Abstract

Rogue waves in optical fibers can be mathematically described by the nonlinear Schrödinger equation and its extensions that take into account third-order dispersion, self-steepening, and self-frequency shift. These equations are integrable in special cases such as the Sasa–Satsuma or the Hirota equations. However, approximate polynomial solutions can also be obtained in cases beyond these integrable ones. We present these solutions and confirm their validity using numerical simulations.

© 2012 Optical Society of America

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  1. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air–silica microstructure optical fiber with anomalous dispersion at 800 nm,” Opt. Lett. 25, 25–27 (2000).
    [CrossRef]
  2. A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001).
    [CrossRef]
  3. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
    [CrossRef]
  4. J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
    [CrossRef]
  5. J. M. Dudley and J. R. Taylor, eds., Supercontinuum Generation in Optical Fibers (Cambridge University, 2010).
  6. K. M. Hilligsoe, H. N. Paulsen, J. Thogersen, S. R. Keiding, and J. J. Larsen, “Initial steps of supercontinuum generation in photonic crystal fibers,” J. Opt. Soc. Am. B 20, 1887–1893 (2003).
    [CrossRef]
  7. J. Bethge, A. Husakou, F. Mitschke, F. Noack, U. Griebner, G. Steinmeyer, and J. Herrmann, “Two-octave supercontinuum generation in a water-filled photonic crystal fiber,” Opt. Express 18, 6230–6240 (2010).
    [CrossRef]
  8. A. Mussot and A. Kudlinski, “19.5 W CW-pumped supercontinuum source from 0.65 to 1.38  μm,” Electron. Lett. 45, 29–30 (2009).
    [CrossRef]
  9. J. C. Travers, “High average power supercontinuum sources,” Pramana J. Phys. 75, 769–785 (2010).
    [CrossRef]
  10. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
    [CrossRef]
  11. N. Akhmediev and E. Pelinovsky, eds., Issue on “Rogue waves—towards a unifying concept?” Eur. Phys. J. Special Topics 185, 1–266 (2010).
  12. D. V. Skryabin and A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287–1299 (2010).
    [CrossRef]
  13. G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010).
    [CrossRef]
  14. J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express 17, 21497–21508 (2009).
    [CrossRef]
  15. A. Mussot, A. Kudlinski, M. Kolobov, E. Louvergneaux, M. Douay, and M. Taki, “Observation of extreme temporal events in CW-pumped supercontinuum,” Opt. Express 17, 17010–17015 (2009).
    [CrossRef]
  16. Ch. Mahnke and F. Mitschke, “Possibility of an Akhmediev breather decaying into solitons,” Phys. Rev. A 85, 033808 (2012).
    [CrossRef]
  17. N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
    [CrossRef]
  18. P. Gaillard, “Families of quasi-rational solutions of the NLS equation and multi-rogue waves,” J. Phys. A 44, 435204(2011).
    [CrossRef]
  19. Y. Ohta and J. Yang, “General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation,” Proc. R. Soc. A 468, 1716–1740 (2012).
    [CrossRef]
  20. A. Ankiewicz, N. Devine, and N. Akhmediev, “Are rogue waves robust against perturbations?” Phys. Lett. A 373, 3997–4000 (2009).
    [CrossRef]
  21. B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
    [CrossRef]
  22. D. H. Peregrine, “Water waves, nonlinear Schrödinger equations and their solutions,” J. Aust. Math. Soc. B 25, 16–43 (1983).
    [CrossRef]
  23. S. B. Cavalcanti, J. C. Cressoni, Heber R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
    [CrossRef]
  24. M. J. Potasek, “Modulation instability in an extended nonlinear Schrödinger equation,” Opt. Lett. 12, 921–923 (1987).
    [CrossRef]
  25. N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams (Chapman and Hall, 1997).
  26. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
    [CrossRef]
  27. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed., Optics and Photonics series (Academic, 2006), Section 5.5.3.
  28. N. Sasa and J. Satsuma, “New type of soliton solutions for a higher-order nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 60, 409–417 (1991).
    [CrossRef]
  29. R. Hirota, “Exact envelope-soliton solutions of a nonlinear wave equation,” J. Math. Phys. 14, 805–809 (1973).
    [CrossRef]
  30. D. Anderson and M. Lisak, “Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,” Phys. Rev. A 27, 1393–1398 (1983).
    [CrossRef]
  31. S. Palacios, A. Guinea, J. M. Fernandez-Diaz, and R. D. Crespo, “Dark solitary waves in the nonlinear Schrödinger equation with third order dispersion, self-steepening, and self-frequency shift,” Phys. Rev. E 60, R45–R47 (1999).
    [CrossRef]
  32. Z. Li, L. Li, H. Tian, and G. Zhou, “New types of solitary wave solutions for the higher order nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
    [CrossRef]
  33. T. Brugarino and M. Sciacca, “Singularity analysis and integrability for a HNLS equation governing pulse propagation in a generic fiber optics,” Opt. Commun. 262, 250–256 (2006).
    [CrossRef]
  34. U. Bandelow and N. Akhmediev, “Persistence of rogue waves in extended nonlinear Schrödinger equations: integrable Sasa–Satsuma case,” Phys. Lett. A 376, 1558–1561 (2012).
    [CrossRef]
  35. A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Rogue waves and rational solutions of the Hirota equation,” Phys. Rev. E 81, 046602 (2010).
    [CrossRef]
  36. N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “Modulation instability of CW signal in an optical fiber with the 3rd-order dispersion,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 33, 95–100 (1990); Radiofiz 33, 111–117 (1990).

