Abstract

Rogue waves can appear in optical fibers and other optical systems as well as in natural events like water waves. Their mathematical description is based on partial differential equations that have solutions that are localized both in time and space. One example is the “Peregrine” solution of the nonlinear Schrödinger equation (NLSE). When higher-order terms in the equation are involved, the solution becomes distorted, but its main features remain localized in space and time. Although exact solutions are not obtained in all cases, approximations that describe the solutions with reasonable accuracy do exist. Here, we consider approximate rogue wave solutions of the NLSE with an optically relevant Raman delay term.

© 2018 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Rogue waves in optical fibers in presence of third-order dispersion, self-steepening, and self-frequency shift

Adrian Ankiewicz, Jose M. Soto-Crespo, M. Amdadul Chowdhury, and Nail Akhmediev
J. Opt. Soc. Am. B 30(1) 87-94 (2013)

Controllable optical rogue waves via nonlinearity management

Zhengping Yang, Wei-Ping Zhong, Milivoj Belić, and Yiqi Zhang
Opt. Express 26(6) 7587-7597 (2018)

Dark three-sister rogue waves in normally dispersive optical fibers with random birefringence

Shihua Chen, Jose M. Soto-Crespo, and Philippe Grelu
Opt. Express 22(22) 27632-27642 (2014)

References

  • View by:
  • |
  • |
  • |

  1. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
    [Crossref]
  2. J. M. Soto-Crespo, Ph. Grelu, and N. Akhmediev, “Dissipative rogue waves: extreme pulses generated by passively mode-locked lasers,” Phys. Rev. E 84, 016604 (2011).
    [Crossref]
  3. A. Zaviyalov, O. Egorov, R. Iliew, and F. Lederer, “Rogue waves in mode-locked fiber lasers,” Phys. Rev. A 85, 013828 (2012).
    [Crossref]
  4. M. G. Kovalsky, A. A. Hnilo, and J. R. Tredicce, “Extreme events in the Ti: sapphire laser,” Opt. Lett. 36, 4449 (2011).
    [Crossref]
  5. C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
    [Crossref]
  6. C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue wave generation in multiple-pulsing mode-locked fiber laser,” J. Opt. 15, 064005 (2013).
    [Crossref]
  7. S. A. Kolpakov, H. Kbashi, and S. V. Sergeyev, “Dynamics of vector rogue waves in a fiber laser with a ring cavity,” Optica 3, 870 (2016).
    [Crossref]
  8. M. Tlidi, K. Panajotov, M. l. Ferré, and M. G. Clerc, “Drifting cavity solitons and dissipative rogue waves induced by time-delayed feedback in Kerr optical frequency comb and in all fiber cavities,” Chaos 27, 114312 (2017).
    [Crossref]
  9. S. Residori, Bortolozzo, A. Montina, F. Lenzini, and F. T. Arecchi, “Rogue waves in spatially extended optical systems,” Fluctuat. Noise Lett. 11, 1240014 (2012).
    [Crossref]
  10. A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian statistics and extreme waves in a nonlinear optical cavity,” Phys. Rev. Lett. 103, 173901 (2009).
    [Crossref]
  11. P.-H. Hanzard, M. Talbi, D. Mallek, A. Kellou, H. Leblond, F. Sanchez, T. Godin, and A. Hideur, “Brillouin scattering-induced rogue waves in self-pulsing fiber lasers,” Sci. Rep. 78, 45868 (2017).
    [Crossref]
  12. A. F. J. Runge, C. Aguergaray, N. G. R. Broderick, and M. Erkintalo, “Raman rogue waves in a partially mode-locked fiber laser,” Opt. Lett. 39, 319–322 (2014).
    [Crossref]
  13. J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8, 755–764 (2014).
    [Crossref]
  14. M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528, 47–89 (2013).
    [Crossref]
  15. N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
    [Crossref]
  16. W. Chang, J. M. Soto-Crespo, P. Vouzas, and N. Akhmediev, “Extreme soliton pulsations in dissipative systems,” Phys. Rev. E 92, 022926 (2015).
    [Crossref]
  17. W. Chang, J. M. Soto-Crespo, P. Vouzas, and N. Akhmediev, “Extreme amplitude spikes in a laser model described by the complex Ginzburg—Landau equation,” Opt. Lett. 40, 2949 (2015).
    [Crossref]
  18. 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]
  19. N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Extreme waves that appear from nowhere: on the nature of rogue waves,” Phys. Lett. A 373, 2137–2145 (2009).
    [Crossref]
  20. 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]
  21. 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]
  22. F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
    [Crossref]
  23. C.-Q. Dai, J. Liu, Y. Fan, and D.-G. Yu, “Two-dimensional localized Peregrine solution and breather excited in a variable-coefficient nonlinear Schrödinger equation with partial nonlocality,” Nonlinear Dyn. 88, 1373–1383 (2017)..
    [Crossref]
  24. Y.-Y. Wang, C.-Q. Dai, G.-Q. Zhou, Y. Fan, and L. Chen, “Rogue wave and combined breather with repeatedly excited behaviors in the dispersion/diffraction decreasing medium,” Nonlinear Dyn. 87, 67–73 (2017).
    [Crossref]
  25. J.-T. Li, X.-T. Zhang, M. Meng, Q.-T. Liu, Y.-Y. Wang, and C.-Q. Dai, “Control and management of the combined Peregrine soliton and Akhmediev breathers in PT -symmetric coupled waveguides,” Nonlinear Dyn. 84, 473–479 (2016).
    [Crossref]
  26. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995).
  27. A. Ankiewicz, J. M. Soto-Crespo, M. A. Chowdhury, and N. Akhmediev, “Rogue waves in optical fibers in presence of third order dispersion, self-steepening and self-frequency shift,” J. Opt. Soc. Am. B 30, 87–94 (2013).
    [Crossref]
  28. V. V. Voronovich, V. I. Shrira, and G. Thomas, “Can bottom friction suppress ‘freak wave’ formation?” J. Fluid Mech. 604, 263–296 (2008).
    [Crossref]
  29. A. Ankiewicz, D. J. Kedziora, A. Chowdury, U. Bandelow, and N. Akhmediev, “Infinite hierarchy of nonlinear Schrödinger equations and their solutions,” Phys. Rev. E 93, 012206 (2016).
    [Crossref]
  30. D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
    [Crossref]
  31. A. I. Maimistov, “Evolution of solitary waves which are approximately solitons of a nonlinear Schrödinger equation,” J. Exp. Theor. Phys. 77, 727 (1993) [Zh. Eksp. Teor. Fiz. 104, 3620 (1993), in Russian].
  32. A. Ankiewicz, N. Akhmediev, and N. Devine, “Dissipative solitons with a Lagrangian approach (Invited paper),” Opt. Fiber Technol. 13, 91–97 (2007).
    [Crossref]
  33. A. Ankiewicz and N. Akhmediev, “Comparison of Lagrangian approach and method of moments for reducing dimensionality of soliton dynamical systems,” Chaos 18, 033129 (2008).
    [Crossref]
  34. W. Chang, J. M. Soto-Crespo, A. Ankiewicz, and N. Akhmediev, “Dissipative soliton resonances in anomalous dispersion regime,” Phys. Rev. A 79, 033840 (2009).
    [Crossref]
  35. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill book co., 1953), pp. 276, 314.
  36. A. D. Boardman and L. Velasco, “Gyroelectric cubic-quintic dissipative solitons,” IEEE J. Sel. Top. Quantum Electron. 12, 388–397, (2006).
    [Crossref]
  37. N. Akhmediev and A. Ankiewicz, “Dissipative solitons in the complex Ginzburg-Landau and Swift-Hohenberg equations,” in Dissipative Solitons (Springer, 2005).
  38. A. Ankiewicz and N. Akhmediev, “Rogue wave solutions for the infinite integrable nonlinear Schrödinger equation hierarchy,” Phys. Rev. E 96, 012219 (2017).
    [Crossref]
  39. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chap. & Hall, 1997).
  40. R. Hirota, “Exact envelope-soliton solutions of a nonlinear wave equation,” J. Math. Phys. 14, 805–809 (1973).
    [Crossref]
  41. M. Lakshmanan, K. Porsezian, and M. Daniel, “Effect of discreteness on the continuum limit of the Heisenberg spin chain,” Phys. Lett. A 133, 483–488 (1988).
    [Crossref]
  42. A. Ankiewicz, Yan Wang, S. Wabnitz, and N. Akhmediev, “Extended nonlinear Schrödinger equation with higher-order odd and even terms and its rogue wave solutions,” Phys. Rev. E 89, 012907 (2014).
    [Crossref]
  43. A. Ankiewicz, “Simplified description of soliton perturbation and interaction using averaged complex potentials,” Int. J. Nonlinear Opt. Phys. Mater. 4, 857–870 (1995).
    [Crossref]
  44. L. Gagnon and P. A. Belanger, “Soliton self-frequency shift versus Galilean-like symmetry,” Opt. Lett. 15, 466–468 (1990); see Eq. (13).
    [Crossref]
  45. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662 (1986).
    [Crossref]
  46. N. Akhmediev, W. Krolikowski, and A. J. Lowery, “Influence of the Raman-effect on solitons in optical fibers,” Opt. Commun. 131, 260–266 (1996).
    [Crossref]
  47. M. Facão, M. I. Carvalho, and D. F. Parker, “Soliton self-frequency shift: Self-similar solutions and their stability,” Phys. Rev. E 81, 046604 (2010).
    [Crossref]
  48. B. Jalali, D. Solli, K. Goda, K. Tsia, and C. Ropers, “Real-time measurements, rare events and photon economics,” Eur. Phys. J. Spec. Top. 185, 145–157 (2010).
    [Crossref]
  49. A. Bhushan, F. Coppinger, and B. Jalali, “Time-stretched analogue-to-digital conversion,” Electron. Lett. 34, 1081 (1998).
    [Crossref]
  50. F. Coppinger, A. Bhushan, and B. Jalali, “Photonic time stretch and its application to analog-to-digital conversion,” IEEE Trans. Microw. Theory Tech. 47, 1309–1314 (1999).
    [Crossref]
  51. P. Suret, R. El Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, “Single-shot observation of optical rogue waves in integrable turbulence using time microscopy,” Nat. Commun. 7, 13136 (2016).
    [Crossref]
  52. M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, “Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability,” Nat. Commun. 7, 13675 (2016).
    [Crossref]

