. C. Kofane, “Modulational instability in the cubicquintic nonlinear Schrödinger equation through the variational approach,” Opt. Commun. 275, 421–428 (2007).
[Crossref]

M. Saha and A. Sarma, “Modulation instability in nonlinear metamaterials induced by cubicquintic nonlinearities and higher order dispersive effects,” Opt. Commun. 291, 321–325 (2013).
[Crossref]

Opt. Express (2)

Opt. Lett. (4)

Phys. Lett. A (2)

Y. B. Zhou, M. L. Wang, and Y. M. Wang, “Periodic wave solutions to a coupled KdV equations with variable coefficients,” Phys. Lett. A 308, 31–36 (2003).
[Crossref]

Y. B. Zhou, M. L. Wang, and T. D. Miao, “The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations,” Phys. Lett. A 323, 77–88 (2004).
[Crossref]

Phys. Rev. A (2)

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, G. Chen, and L. Yi, “Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrdinger equation with distributed coefficients,” Phys. Rev. A 78, 023821 (2008).
[Crossref]

Y. S. Kivshar and M. Peyrard, “Modulational instabilities in discrete lattices,” Phys. Rev. A 46, 3198–3205 (1992).
[Crossref] [PubMed]

Phys. Rev. E (9)

Z. Rapti, P. G. Kevrekidis, A. Smerzi, and A. R. Bishop, “Variational approach to the modulational instability,” Phys. Rev. E 69, 017601 (2004).
[Crossref]

Y. S. Kivshar and M. Salerno, “Modulational instabilities in the discrete deformable nonlinear Schrodinger equation,” Phys. Rev. E 49, 3543–3546 (1994).
[Crossref]

H. Sakaguchi and Y. Kageyama, “Modulational instability and breathing motion in the two-dimensional nonlinear Schrodinger equation with a one-dimensional harmonic potential,” Phys. Rev. E 88, 053203 (2013).
[Crossref]

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
[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]

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

I. Kourakis, N. Lazarides, and G. P. Tsironis, “Self-focusing and envelope pulse generation in nonlinear magnetic metamaterials,” Phys. Rev. E 75, 067601 (2007).
[Crossref]

N. Z. Petrović, M. Belić, and W. P. Zhong, “Spatiotemporal wave and soliton solutions to the generalized (3+1)-dimensional Gross-Pitaevskii equation,” Phys. Rev. E 81, 016610 (2010).
[Crossref]

N. Z. Petrović, N. B. Aleksić, A. A. Bastami, and M. R. Belić, “Analytical traveling-wave and solitary solutions to the generalized Gross-Pitaevskii equation with sinusoidal time-varying diffraction and potential,” Phys. Rev. E 83, 036609 (2011).
[Crossref]

Phys. Rev. Lett. (3)

G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[Crossref] [PubMed]

M. Centurion, M. A. Porter, Y. Pu, P. G. Kevrekidis, D. J. Frantzeskakis, and D. Psaltis, “Modulational instability in a layered Kerr medium: theory and experiment,” Phys. Rev. Lett. 97, 234101 (2006).
[Crossref]

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (3+ 1)-dimensional nonlinear Schrdinger equation,” Phys. Rev. Lett. 101, 123904 (2008).
[Crossref]

Phys. Scripta (1)

N. Petrović, H. Zahreddine, and M. Belić, “Exact spatiotemporal wave and soliton solutions to the generalized (3+ 1)-dimensional nonlinear Schrödinger equation with linear potential,” Phys. Scripta 83, 065001 (2011).
[Crossref]

Sov. Phys. JETP (1)

L. Ostrovskii, “Propagation of wave packets and space-time self focusing in a nonlinear medium,” Sov. Phys. JETP 24, 797–800 (1969).

Other (6)

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

N. Akhmediev and A. Ankiewicz, Solitons (Chapman and Hall, 1997).

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

A. Hasegava and M. Matsumoto, Optical Solitons in Fibers (Springer, 2003).
[Crossref]

B. Malomed, Soliton Management in Periodic Systems (SpringerUS, 2006).

Y. Castin, Coherent Atomic Matter Waves, R. Kaiser, C. Westbrook, and F. David, eds. (Les Houches Session LXXII, Springer, 2001).