2012 (3)

Ch. Mahnke and F. Mitschke, “Possibility of an Akhmediev breather decaying into solitons,” Phys. Rev. A 85, 033808 (2012).
[CrossRef]

Y. Ohta and J. Yang, “General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation,” Proc. R. Soc. A 468, 1716–1740 (2012).
[CrossRef]

U. Bandelow and N. Akhmediev, “Persistence of rogue waves in extended nonlinear Schrödinger equations: integrable Sasa–Satsuma case,” Phys. Lett. A 376, 1558–1561 (2012).
[CrossRef]

2011 (1)

P. Gaillard, “Families of quasi-rational solutions of the NLS equation and multi-rogue waves,” J. Phys. A 44, 435204(2011).
[CrossRef]

2010 (7)

A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Rogue waves and rational solutions of the Hirota equation,” Phys. Rev. E 81, 046602 (2010).
[CrossRef]

J. Bethge, A. Husakou, F. Mitschke, F. Noack, U. Griebner, G. Steinmeyer, and J. Herrmann, “Two-octave supercontinuum generation in a water-filled photonic crystal fiber,” Opt. Express 18, 6230–6240 (2010).
[CrossRef]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

J. C. Travers, “High average power supercontinuum sources,” Pramana J. Phys. 75, 769–785 (2010).
[CrossRef]

N. Akhmediev and E. Pelinovsky, eds., Issue on “Rogue waves—towards a unifying concept?” Eur. Phys. J. Special Topics 185, 1–266 (2010).

D. V. Skryabin and A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287–1299 (2010).
[CrossRef]

G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010).
[CrossRef]

2009 (5)

A. Mussot and A. Kudlinski, “19.5 W CW-pumped supercontinuum source from 0.65 to 1.38  μm,” Electron. Lett. 45, 29–30 (2009).
[CrossRef]

A. Ankiewicz, N. Devine, and N. Akhmediev, “Are rogue waves robust against perturbations?” Phys. Lett. A 373, 3997–4000 (2009).
[CrossRef]

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
[CrossRef]

A. Mussot, A. Kudlinski, M. Kolobov, E. Louvergneaux, M. Douay, and M. Taki, “Observation of extreme temporal events in CW-pumped supercontinuum,” Opt. Express 17, 17010–17015 (2009).
[CrossRef]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express 17, 21497–21508 (2009).
[CrossRef]

2007 (1)

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

2006 (2)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

T. Brugarino and M. Sciacca, “Singularity analysis and integrability for a HNLS equation governing pulse propagation in a generic fiber optics,” Opt. Commun. 262, 250–256 (2006).
[CrossRef]

2003 (1)

2002 (1)

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef]

2001 (1)

A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef]

2000 (2)

J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air–silica microstructure optical fiber with anomalous dispersion at 800 nm,” Opt. Lett. 25, 25–27 (2000).
[CrossRef]

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

1999 (1)

S. Palacios, A. Guinea, J. M. Fernandez-Diaz, and R. D. Crespo, “Dark solitary waves in the nonlinear Schrödinger equation with third order dispersion, self-steepening, and self-frequency shift,” Phys. Rev. E 60, R45–R47 (1999).
[CrossRef]

1991 (2)

N. Sasa and J. Satsuma, “New type of soliton solutions for a higher-order nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 60, 409–417 (1991).
[CrossRef]

S. B. Cavalcanti, J. C. Cressoni, Heber R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef]

1990 (1)

N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “Modulation instability of CW signal in an optical fiber with the 3rd-order dispersion,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 33, 95–100 (1990); Radiofiz 33, 111–117 (1990).