2017 (5)

M. Tlidi, K. Panajotov, M. l. Ferré, and M. G. Clerc, “Drifting cavity solitons and dissipative rogue waves induced by time-delayed feedback in Kerr optical frequency comb and in all fiber cavities,” Chaos 27, 114312 (2017).
[Crossref]

P.-H. Hanzard, M. Talbi, D. Mallek, A. Kellou, H. Leblond, F. Sanchez, T. Godin, and A. Hideur, “Brillouin scattering-induced rogue waves in self-pulsing fiber lasers,” Sci. Rep. 78, 45868 (2017).
[Crossref]

C.-Q. Dai, J. Liu, Y. Fan, and D.-G. Yu, “Two-dimensional localized Peregrine solution and breather excited in a variable-coefficient nonlinear Schrödinger equation with partial nonlocality,” Nonlinear Dyn. 88, 1373–1383 (2017)..
[Crossref]

Y.-Y. Wang, C.-Q. Dai, G.-Q. Zhou, Y. Fan, and L. Chen, “Rogue wave and combined breather with repeatedly excited behaviors in the dispersion/diffraction decreasing medium,” Nonlinear Dyn. 87, 67–73 (2017).
[Crossref]

A. Ankiewicz and N. Akhmediev, “Rogue wave solutions for the infinite integrable nonlinear Schrödinger equation hierarchy,” Phys. Rev. E 96, 012219 (2017).
[Crossref]

2016 (6)

A. Ankiewicz, D. J. Kedziora, A. Chowdury, U. Bandelow, and N. Akhmediev, “Infinite hierarchy of nonlinear Schrödinger equations and their solutions,” Phys. Rev. E 93, 012206 (2016).
[Crossref]

J.-T. Li, X.-T. Zhang, M. Meng, Q.-T. Liu, Y.-Y. Wang, and C.-Q. Dai, “Control and management of the combined Peregrine soliton and Akhmediev breathers in PT -symmetric coupled waveguides,” Nonlinear Dyn. 84, 473–479 (2016).
[Crossref]

S. A. Kolpakov, H. Kbashi, and S. V. Sergeyev, “Dynamics of vector rogue waves in a fiber laser with a ring cavity,” Optica 3, 870 (2016).
[Crossref]

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

P. Suret, R. El Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, “Single-shot observation of optical rogue waves in integrable turbulence using time microscopy,” Nat. Commun. 7, 13136 (2016).
[Crossref]

M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, “Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability,” Nat. Commun. 7, 13675 (2016).
[Crossref]

2015 (2)

W. Chang, J. M. Soto-Crespo, P. Vouzas, and N. Akhmediev, “Extreme soliton pulsations in dissipative systems,” Phys. Rev. E 92, 022926 (2015).
[Crossref]