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Figures (7)

Fig. 1
Fig. 1 (a) Nonlinearity parameter d for solutions cn, sn and dn. (b) The growth rate parameter γ for dark an bright SWs, as a function of K for the case κσ = 1. Modulational instability occurs for values of K depicted in the respective graphs. The solid lines represent the theoretical calculation of K using Eq. (30), and the square and circle dots are values of γ measured using numerical simulations, in which the dark and bright SWs, respectively, were perturbed by a small wave of the given wave number K.
Fig. 2
Fig. 2 Perturbation amplitude growth for κσ = 1, a0 = 0 and d = 8/3 as a function of propagation distance z. Black curves are numerical results, while the red curves are analytical results: (a) β0 = 1, β1 = 0, top to bottom: numerical results for K = d / 2, analytic results for K = d / 2, analytical results for K = 2, numerical results for K = 2, (b) β0 = 0, β1 = 1, Z = 1, top to bottom: numerical results for K = d / 2, analytical results for K = d / 2, analytical results for K = 2, numerical results for K = 2.
Fig. 3
Fig. 3 Perturbation amplitude growth for κσ = 1, d = 8/3 and K = d / 2 as a function of propagation distance z for systems with chirp. Black curves are numerical results, while the red curves are analytical results: (a) β0 = 1, β1 = 0, a0 = 0.1, top: numerical results, bottom: analytical results, (b) β0 = 0, β1 = 1, Z = 1, dashed lines represent plots for a0 = 0.1, top to bottom: analytical results for a0 = 0.1, numerical results for a0 = 0.1, analytical results for a0 = 0.3, numerical results for a0 = 0.3.
Fig. 4
Fig. 4 Maximum amplitude of perturbation for K = d / 2 plotted against z = β1Z/π and: (a) d for a0 = 0.05, (b) a0 for d = 8/3.
Fig. 5
Fig. 5 Development of modulation instability for the bright SW for three different values of z. Here, x is the direction of perturbation, y is the direction of the SW and t is the remaining transverse direction. Bright colors, i.e. towards the color red (the center in Fig. 5(a)), indicate a higher value of |u|2.
Fig. 6
Fig. 6 Development of modulational instability for the dark SW for three different values of z. Here, x is the direction of perturbation, y is the direction of the SW and t is the remaining transverse direction. Red color (away from the center in Fig. 6(a)), indicates a higher value of |u|2.
Fig. 7
Fig. 7 Development of modulational instability for the dark traveling wave (F = sn) for three different values of z. Here, x is the direction of perturbation, y is the direction of the traveling wave and t is the remaining transverse direction. Parameters are M = 0.5 and K = d / 2. Blue color (at the top, bottom and the three central stripes in Fig. 7(a)), indicates a lower value of |u|2.

Tables (1)

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Table 1 Stability cases

Equations (63)