1987 (1)

1983 (2)

D. Anderson and M. Lisak, “Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,” Phys. Rev. A 27, 1393–1398 (1983).
[CrossRef]

D. H. Peregrine, “Water waves, nonlinear Schrödinger equations and their solutions,” J. Aust. Math. Soc. B 25, 16–43 (1983).
[CrossRef]

1973 (2)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

R. Hirota, “Exact envelope-soliton solutions of a nonlinear wave equation,” J. Math. Phys. 14, 805–809 (1973).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed., Optics and Photonics series (Academic, 2006), Section 5.5.3.

Akhmediev, N.

U. Bandelow and N. Akhmediev, “Persistence of rogue waves in extended nonlinear Schrödinger equations: integrable Sasa–Satsuma case,” Phys. Lett. A 376, 1558–1561 (2012).
[CrossRef]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Rogue waves and rational solutions of the Hirota equation,” Phys. Rev. E 81, 046602 (2010).
[CrossRef]

G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010).
[CrossRef]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express 17, 21497–21508 (2009).
[CrossRef]

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
[CrossRef]

A. Ankiewicz, N. Devine, and N. Akhmediev, “Are rogue waves robust against perturbations?” Phys. Lett. A 373, 3997–4000 (2009).
[CrossRef]

N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “Modulation instability of CW signal in an optical fiber with the 3rd-order dispersion,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 33, 95–100 (1990); Radiofiz 33, 111–117 (1990).

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

Anderson, D.

D. Anderson and M. Lisak, “Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,” Phys. Rev. A 27, 1393–1398 (1983).
[CrossRef]

Ankiewicz, A.

A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Rogue waves and rational solutions of the Hirota equation,” Phys. Rev. E 81, 046602 (2010).
[CrossRef]

A. Ankiewicz, N. Devine, and N. Akhmediev, “Are rogue waves robust against perturbations?” Phys. Lett. A 373, 3997–4000 (2009).
[CrossRef]

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
[CrossRef]

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

Bandelow, U.

U. Bandelow and N. Akhmediev, “Persistence of rogue waves in extended nonlinear Schrödinger equations: integrable Sasa–Satsuma case,” Phys. Lett. A 376, 1558–1561 (2012).
[CrossRef]

Bang, O.

G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010).
[CrossRef]

Bethge, J.

Brugarino, T.

T. Brugarino and M. Sciacca, “Singularity analysis and integrability for a HNLS equation governing pulse propagation in a generic fiber optics,” Opt. Commun. 262, 250–256 (2006).
[CrossRef]

Cavalcanti, S. B.

S. B. Cavalcanti, J. C. Cressoni, Heber R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef]

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

Crespo, R. D.

S. Palacios, A. Guinea, J. M. Fernandez-Diaz, and R. D. Crespo, “Dark solitary waves in the nonlinear Schrödinger equation with third order dispersion, self-steepening, and self-frequency shift,” Phys. Rev. E 60, R45–R47 (1999).
[CrossRef]

Cressoni, J. C.

S. B. Cavalcanti, J. C. Cressoni, Heber R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef]

da Cruz, Heber R.

S. B. Cavalcanti, J. C. Cressoni, Heber R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef]

de Sterke, C. M.

G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010).
[CrossRef]

Devine, N.

A. Ankiewicz, N. Devine, and N. Akhmediev, “Are rogue waves robust against perturbations?” Phys. Lett. A 373, 3997–4000 (2009).
[CrossRef]

Dias, F.

G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010).
[CrossRef]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express 17, 21497–21508 (2009).
[CrossRef]

Douay, M.

Dudley, J. M.

G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010).
[CrossRef]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express 17, 21497–21508 (2009).
[CrossRef]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

Fatome, J.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

Fernandez-Diaz, J. M.