W. Chang, J. M. Soto-Crespo, P. Vouzas, and N. Akhmediev, “Extreme amplitude spikes in a laser model described by the complex Ginzburg—Landau equation,” Opt. Lett. 40, 2949 (2015).
[Crossref]

2014 (3)

A. F. J. Runge, C. Aguergaray, N. G. R. Broderick, and M. Erkintalo, “Raman rogue waves in a partially mode-locked fiber laser,” Opt. Lett. 39, 319–322 (2014).
[Crossref]

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8, 755–764 (2014).
[Crossref]

A. Ankiewicz, Yan Wang, S. Wabnitz, and N. Akhmediev, “Extended nonlinear Schrödinger equation with higher-order odd and even terms and its rogue wave solutions,” Phys. Rev. E 89, 012907 (2014).
[Crossref]

2013 (3)

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528, 47–89 (2013).
[Crossref]

C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue wave generation in multiple-pulsing mode-locked fiber laser,” J. Opt. 15, 064005 (2013).
[Crossref]

A. Ankiewicz, J. M. Soto-Crespo, M. A. Chowdhury, and N. Akhmediev, “Rogue waves in optical fibers in presence of third order dispersion, self-steepening and self-frequency shift,” J. Opt. Soc. Am. B 30, 87–94 (2013).
[Crossref]

2012 (5)

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]

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[Crossref]

C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
[Crossref]

S. Residori, Bortolozzo, A. Montina, F. Lenzini, and F. T. Arecchi, “Rogue waves in spatially extended optical systems,” Fluctuat. Noise Lett. 11, 1240014 (2012).
[Crossref]

A. Zaviyalov, O. Egorov, R. Iliew, and F. Lederer, “Rogue waves in mode-locked fiber lasers,” Phys. Rev. A 85, 013828 (2012).
[Crossref]

2011 (2)

M. G. Kovalsky, A. A. Hnilo, and J. R. Tredicce, “Extreme events in the Ti: sapphire laser,” Opt. Lett. 36, 4449 (2011).
[Crossref]

J. M. Soto-Crespo, Ph. Grelu, and N. Akhmediev, “Dissipative rogue waves: extreme pulses generated by passively mode-locked lasers,” Phys. Rev. E 84, 016604 (2011).
[Crossref]

2010 (3)

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]

M. Facão, M. I. Carvalho, and D. F. Parker, “Soliton self-frequency shift: Self-similar solutions and their stability,” Phys. Rev. E 81, 046604 (2010).
[Crossref]

B. Jalali, D. Solli, K. Goda, K. Tsia, and C. Ropers, “Real-time measurements, rare events and photon economics,” Eur. Phys. J. Spec. Top. 185, 145–157 (2010).
[Crossref]

2009 (4)

W. Chang, J. M. Soto-Crespo, A. Ankiewicz, and N. Akhmediev, “Dissipative soliton resonances in anomalous dispersion regime,” Phys. Rev. A 79, 033840 (2009).
[Crossref]

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian statistics and extreme waves in a nonlinear optical cavity,” Phys. Rev. Lett. 103, 173901 (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, J. M. Soto-Crespo, and A. Ankiewicz, “Extreme waves that appear from nowhere: on the nature of rogue waves,” Phys. Lett. A 373, 2137–2145 (2009).
[Crossref]

2008 (2)

V. V. Voronovich, V. I. Shrira, and G. Thomas, “Can bottom friction suppress ‘freak wave’ formation?” J. Fluid Mech. 604, 263–296 (2008).
[Crossref]

A. Ankiewicz and N. Akhmediev, “Comparison of Lagrangian approach and method of moments for reducing dimensionality of soliton dynamical systems,” Chaos 18, 033129 (2008).
[Crossref]

2007 (2)

A. Ankiewicz, N. Akhmediev, and N. Devine, “Dissipative solitons with a Lagrangian approach (Invited paper),” Opt. Fiber Technol. 13, 91–97 (2007).
[Crossref]

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

2006 (1)

A. D. Boardman and L. Velasco, “Gyroelectric cubic-quintic dissipative solitons,” IEEE J. Sel. Top. Quantum Electron. 12, 388–397, (2006).
[Crossref]

1999 (1)

F. Coppinger, A. Bhushan, and B. Jalali, “Photonic time stretch and its application to analog-to-digital conversion,” IEEE Trans. Microw. Theory Tech. 47, 1309–1314 (1999).
[Crossref]

1998 (1)

A. Bhushan, F. Coppinger, and B. Jalali, “Time-stretched analogue-to-digital conversion,” Electron. Lett. 34, 1081 (1998).
[Crossref]

1996 (1)

N. Akhmediev, W. Krolikowski, and A. J. Lowery, “Influence of the Raman-effect on solitons in optical fibers,” Opt. Commun. 131, 260–266 (1996).
[Crossref]

1995 (1)

A. Ankiewicz, “Simplified description of soliton perturbation and interaction using averaged complex potentials,” Int. J. Nonlinear Opt. Phys. Mater. 4, 857–870 (1995).
[Crossref]

1993 (1)

A. I. Maimistov, “Evolution of solitary waves which are approximately solitons of a nonlinear Schrödinger equation,” J. Exp. Theor. Phys. 77, 727 (1993) [Zh. Eksp. Teor. Fiz. 104, 3620 (1993), in Russian].

1990 (1)

1988 (1)

M. Lakshmanan, K. Porsezian, and M. Daniel, “Effect of discreteness on the continuum limit of the Heisenberg spin chain,” Phys. Lett. A 133, 483–488 (1988).
[Crossref]

1986 (1)

1983 (1)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[Crossref]

1973 (1)

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, 2nd ed. (Academic, 1995).

Aguergaray, C.

Akhmediev, N.