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i u z + β ( z ) 2 ( 2 u x 2 + 2 u y 2 + s 2 u t 2 ) + χ ( z ) | u | 2 u = i δ ( z ) u .
u = ( α ) 3 / 2 f 0 e 0 z δ d z ( F ( θ ) + ε c 0 c 4 1 F ( θ ) ) exp ( i ( a ( z ) ( x 2 + y 2 + s t 2 ) + b ( z ) ( x + y + t ) + e ( z ) ) ) ,
α = 1 1 + 2 a 0 0 z β d z
θ = k ( z ) x + l ( z ) y + m ( z ) t + ω ( z ) ,
χ ( z ) = c 4 β ( z ) α f 0 2 χ 0 exp ( 2 0 z γ d z ) ,
χ 0 = ( k 0 2 + l 0 2 + s m 0 2 ) .
a = α a 0 , b = α b 0 ,
k = α k 0 , l = α l 0 , m = α m 0 ,
ω = ω 0 α ( k 0 + l 0 + s m 0 ) b 0 0 z β d z ,
e = e 0 + ( α / 2 ) ( c χ 0 ( 2 + s ) b 0 2 ) 0 z β d z ,
e ¯ = e c 2 0 z β α 2 d z .
u G = u exp ( 0 z δ z ) exp ( i ( a ( x 2 + y 2 + s t 2 ) + b ( x + y + t ) + e ¯ ) ) / ( f 0 α 3 / 2 | χ 0 c 4 | 1 / 2 ) ,
x x = α ( x ς ) ,
y y = α ( y ς ) ,
t t = α ( t s ς ) ,
z z = 0 z α 2 β d z ,
ς ( z ) z = β ( z ) ( 2 a ( z ) ς ( z ) + b ( z ) ) ,
i G z + 1 2 ( 2 G x 2 + 2 G y 2 + s 2 G t 2 ) + σ | G | 2 G = 0 ,
G = G 0 ( 1 + U ( z ) cos ( K x ) ) ,
L = i 2 ( G G * z G * G z ) + 1 2 | G | 2 σ | G | 4 ,
L = ( i 2 ( G G * z G * G z ) + 1 2 | G | 2 σ | G | 4 ) d x d y d t .
Λ = L d z .
z U r = 1 2 K 2 α 2 β ( κ U i ) ,
z ( κ U i ) = 1 2 ( K 2 κ σ d ) α 2 β U r ,
d = d cn ( M ) = 8 3 ( 2 M 1 ) ( E ( M ) E ( am ( 5 K ( M ) | M ) | M ) ) 2 ( 2 5 M + 3 M 2 ) K ( M ) E ( M ) E ( am ( 5 K ( M ) | M ) | M ) ) 4 ( M 1 ) K ( M ) )
d = d dn ( M ) = 8 3 ( 2 M + ( 1 M ) K ( M ) 2 E ( M ) )
d = d sn ( M ) = 8 3 ( M + 1 ) ( E ( am ( 4 K ( M ) | M ) | M ) ) 2 ( 2 + M ) K ( M ) ( E ( am ( 4 K ( M ) | M ) | M ) 4 K ( M ) )
U r = U 0 cosh ( γ ξ 1 + 2 a 0 ξ ) ,
U i = U 0 2 γ K 2 sinh ( γ ξ 1 + 2 a 0 ξ ) ,
γ = K ( σ κ d K 2 ) / 2
ξ = 0 z β d z .
| U | = U 0 ( 1 + σ κ d γ 2 2 K 2 ( 1 4 a 0 ξ ) ξ 2 ) .
U r = U 0 cosh ( γ ξ ) ,
U i = U 0 2 γ K 2 sinh ( γ ξ ) .
1 d K 2 sin 2 ( γ ¯ β 1 Z / π ) | U / U 0 | 1 .
1 | U / U 0 | 1 + d K 2 sinh 2 ( γ ¯ β 1 Z / π ) .
| U | = U 0 1 + σ κ d K 2 sinh 2 ( γ 2 a 0 ) ( 1 C a 0 ξ ) ,
C = γ 8 a 0 σ κ d K 2 sinh ( γ a 0 ) 1 + σ κ d K 2 sinh 2 ( γ 2 a 0 ) .
i t u + β ( t ) 2 ( 2 u x 2 + 2 u y 2 + 2 u z 2 ) + χ ( t ) | u | 2 u + η ( t ) ( x 2 + y 2 + z 2 ) u = i δ ( t ) u .
b = p b 0 ,
k = p k 0 , l = p l 0 , m = p m 0 ,
ω = ω 0 q ( k 0 + l 0 + m 0 ) b 0 .
p = η 0 η 0 2 a 0 2 β 0 sech ( τ ( t ) + τ 0 ) ,
q = η 0 β 0 2 ( η 0 2 a 0 2 β 0 ) ( tanh ( τ ( t ) + τ 0 ) tanh τ 0 ) ,
a = η 0 2 β 0 tanh ( τ ( t ) + τ 0 ) ,
τ 0 = arctanh ( a 0 2 β 0 η 0 )
τ ( t ) = 2 α 0 β 0 0 t β ( t ) d t .
χ ( t ) = c 4 β ( t ) p f 0 2 χ 0 exp ( 2 0 t δ d t ) ,
e ¯ = e q c 2 ,
G = u p 3 / 2 f 0 | c 4 χ 0 | exp ( 0 t δ d t ) exp ( i a ( x 2 + y 2 + z 2 ) i b ( x + y + z ) i e ¯ )
x x = p ( x ζ )
y y = p ( y ζ )
z z = p ( z ζ )
t t = 0 t p 2 β d t ,
i G t + ( 2 G x 2 + 2 G y 2 + 2 G z 2 ) σ | G | 2 G = 0 ,
G = G 0 ( 1 + U ( t ) cos ( K x ) ) ,
U r t = 1 2 K 2 p 2 β U i ,
U i t = 1 2 ( K 2 σ d ) p 2 β U r ,
U r ( t ) = U 0 cosh ( γ q ( τ ) ) ,
U i ( t ) = U 0 γ K 2 sinh ( γ q ( τ ) ) ,
γ = K ( d K 2 ) / 2 .
| U | = U 0 ( 1 + σ d K 2 sinh 2 ( γ q ( τ ) ) ) 1 / 2 .
γ q = d 4 2 ( η 0 2 a 0 ) .

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