S. Palacios, A. Guinea, J. M. Fernandez-Diaz, and R. D. Crespo, “Dark solitary waves in the nonlinear Schrödinger equation with third order dispersion, self-steepening, and self-frequency shift,” Phys. Rev. E 60, R45–R47 (1999).
[CrossRef]

Finot, C.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

Gaillard, P.

P. Gaillard, “Families of quasi-rational solutions of the NLS equation and multi-rogue waves,” J. Phys. A 44, 435204(2011).
[CrossRef]

Genty, G.

G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010).
[CrossRef]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express 17, 21497–21508 (2009).
[CrossRef]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

Gorbach, A. V.

D. V. Skryabin and A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287–1299 (2010).
[CrossRef]

Gouveia-Neto, A. S.

S. B. Cavalcanti, J. C. Cressoni, Heber R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef]

Griebner, U.

J. Bethge, A. Husakou, F. Mitschke, F. Noack, U. Griebner, G. Steinmeyer, and J. Herrmann, “Two-octave supercontinuum generation in a water-filled photonic crystal fiber,” Opt. Express 18, 6230–6240 (2010).
[CrossRef]

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef]

Guinea, A.

S. Palacios, A. Guinea, J. M. Fernandez-Diaz, and R. D. Crespo, “Dark solitary waves in the nonlinear Schrödinger equation with third order dispersion, self-steepening, and self-frequency shift,” Phys. Rev. E 60, R45–R47 (1999).
[CrossRef]

Hasegawa, A.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Herrmann, J.

J. Bethge, A. Husakou, F. Mitschke, F. Noack, U. Griebner, G. Steinmeyer, and J. Herrmann, “Two-octave supercontinuum generation in a water-filled photonic crystal fiber,” Opt. Express 18, 6230–6240 (2010).
[CrossRef]

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef]

A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef]

Hilligsoe, K. M.

Hirota, R.

R. Hirota, “Exact envelope-soliton solutions of a nonlinear wave equation,” J. Math. Phys. 14, 805–809 (1973).
[CrossRef]

Husakou, A.

J. Bethge, A. Husakou, F. Mitschke, F. Noack, U. Griebner, G. Steinmeyer, and J. Herrmann, “Two-octave supercontinuum generation in a water-filled photonic crystal fiber,” Opt. Express 18, 6230–6240 (2010).
[CrossRef]

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef]

Husakou, A. V.

A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef]

Jalali, B.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

Keiding, S. R.

Kibler, B.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express 17, 21497–21508 (2009).
[CrossRef]

Knight, J. C.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef]

Kolobov, M.

Koonath, P.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

Korn, G.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef]

Korneev, V. I.

N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “Modulation instability of CW signal in an optical fiber with the 3rd-order dispersion,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 33, 95–100 (1990); Radiofiz 33, 111–117 (1990).

Kudlinski, A.

Larsen, J. J.

Li, L.

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

Li, Z.

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

Lisak, M.

D. Anderson and M. Lisak, “Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,” Phys. Rev. A 27, 1393–1398 (1983).
[CrossRef]

Louvergneaux, E.

Mahnke, Ch.

Ch. Mahnke and F. Mitschke, “Possibility of an Akhmediev breather decaying into solitons,” Phys. Rev. A 85, 033808 (2012).
[CrossRef]

Millot, G.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

Mitschke, F.

Mitskevich, N. V.

N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “Modulation instability of CW signal in an optical fiber with the 3rd-order dispersion,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 33, 95–100 (1990); Radiofiz 33, 111–117 (1990).

Mussot, A.

Nickel, D.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef]

Noack, F.

Ohta, Y.

Y. Ohta and J. Yang, “General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation,” Proc. R. Soc. A 468, 1716–1740 (2012).
[CrossRef]

Palacios, S.

S. Palacios, A. Guinea, J. M. Fernandez-Diaz, and R. D. Crespo, “Dark solitary waves in the nonlinear Schrödinger equation with third order dispersion, self-steepening, and self-frequency shift,” Phys. Rev. E 60, R45–R47 (1999).
[CrossRef]

Paulsen, H. N.

Peregrine, D. H.

D. H. Peregrine, “Water waves, nonlinear Schrödinger equations and their solutions,” J. Aust. Math. Soc. B 25, 16–43 (1983).
[CrossRef]

Potasek, M. J.