A. Ankiewicz and N. Akhmediev, “Rogue wave solutions for the infinite integrable nonlinear Schrödinger equation hierarchy,” Phys. Rev. E 96, 012219 (2017).
[Crossref]

A. Ankiewicz, D. J. Kedziora, A. Chowdury, U. Bandelow, and N. Akhmediev, “Infinite hierarchy of nonlinear Schrödinger equations and their solutions,” Phys. Rev. E 93, 012206 (2016).
[Crossref]

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

W. Chang, J. M. Soto-Crespo, P. Vouzas, and N. Akhmediev, “Extreme soliton pulsations in dissipative systems,” Phys. Rev. E 92, 022926 (2015).
[Crossref]

W. Chang, J. M. Soto-Crespo, P. Vouzas, and N. Akhmediev, “Extreme amplitude spikes in a laser model described by the complex Ginzburg—Landau equation,” Opt. Lett. 40, 2949 (2015).
[Crossref]

A. Ankiewicz, Yan Wang, S. Wabnitz, and N. Akhmediev, “Extended nonlinear Schrödinger equation with higher-order odd and even terms and its rogue wave solutions,” Phys. Rev. E 89, 012907 (2014).
[Crossref]

A. Ankiewicz, J. M. Soto-Crespo, M. A. Chowdhury, and N. Akhmediev, “Rogue waves in optical fibers in presence of third order dispersion, self-steepening and self-frequency shift,” J. Opt. Soc. Am. B 30, 87–94 (2013).
[Crossref]

C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue wave generation in multiple-pulsing mode-locked fiber laser,” J. Opt. 15, 064005 (2013).
[Crossref]

C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (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]

J. M. Soto-Crespo, Ph. Grelu, and N. Akhmediev, “Dissipative rogue waves: extreme pulses generated by passively mode-locked lasers,” Phys. Rev. E 84, 016604 (2011).
[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]

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Extreme waves that appear from nowhere: on the nature of rogue waves,” Phys. Lett. A 373, 2137–2145 (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]

W. Chang, J. M. Soto-Crespo, A. Ankiewicz, and N. Akhmediev, “Dissipative soliton resonances in anomalous dispersion regime,” Phys. Rev. A 79, 033840 (2009).
[Crossref]

A. Ankiewicz and N. Akhmediev, “Comparison of Lagrangian approach and method of moments for reducing dimensionality of soliton dynamical systems,” Chaos 18, 033129 (2008).
[Crossref]

A. Ankiewicz, N. Akhmediev, and N. Devine, “Dissipative solitons with a Lagrangian approach (Invited paper),” Opt. Fiber Technol. 13, 91–97 (2007).
[Crossref]

N. Akhmediev, W. Krolikowski, and A. J. Lowery, “Influence of the Raman-effect on solitons in optical fibers,” Opt. Commun. 131, 260–266 (1996).
[Crossref]

N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chap. & Hall, 1997).

N. Akhmediev and A. Ankiewicz, “Dissipative solitons in the complex Ginzburg-Landau and Swift-Hohenberg equations,” in Dissipative Solitons (Springer, 2005).

Amiraranashvili, S.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Anderson, D.

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[Crossref]

Ankiewicz, A.

A. Ankiewicz and N. Akhmediev, “Rogue wave solutions for the infinite integrable nonlinear Schrödinger equation hierarchy,” Phys. Rev. E 96, 012219 (2017).
[Crossref]

A. Ankiewicz, D. J. Kedziora, A. Chowdury, U. Bandelow, and N. Akhmediev, “Infinite hierarchy of nonlinear Schrödinger equations and their solutions,” Phys. Rev. E 93, 012206 (2016).
[Crossref]

A. Ankiewicz, Yan Wang, S. Wabnitz, and N. Akhmediev, “Extended nonlinear Schrödinger equation with higher-order odd and even terms and its rogue wave solutions,” Phys. Rev. E 89, 012907 (2014).
[Crossref]

A. Ankiewicz, J. M. Soto-Crespo, M. A. Chowdhury, and N. Akhmediev, “Rogue waves in optical fibers in presence of third order dispersion, self-steepening and self-frequency shift,” J. Opt. Soc. Am. B 30, 87–94 (2013).
[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]

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Extreme waves that appear from nowhere: on the nature of rogue waves,” Phys. Lett. A 373, 2137–2145 (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]

W. Chang, J. M. Soto-Crespo, A. Ankiewicz, and N. Akhmediev, “Dissipative soliton resonances in anomalous dispersion regime,” Phys. Rev. A 79, 033840 (2009).
[Crossref]

A. Ankiewicz and N. Akhmediev, “Comparison of Lagrangian approach and method of moments for reducing dimensionality of soliton dynamical systems,” Chaos 18, 033129 (2008).
[Crossref]

A. Ankiewicz, N. Akhmediev, and N. Devine, “Dissipative solitons with a Lagrangian approach (Invited paper),” Opt. Fiber Technol. 13, 91–97 (2007).
[Crossref]

A. Ankiewicz, “Simplified description of soliton perturbation and interaction using averaged complex potentials,” Int. J. Nonlinear Opt. Phys. Mater. 4, 857–870 (1995).
[Crossref]

N. Akhmediev and A. Ankiewicz, “Dissipative solitons in the complex Ginzburg-Landau and Swift-Hohenberg equations,” in Dissipative Solitons (Springer, 2005).

N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chap. & Hall, 1997).

Arecchi, F.

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528, 47–89 (2013).
[Crossref]

Arecchi, F. T.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

S. Residori, Bortolozzo, A. Montina, F. Lenzini, and F. T. Arecchi, “Rogue waves in spatially extended optical systems,” Fluctuat. Noise Lett. 11, 1240014 (2012).
[Crossref]

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian statistics and extreme waves in a nonlinear optical cavity,” Phys. Rev. Lett. 103, 173901 (2009).
[Crossref]

Bandelow, U.

A. Ankiewicz, D. J. Kedziora, A. Chowdury, U. Bandelow, and N. Akhmediev, “Infinite hierarchy of nonlinear Schrödinger equations and their solutions,” Phys. Rev. E 93, 012206 (2016).
[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]

Baronio, F.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[Crossref]

Belanger, P. A.

Belic, M.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Bendahmane, A.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Bhushan, A.

F. Coppinger, A. Bhushan, and B. Jalali, “Photonic time stretch and its application to analog-to-digital conversion,” IEEE Trans. Microw. Theory Tech. 47, 1309–1314 (1999).
[Crossref]

A. Bhushan, F. Coppinger, and B. Jalali, “Time-stretched analogue-to-digital conversion,” Electron. Lett. 34, 1081 (1998).
[Crossref]

Bielawski, S.

P. Suret, R. El Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, “Single-shot observation of optical rogue waves in integrable turbulence using time microscopy,” Nat. Commun. 7, 13136 (2016).
[Crossref]

Billet, C.

M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, “Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability,” Nat. Commun. 7, 13675 (2016).
[Crossref]

Boardman, A. D.

A. D. Boardman and L. Velasco, “Gyroelectric cubic-quintic dissipative solitons,” IEEE J. Sel. Top. Quantum Electron. 12, 388–397, (2006).
[Crossref]

Bortolozzo,

S. Residori, Bortolozzo, A. Montina, F. Lenzini, and F. T. Arecchi, “Rogue waves in spatially extended optical systems,” Fluctuat. Noise Lett. 11, 1240014 (2012).
[Crossref]

Bortolozzo, U.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528, 47–89 (2013).
[Crossref]

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian statistics and extreme waves in a nonlinear optical cavity,” Phys. Rev. Lett. 103, 173901 (2009).
[Crossref]

Bree, C.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Broderick, N. G. R.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

A. F. J. Runge, C. Aguergaray, N. G. R. Broderick, and M. Erkintalo, “Raman rogue waves in a partially mode-locked fiber laser,” Opt. Lett. 39, 319–322 (2014).
[Crossref]

Carvalho, M. I.

M. Facão, M. I. Carvalho, and D. F. Parker, “Soliton self-frequency shift: Self-similar solutions and their stability,” Phys. Rev. E 81, 046604 (2010).
[Crossref]

Chang, W.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

W. Chang, J. M. Soto-Crespo, P. Vouzas, and N. Akhmediev, “Extreme soliton pulsations in dissipative systems,” Phys. Rev. E 92, 022926 (2015).
[Crossref]

W. Chang, J. M. Soto-Crespo, P. Vouzas, and N. Akhmediev, “Extreme amplitude spikes in a laser model described by the complex Ginzburg—Landau equation,” Opt. Lett. 40, 2949 (2015).
[Crossref]

W. Chang, J. M. Soto-Crespo, A. Ankiewicz, and N. Akhmediev, “Dissipative soliton resonances in anomalous dispersion regime,” Phys. Rev. A 79, 033840 (2009).
[Crossref]

Chen, L.

Y.-Y. Wang, C.-Q. Dai, G.-Q. Zhou, Y. Fan, and L. Chen, “Rogue wave and combined breather with repeatedly excited behaviors in the dispersion/diffraction decreasing medium,” Nonlinear Dyn. 87, 67–73 (2017).
[Crossref]

Chowdhury, M. A.

Chowdury, A.

A. Ankiewicz, D. J. Kedziora, A. Chowdury, U. Bandelow, and N. Akhmediev, “Infinite hierarchy of nonlinear Schrödinger equations and their solutions,” Phys. Rev. E 93, 012206 (2016).
[Crossref]

Clerc, M. G.

M. Tlidi, K. Panajotov, M. l. Ferré, and M. G. Clerc, “Drifting cavity solitons and dissipative rogue waves induced by time-delayed feedback in Kerr optical frequency comb and in all fiber cavities,” Chaos 27, 114312 (2017).
[Crossref]

Conforti, M.

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[Crossref]

Coppinger, F.

F. Coppinger, A. Bhushan, and B. Jalali, “Photonic time stretch and its application to analog-to-digital conversion,” IEEE Trans. Microw. Theory Tech. 47, 1309–1314 (1999).
[Crossref]

A. Bhushan, F. Coppinger, and B. Jalali, “Time-stretched analogue-to-digital conversion,” Electron. Lett. 34, 1081 (1998).
[Crossref]

Coulibaly, S.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Dai, C.-Q.

C.-Q. Dai, J. Liu, Y. Fan, and D.-G. Yu, “Two-dimensional localized Peregrine solution and breather excited in a variable-coefficient nonlinear Schrödinger equation with partial nonlocality,” Nonlinear Dyn. 88, 1373–1383 (2017)..
[Crossref]

Y.-Y. Wang, C.-Q. Dai, G.-Q. Zhou, Y. Fan, and L. Chen, “Rogue wave and combined breather with repeatedly excited behaviors in the dispersion/diffraction decreasing medium,” Nonlinear Dyn. 87, 67–73 (2017).
[Crossref]

J.-T. Li, X.-T. Zhang, M. Meng, Q.-T. Liu, Y.-Y. Wang, and C.-Q. Dai, “Control and management of the combined Peregrine soliton and Akhmediev breathers in PT -symmetric coupled waveguides,” Nonlinear Dyn. 84, 473–479 (2016).
[Crossref]

Daniel, M.

M. Lakshmanan, K. Porsezian, and M. Daniel, “Effect of discreteness on the continuum limit of the Heisenberg spin chain,” Phys. Lett. A 133, 483–488 (1988).
[Crossref]

Degasperis, A.

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[Crossref]

Demircan, A.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Devine, N.

A. Ankiewicz, N. Akhmediev, and N. Devine, “Dissipative solitons with a Lagrangian approach (Invited paper),” Opt. Fiber Technol. 13, 91–97 (2007).
[Crossref]

Dias, F.

M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, “Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability,” Nat. Commun. 7, 13675 (2016).
[Crossref]

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8, 755–764 (2014).
[Crossref]

Dudley, J.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Dudley, J. M.

M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, “Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability,” Nat. Commun. 7, 13675 (2016).
[Crossref]

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8, 755–764 (2014).
[Crossref]

Egorov, O.

A. Zaviyalov, O. Egorov, R. Iliew, and F. Lederer, “Rogue waves in mode-locked fiber lasers,” Phys. Rev. A 85, 013828 (2012).
[Crossref]

El Koussaifi, R.

P. Suret, R. El Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, “Single-shot observation of optical rogue waves in integrable turbulence using time microscopy,” Nat. Commun. 7, 13136 (2016).
[Crossref]

Erkintalo, M.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8, 755–764 (2014).
[Crossref]

A. F. J. Runge, C. Aguergaray, N. G. R. Broderick, and M. Erkintalo, “Raman rogue waves in a partially mode-locked fiber laser,” Opt. Lett. 39, 319–322 (2014).
[Crossref]

Evain, C.

P. Suret, R. El Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, “Single-shot observation of optical rogue waves in integrable turbulence using time microscopy,” Nat. Commun. 7, 13136 (2016).
[Crossref]

Facão, M.

M. Facão, M. I. Carvalho, and D. F. Parker, “Soliton self-frequency shift: Self-similar solutions and their stability,” Phys. Rev. E 81, 046604 (2010).
[Crossref]

Fan, Y.

C.-Q. Dai, J. Liu, Y. Fan, and D.-G. Yu, “Two-dimensional localized Peregrine solution and breather excited in a variable-coefficient nonlinear Schrödinger equation with partial nonlocality,” Nonlinear Dyn. 88, 1373–1383 (2017)..
[Crossref]

Y.-Y. Wang, C.-Q. Dai, G.-Q. Zhou, Y. Fan, and L. Chen, “Rogue wave and combined breather with repeatedly excited behaviors in the dispersion/diffraction decreasing medium,” Nonlinear Dyn. 87, 67–73 (2017).
[Crossref]

Ferré, M. l.