Ranka, J. K.

Ropers, C.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

Russell, P. St. J.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef]

Sasa, N.

N. Sasa and J. Satsuma, “New type of soliton solutions for a higher-order nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 60, 409–417 (1991).
[CrossRef]

Satsuma, J.

N. Sasa and J. Satsuma, “New type of soliton solutions for a higher-order nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 60, 409–417 (1991).
[CrossRef]

Sciacca, M.

T. Brugarino and M. Sciacca, “Singularity analysis and integrability for a HNLS equation governing pulse propagation in a generic fiber optics,” Opt. Commun. 262, 250–256 (2006).
[CrossRef]

Skryabin, D. V.

D. V. Skryabin and A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287–1299 (2010).
[CrossRef]

Solli, D. R.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

Soto-Crespo, J. M.

A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Rogue waves and rational solutions of the Hirota equation,” Phys. Rev. E 81, 046602 (2010).
[CrossRef]

Steinmeyer, G.

Stentz, A. J.

Taki, M.

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
[CrossRef]

A. Mussot, A. Kudlinski, M. Kolobov, E. Louvergneaux, M. Douay, and M. Taki, “Observation of extreme temporal events in CW-pumped supercontinuum,” Opt. Express 17, 17010–17015 (2009).
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A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
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Tian, H.

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

Travers, J. C.

J. C. Travers, “High average power supercontinuum sources,” Pramana J. Phys. 75, 769–785 (2010).
[CrossRef]

Wadsworth, W. J.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef]

Windeler, R. S.

Yang, J.

Y. Ohta and J. Yang, “General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation,” Proc. R. Soc. A 468, 1716–1740 (2012).
[CrossRef]

Zhavoronkov, N.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef]

Zhou, G.

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

Appl. Phys. Lett. (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Electron. Lett. (1)

A. Mussot and A. Kudlinski, “19.5 W CW-pumped supercontinuum source from 0.65 to 1.38  μm,” Electron. Lett. 45, 29–30 (2009).
[CrossRef]

Eur. Phys. J. Special Topics (1)

N. Akhmediev and E. Pelinovsky, eds., Issue on “Rogue waves—towards a unifying concept?” Eur. Phys. J. Special Topics 185, 1–266 (2010).

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (1)

N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “Modulation instability of CW signal in an optical fiber with the 3rd-order dispersion,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 33, 95–100 (1990); Radiofiz 33, 111–117 (1990).

J. Aust. Math. Soc. B (1)

D. H. Peregrine, “Water waves, nonlinear Schrödinger equations and their solutions,” J. Aust. Math. Soc. B 25, 16–43 (1983).
[CrossRef]

J. Math. Phys. (1)

R. Hirota, “Exact envelope-soliton solutions of a nonlinear wave equation,” J. Math. Phys. 14, 805–809 (1973).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. A (1)

P. Gaillard, “Families of quasi-rational solutions of the NLS equation and multi-rogue waves,” J. Phys. A 44, 435204(2011).
[CrossRef]

J. Phys. Soc. Jpn. (1)

N. Sasa and J. Satsuma, “New type of soliton solutions for a higher-order nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 60, 409–417 (1991).
[CrossRef]

Nat. Phys. (1)

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
[CrossRef]

Nature (1)

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

Opt. Commun. (1)

T. Brugarino and M. Sciacca, “Singularity analysis and integrability for a HNLS equation governing pulse propagation in a generic fiber optics,” Opt. Commun. 262, 250–256 (2006).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys. Lett. A (4)

U. Bandelow and N. Akhmediev, “Persistence of rogue waves in extended nonlinear Schrödinger equations: integrable Sasa–Satsuma case,” Phys. Lett. A 376, 1558–1561 (2012).
[CrossRef]

N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace,” Phys. Lett. A 373, 675–678 (2009).
[CrossRef]

G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010).
[CrossRef]

A. Ankiewicz, N. Devine, and N. Akhmediev, “Are rogue waves robust against perturbations?” Phys. Lett. A 373, 3997–4000 (2009).
[CrossRef]

Phys. Rev. A (3)

Ch. Mahnke and F. Mitschke, “Possibility of an Akhmediev breather decaying into solitons,” Phys. Rev. A 85, 033808 (2012).
[CrossRef]

D. Anderson and M. Lisak, “Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,” Phys. Rev. A 27, 1393–1398 (1983).
[CrossRef]

S. B. Cavalcanti, J. C. Cressoni, Heber R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef]

Phys. Rev. E (2)

S. Palacios, A. Guinea, J. M. Fernandez-Diaz, and R. D. Crespo, “Dark solitary waves in the nonlinear Schrödinger equation with third order dispersion, self-steepening, and self-frequency shift,” Phys. Rev. E 60, R45–R47 (1999).
[CrossRef]

A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, “Rogue waves and rational solutions of the Hirota equation,” Phys. Rev. E 81, 046602 (2010).
[CrossRef]

Phys. Rev. Lett. (3)

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

A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef]

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef]

Pramana J. Phys. (1)

J. C. Travers, “High average power supercontinuum sources,” Pramana J. Phys. 75, 769–785 (2010).
[CrossRef]

Proc. R. Soc. A (1)

Y. Ohta and J. Yang, “General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation,” Proc. R. Soc. A 468, 1716–1740 (2012).
[CrossRef]

Rev. Mod. Phys. (2)

D. V. Skryabin and A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287–1299 (2010).
[CrossRef]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

Other (3)

J. M. Dudley and J. R. Taylor, eds., Supercontinuum Generation in Optical Fibers (Cambridge University, 2010).

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

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed., Optics and Photonics series (Academic, 2006), Section 5.5.3.

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Figures (11)

Fig. 1.
Fig. 1.

Two-dimensional profile of the exact rogue wave solution of the Sasa–Satsuma equation based on Eqs. (10) and (11) of [34]. Here k=0 and (a) c=0.18, ϵ=s=0.09, (b) c=0.24, ϵ=s=0.12.

Fig. 2.
Fig. 2.

Excitation of the rogue wave of the Sasa–Satsuma equation (within the blue region, 4<x<8) in propagation starting from the initial condition given by Eq. (3) at x=6. Further evolution is chaotic with repeated double peaks, characteristic of the wave fields governed by the SSE.

Fig. 3.
Fig. 3.

Wave profiles of the rogue waves of the Sasa–Satsuma equation within (a) the blue and (b) red regions of Fig. 2. Each one has a shape very close to that given by the exact solution and shown in Fig. 3(a).

Fig. 4.
Fig. 4.

(a) Small part of the chaotic wave field generated by the SSE for an initial condition consisting of a plane wave randomly perturbed. Two rogue waves are marked by black circles. (b) and (c) represent the magnification of the encircled parts of the plot in (a). The value of the parameters is the same as in the preceding figure, namely γ3=1, af=3, s=0.09, and sa=6.

Fig. 5.
Fig. 5.

Plane of the parameters (sa,af) considered in this paper. With no loss of generality, we have set γ3=1. The blue diamond on the sloping dashed line corresponds to the parameters of the Sasa–Satsuma equation, while the triangle directly above it indicates the parameters of the Hirota equation. The dashed blue line with negative slope gives the extension of the Sasa–Satsuma equation, while the horizontal dotted red line gives the extension of the Hirota equation. The intersection point of the lines is marked by the green square.

Fig. 6.
Fig. 6.

Approximation of the rogue wave near the Sasa–Satsuma equation given by Eq. (6). Here s=0.09.

Fig. 7.
Fig. 7.

Wave propagation starting with the approximate rogue wave solution given by Eq. (6) at x=6 for the parameters sa=4 and af=1. The three plots are made for (a) s=0.01, (b) s=0.05, (c) s=0.08. The blue dotted curve corresponds to the maximum amplitude of the field obtained in the numerical simulation, while the red dashed curve corresponds to the approximate solution Eq. (6).

Fig. 8.
Fig. 8.

Contour plots of (a) approximate and (b) numerical profiles located within the yellow panel of Fig. 7(b).

Fig. 9.
Fig. 9.

Profiles of (a) the exact rogue wave solution of the Hirota equation (9). (b) Approximation of the rogue wave of Hirota equation defined by Eq. (12). In each case, s=0.12.

Fig. 10.
Fig. 10.