M. Tlidi, K. Panajotov, M. l. Ferré, and M. G. Clerc, “Drifting cavity solitons and dissipative rogue waves induced by time-delayed feedback in Kerr optical frequency comb and in all fiber cavities,” Chaos 27, 114312 (2017).
[Crossref]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill book co., 1953), pp. 276, 314.

Gagnon, L.

Genty, G.

M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, “Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability,” Nat. Commun. 7, 13675 (2016).
[Crossref]

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8, 755–764 (2014).
[Crossref]

Goda, K.

B. Jalali, D. Solli, K. Goda, K. Tsia, and C. Ropers, “Real-time measurements, rare events and photon economics,” Eur. Phys. J. Spec. Top. 185, 145–157 (2010).
[Crossref]

Godin, T.

P.-H. Hanzard, M. Talbi, D. Mallek, A. Kellou, H. Leblond, F. Sanchez, T. Godin, and A. Hideur, “Brillouin scattering-induced rogue waves in self-pulsing fiber lasers,” Sci. Rep. 78, 45868 (2017).
[Crossref]

Gordon, J. P.

Grelu, P.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Grelu, Ph.

C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue wave generation in multiple-pulsing mode-locked fiber laser,” J. Opt. 15, 064005 (2013).
[Crossref]

C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
[Crossref]

J. M. Soto-Crespo, Ph. Grelu, and N. Akhmediev, “Dissipative rogue waves: extreme pulses generated by passively mode-locked lasers,” Phys. Rev. E 84, 016604 (2011).
[Crossref]

Hammani, K.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Hanzard, P.-H.

P.-H. Hanzard, M. Talbi, D. Mallek, A. Kellou, H. Leblond, F. Sanchez, T. Godin, and A. Hideur, “Brillouin scattering-induced rogue waves in self-pulsing fiber lasers,” Sci. Rep. 78, 45868 (2017).
[Crossref]

Hideur, A.

P.-H. Hanzard, M. Talbi, D. Mallek, A. Kellou, H. Leblond, F. Sanchez, T. Godin, and A. Hideur, “Brillouin scattering-induced rogue waves in self-pulsing fiber lasers,” Sci. Rep. 78, 45868 (2017).
[Crossref]

Hirota, R.

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

Hnilo, A. A.

Iliew, R.

A. Zaviyalov, O. Egorov, R. Iliew, and F. Lederer, “Rogue waves in mode-locked fiber lasers,” Phys. Rev. A 85, 013828 (2012).
[Crossref]

Jalali, B.

B. Jalali, D. Solli, K. Goda, K. Tsia, and C. Ropers, “Real-time measurements, rare events and photon economics,” Eur. Phys. J. Spec. Top. 185, 145–157 (2010).
[Crossref]

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

F. Coppinger, A. Bhushan, and B. Jalali, “Photonic time stretch and its application to analog-to-digital conversion,” IEEE Trans. Microw. Theory Tech. 47, 1309–1314 (1999).
[Crossref]

A. Bhushan, F. Coppinger, and B. Jalali, “Time-stretched analogue-to-digital conversion,” Electron. Lett. 34, 1081 (1998).
[Crossref]

Kbashi, H.

Kedziora, D. J.

A. Ankiewicz, D. J. Kedziora, A. Chowdury, U. Bandelow, and N. Akhmediev, “Infinite hierarchy of nonlinear Schrödinger equations and their solutions,” Phys. Rev. E 93, 012206 (2016).
[Crossref]

Kellou, A.

P.-H. Hanzard, M. Talbi, D. Mallek, A. Kellou, H. Leblond, F. Sanchez, T. Godin, and A. Hideur, “Brillouin scattering-induced rogue waves in self-pulsing fiber lasers,” Sci. Rep. 78, 45868 (2017).
[Crossref]

Kibler, B.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Kolpakov, S. A.

Koonath, P.

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

Kovalsky, M. G.

Krolikowski, W.

N. Akhmediev, W. Krolikowski, and A. J. Lowery, “Influence of the Raman-effect on solitons in optical fibers,” Opt. Commun. 131, 260–266 (1996).
[Crossref]

Kudlinski, A.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Lakshmanan, M.

M. Lakshmanan, K. Porsezian, and M. Daniel, “Effect of discreteness on the continuum limit of the Heisenberg spin chain,” Phys. Lett. A 133, 483–488 (1988).
[Crossref]

Leblond, H.

P.-H. Hanzard, M. Talbi, D. Mallek, A. Kellou, H. Leblond, F. Sanchez, T. Godin, and A. Hideur, “Brillouin scattering-induced rogue waves in self-pulsing fiber lasers,” Sci. Rep. 78, 45868 (2017).
[Crossref]

Lecaplain, C.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue wave generation in multiple-pulsing mode-locked fiber laser,” J. Opt. 15, 064005 (2013).
[Crossref]

C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
[Crossref]

Lederer, F.

A. Zaviyalov, O. Egorov, R. Iliew, and F. Lederer, “Rogue waves in mode-locked fiber lasers,” Phys. Rev. A 85, 013828 (2012).
[Crossref]

Lenzini, F.

S. Residori, Bortolozzo, A. Montina, F. Lenzini, and F. T. Arecchi, “Rogue waves in spatially extended optical systems,” Fluctuat. Noise Lett. 11, 1240014 (2012).
[Crossref]

Li, J.-T.

J.-T. Li, X.-T. Zhang, M. Meng, Q.-T. Liu, Y.-Y. Wang, and C.-Q. Dai, “Control and management of the combined Peregrine soliton and Akhmediev breathers in PT -symmetric coupled waveguides,” Nonlinear Dyn. 84, 473–479 (2016).
[Crossref]

Liu, J.

C.-Q. Dai, J. Liu, Y. Fan, and D.-G. Yu, “Two-dimensional localized Peregrine solution and breather excited in a variable-coefficient nonlinear Schrödinger equation with partial nonlocality,” Nonlinear Dyn. 88, 1373–1383 (2017)..
[Crossref]

Liu, Q.-T.

J.-T. Li, X.-T. Zhang, M. Meng, Q.-T. Liu, Y.-Y. Wang, and C.-Q. Dai, “Control and management of the combined Peregrine soliton and Akhmediev breathers in PT -symmetric coupled waveguides,” Nonlinear Dyn. 84, 473–479 (2016).
[Crossref]

Lowery, A. J.

N. Akhmediev, W. Krolikowski, and A. J. Lowery, “Influence of the Raman-effect on solitons in optical fibers,” Opt. Commun. 131, 260–266 (1996).
[Crossref]

Maimistov, A. I.