(a) Evolution of the maximum field along x governed by the “extended-Hirota” equation. The solid black curve shows the maximum amplitude of the approximate solution centered at x=0. The dashed blue curve shows the evolution of the maximum value of the field with the initial condition Eq. (10) at x=6, while the dotted red curve is for the initial condition Eq. (4). (b) Contour plot of the approximate solution in the (x,t) plane [yellow part in (a), 3<x<3]. (c) Contour plot of the first peak of the blue dashed curve [light blue colored part in (a)].

Fig. 11.
Fig. 11.

(a) Maximum of the field governed by Eq. (14), with the initial condition Eq. (15) at x=6. (b) Maximum of the field governed by Eq. (14), with the initial condition as small amplitude chaotic perturbation of a constant background. Here af=sa, γ3=0, and ssa=0.3.

Equations (23)

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iψx+122ψt2+|ψ|2ψ=0,
iψx+122ψt2+|ψ|2ψ+is[afψt(|ψ|2)+sat(|ψ|2ψ)γ3ψttt]=0.
ψ=[41+2ix1+4x2+4t21]eix,
ψ(t,0)=1+μR(t),
kn=6γ3+3af+5sa.
ψ(x,t)eix=1+4D(1+2ix)+8tsD2[4knx+ifa(x,t)]+4s2D3[kb(x,t)+2ixfb(x,t)],
fa(x,t)=3sa+4t2(sa+3)+12x2(sa1),kb(x,t)=9[1+64t4+44x2+192x4+4t2(3+80x2)]+3sa[1+48t4+24x2+80x416t2(3+8x2)]+4sa2[4t4+t2(28x23)6x2(1+4x2)],
fb(x,t)=9[9+32t420x2+64x4+4t2(40x27)]6sa[116x4+4t2(5+12x2)]+sa2[348x4+4t2(4x29)].
216D4s3t[8xkd(x,t)ifd(x,t)],
kd(x,t)=(4x2+1)(344x2+55)32t44t2(312x2+35),fd(x,t)=16t4(16x2+23)+8t2(736x4212x213)8192x61872x4+104x2+15.
v(43sa2)s.
ψex(x,t)eix=1+4(1+2ix)1+4(t+6sx)2+4x2.
ψ(x,t)eix=1+4D(1+2ix)+32tsD2[knx+ife(x,t)]+16s2D3[kc(x,t)+2ixfc(x,t)]+,
fe(x,t)=(sa+6)(2t21)+6(sa+4)x2,kc(x,t)=sa2(32t4+12t2(17x21)21x2(4x2+1))+3sa(144t4+8t2(80x29)208x448x2+1)+18[1+80t4+16t2(16x23)8(6x4+x2)],
fc(x,t)=sa2(8t4+18t2(2x23)36x4+3x2+3)+3sa(32t4+4t2(20x253)80x4+16x2+9)+18(16t4+8t2(4x213)16x4+8x2+3).
v(83sa+10)s.
ψ(x,t)eix=1+4D(1+2ix)[148xtsD144x2s2D2(112t2+4x2)+13824tx3s3D3(14t2+4x2)+].
ψex(x,t)eix=1+4D(1+2ix)[148xtsD144x2s2D2(112t2+4x2)+13824tx3s3D3(14t2+4x2)+20736x4s4D4(140t2+80t4+8x2160t2x2+16x4)+].
iψx+122ψt2+|ψ|2ψ+issaψt|ψ|2=0.
ψ(x,t)eix=1+4D(1+2ix)+8tssaD2[8x+ifr(x,t)]+8(ssa)2D3[2kg(x,t)+ixfg(x,t)]+32t(ssa)3D4[2xkh(x,t)+ifh(x,t)]+16(ssa)4D5[kj(x,t)+2xifj(x,t)]+,
fr(x,t)=4t2+12x21,kg(x,t)=4t4+t2(28x23)6x2(4x2+1),fg(x,t)=4t2(4x29)48x4+3,kh(x,t)=16t48(4t2+3)x224t2144x4+3,fh(x,t)=16t68t4(8x2+1)+t2(80x456x2+7)+2x2(1+4x2)(1920x2),
kj(x,t)=192t832t6(38x2+1)4t4(544x4+84x217)12t2x2(400x488x247)+(92x213)x2(4x2+1)2,
fj(x,t)=304t6+8t4(8x4+110x2+29)+t2(640x6+2480x4+496x241)+(20x231)x2(4x2+1)2.

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