A. I. Maimistov, “Evolution of solitary waves which are approximately solitons of a nonlinear Schrödinger equation,” J. Exp. Theor. Phys. 77, 727 (1993) [Zh. Eksp. Teor. Fiz. 104, 3620 (1993), in Russian].

Mallek, D.

P.-H. Hanzard, M. Talbi, D. Mallek, A. Kellou, H. Leblond, F. Sanchez, T. Godin, and A. Hideur, “Brillouin scattering-induced rogue waves in self-pulsing fiber lasers,” Sci. Rep. 78, 45868 (2017).
[Crossref]

Masoller, C.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Meng, M.

J.-T. Li, X.-T. Zhang, M. Meng, Q.-T. Liu, Y.-Y. Wang, and C.-Q. Dai, “Control and management of the combined Peregrine soliton and Akhmediev breathers in PT -symmetric coupled waveguides,” Nonlinear Dyn. 84, 473–479 (2016).
[Crossref]

Merolla, J.-M.

M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, “Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability,” Nat. Commun. 7, 13675 (2016).
[Crossref]

Montina, A.

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528, 47–89 (2013).
[Crossref]

S. Residori, Bortolozzo, A. Montina, F. Lenzini, and F. T. Arecchi, “Rogue waves in spatially extended optical systems,” Fluctuat. Noise Lett. 11, 1240014 (2012).
[Crossref]

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian statistics and extreme waves in a nonlinear optical cavity,” Phys. Rev. Lett. 103, 173901 (2009).
[Crossref]

Morandotti, R.

M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, “Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability,” Nat. Commun. 7, 13675 (2016).
[Crossref]

Morgner, U.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill book co., 1953), pp. 276, 314.

Mussot, A.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Närhi, M.

M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, “Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability,” Nat. Commun. 7, 13675 (2016).
[Crossref]

Onorato, M.

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528, 47–89 (2013).
[Crossref]

Panajotov, K.

M. Tlidi, K. Panajotov, M. l. Ferré, and M. G. Clerc, “Drifting cavity solitons and dissipative rogue waves induced by time-delayed feedback in Kerr optical frequency comb and in all fiber cavities,” Chaos 27, 114312 (2017).
[Crossref]

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Parker, D. F.

M. Facão, M. I. Carvalho, and D. F. Parker, “Soliton self-frequency shift: Self-similar solutions and their stability,” Phys. Rev. E 81, 046604 (2010).
[Crossref]

Picozzi, A.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Porsezian, K.

M. Lakshmanan, K. Porsezian, and M. Daniel, “Effect of discreteness on the continuum limit of the Heisenberg spin chain,” Phys. Lett. A 133, 483–488 (1988).
[Crossref]

Randoux, S.

P. Suret, R. El Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, “Single-shot observation of optical rogue waves in integrable turbulence using time microscopy,” Nat. Commun. 7, 13136 (2016).
[Crossref]

Residori, S.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528, 47–89 (2013).
[Crossref]

S. Residori, Bortolozzo, A. Montina, F. Lenzini, and F. T. Arecchi, “Rogue waves in spatially extended optical systems,” Fluctuat. Noise Lett. 11, 1240014 (2012).
[Crossref]

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian statistics and extreme waves in a nonlinear optical cavity,” Phys. Rev. Lett. 103, 173901 (2009).
[Crossref]

Rica, S.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Ropers, C.

B. Jalali, D. Solli, K. Goda, K. Tsia, and C. Ropers, “Real-time measurements, rare events and photon economics,” Eur. Phys. J. Spec. Top. 185, 145–157 (2010).
[Crossref]

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

Runge, A. F. J.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

A. F. J. Runge, C. Aguergaray, N. G. R. Broderick, and M. Erkintalo, “Raman rogue waves in a partially mode-locked fiber laser,” Opt. Lett. 39, 319–322 (2014).
[Crossref]

Sanchez, F.

P.-H. Hanzard, M. Talbi, D. Mallek, A. Kellou, H. Leblond, F. Sanchez, T. Godin, and A. Hideur, “Brillouin scattering-induced rogue waves in self-pulsing fiber lasers,” Sci. Rep. 78, 45868 (2017).
[Crossref]

Sergeyev, S. V.

Shrira, V. I.

V. V. Voronovich, V. I. Shrira, and G. Thomas, “Can bottom friction suppress ‘freak wave’ formation?” J. Fluid Mech. 604, 263–296 (2008).
[Crossref]

Solli, D.

B. Jalali, D. Solli, K. Goda, K. Tsia, and C. Ropers, “Real-time measurements, rare events and photon economics,” Eur. Phys. J. Spec. Top. 185, 145–157 (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.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

W. Chang, J. M. Soto-Crespo, P. Vouzas, and N. Akhmediev, “Extreme soliton pulsations in dissipative systems,” Phys. Rev. E 92, 022926 (2015).
[Crossref]

W. Chang, J. M. Soto-Crespo, P. Vouzas, and N. Akhmediev, “Extreme amplitude spikes in a laser model described by the complex Ginzburg—Landau equation,” Opt. Lett. 40, 2949 (2015).
[Crossref]

A. Ankiewicz, J. M. Soto-Crespo, M. A. Chowdhury, and N. Akhmediev, “Rogue waves in optical fibers in presence of third order dispersion, self-steepening and self-frequency shift,” J. Opt. Soc. Am. B 30, 87–94 (2013).
[Crossref]

C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue wave generation in multiple-pulsing mode-locked fiber laser,” J. Opt. 15, 064005 (2013).
[Crossref]

C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
[Crossref]

J. M. Soto-Crespo, Ph. Grelu, and N. Akhmediev, “Dissipative rogue waves: extreme pulses generated by passively mode-locked lasers,” Phys. Rev. E 84, 016604 (2011).
[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]

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, “Extreme waves that appear from nowhere: on the nature of rogue waves,” Phys. Lett. A 373, 2137–2145 (2009).
[Crossref]

W. Chang, J. M. Soto-Crespo, A. Ankiewicz, and N. Akhmediev, “Dissipative soliton resonances in anomalous dispersion regime,” Phys. Rev. A 79, 033840 (2009).
[Crossref]

Steinmeyer, G.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Suret, P.

P. Suret, R. El Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, “Single-shot observation of optical rogue waves in integrable turbulence using time microscopy,” Nat. Commun. 7, 13136 (2016).
[Crossref]

Sylvestre, T.

M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, “Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability,” Nat. Commun. 7, 13675 (2016).
[Crossref]

Szriftgiser, P.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Szwaj, C.

P. Suret, R. El Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, “Single-shot observation of optical rogue waves in integrable turbulence using time microscopy,” Nat. Commun. 7, 13136 (2016).
[Crossref]

Taki, M.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[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]

Talbi, M.

P.-H. Hanzard, M. Talbi, D. Mallek, A. Kellou, H. Leblond, F. Sanchez, T. Godin, and A. Hideur, “Brillouin scattering-induced rogue waves in self-pulsing fiber lasers,” Sci. Rep. 78, 45868 (2017).
[Crossref]

Thomas, G.

V. V. Voronovich, V. I. Shrira, and G. Thomas, “Can bottom friction suppress ‘freak wave’ formation?” J. Fluid Mech. 604, 263–296 (2008).
[Crossref]

Tikan, A.

P. Suret, R. El Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, “Single-shot observation of optical rogue waves in integrable turbulence using time microscopy,” Nat. Commun. 7, 13136 (2016).
[Crossref]

Tiofack, C. G.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Tlidi, M.

M. Tlidi, K. Panajotov, M. l. Ferré, and M. G. Clerc, “Drifting cavity solitons and dissipative rogue waves induced by time-delayed feedback in Kerr optical frequency comb and in all fiber cavities,” Chaos 27, 114312 (2017).
[Crossref]

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Toenger, S.

M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, “Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability,” Nat. Commun. 7, 13675 (2016).
[Crossref]

Tredicce, J. R.

Tsia, K.

B. Jalali, D. Solli, K. Goda, K. Tsia, and C. Ropers, “Real-time measurements, rare events and photon economics,” Eur. Phys. J. Spec. Top. 185, 145–157 (2010).
[Crossref]

Velasco, L.

A. D. Boardman and L. Velasco, “Gyroelectric cubic-quintic dissipative solitons,” IEEE J. Sel. Top. Quantum Electron. 12, 388–397, (2006).
[Crossref]

Voronovich, V. V.

V. V. Voronovich, V. I. Shrira, and G. Thomas, “Can bottom friction suppress ‘freak wave’ formation?” J. Fluid Mech. 604, 263–296 (2008).
[Crossref]

Vouzas, P.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

W. Chang, J. M. Soto-Crespo, P. Vouzas, and N. Akhmediev, “Extreme amplitude spikes in a laser model described by the complex Ginzburg—Landau equation,” Opt. Lett. 40, 2949 (2015).
[Crossref]

W. Chang, J. M. Soto-Crespo, P. Vouzas, and N. Akhmediev, “Extreme soliton pulsations in dissipative systems,” Phys. Rev. E 92, 022926 (2015).
[Crossref]

Wabnitz, S.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

A. Ankiewicz, Yan Wang, S. Wabnitz, and N. Akhmediev, “Extended nonlinear Schrödinger equation with higher-order odd and even terms and its rogue wave solutions,” Phys. Rev. E 89, 012907 (2014).
[Crossref]

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[Crossref]

Wang, Y.-Y.

Y.-Y. Wang, C.-Q. Dai, G.-Q. Zhou, Y. Fan, and L. Chen, “Rogue wave and combined breather with repeatedly excited behaviors in the dispersion/diffraction decreasing medium,” Nonlinear Dyn. 87, 67–73 (2017).
[Crossref]

J.-T. Li, X.-T. Zhang, M. Meng, Q.-T. Liu, Y.-Y. Wang, and C.-Q. Dai, “Control and management of the combined Peregrine soliton and Akhmediev breathers in PT -symmetric coupled waveguides,” Nonlinear Dyn. 84, 473–479 (2016).
[Crossref]

Wetzel, B.

M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, “Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability,” Nat. Commun. 7, 13675 (2016).
[Crossref]

Yan Wang,

A. Ankiewicz, Yan Wang, S. Wabnitz, and N. Akhmediev, “Extended nonlinear Schrödinger equation with higher-order odd and even terms and its rogue wave solutions,” Phys. Rev. E 89, 012907 (2014).
[Crossref]

Yu, D.-G.

C.-Q. Dai, J. Liu, Y. Fan, and D.-G. Yu, “Two-dimensional localized Peregrine solution and breather excited in a variable-coefficient nonlinear Schrödinger equation with partial nonlocality,” Nonlinear Dyn. 88, 1373–1383 (2017)..
[Crossref]

Zaviyalov, A.

A. Zaviyalov, O. Egorov, R. Iliew, and F. Lederer, “Rogue waves in mode-locked fiber lasers,” Phys. Rev. A 85, 013828 (2012).
[Crossref]

Zhang, X.-T.

J.-T. Li, X.-T. Zhang, M. Meng, Q.-T. Liu, Y.-Y. Wang, and C.-Q. Dai, “Control and management of the combined Peregrine soliton and Akhmediev breathers in PT -symmetric coupled waveguides,” Nonlinear Dyn. 84, 473–479 (2016).
[Crossref]

Zhang, Y.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Zhong, W.-P.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Zhou, G.-Q.

Y.-Y. Wang, C.-Q. Dai, G.-Q. Zhou, Y. Fan, and L. Chen, “Rogue wave and combined breather with repeatedly excited behaviors in the dispersion/diffraction decreasing medium,” Nonlinear Dyn. 87, 67–73 (2017).
[Crossref]

Chaos (2)

M. Tlidi, K. Panajotov, M. l. Ferré, and M. G. Clerc, “Drifting cavity solitons and dissipative rogue waves induced by time-delayed feedback in Kerr optical frequency comb and in all fiber cavities,” Chaos 27, 114312 (2017).
[Crossref]

A. Ankiewicz and N. Akhmediev, “Comparison of Lagrangian approach and method of moments for reducing dimensionality of soliton dynamical systems,” Chaos 18, 033129 (2008).
[Crossref]

Electron. Lett. (1)

A. Bhushan, F. Coppinger, and B. Jalali, “Time-stretched analogue-to-digital conversion,” Electron. Lett. 34, 1081 (1998).
[Crossref]

Eur. Phys. J. Spec. Top. (1)

B. Jalali, D. Solli, K. Goda, K. Tsia, and C. Ropers, “Real-time measurements, rare events and photon economics,” Eur. Phys. J. Spec. Top. 185, 145–157 (2010).
[Crossref]

Fluctuat. Noise Lett. (1)

S. Residori, Bortolozzo, A. Montina, F. Lenzini, and F. T. Arecchi, “Rogue waves in spatially extended optical systems,” Fluctuat. Noise Lett. 11, 1240014 (2012).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

A. D. Boardman and L. Velasco, “Gyroelectric cubic-quintic dissipative solitons,” IEEE J. Sel. Top. Quantum Electron. 12, 388–397, (2006).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

F. Coppinger, A. Bhushan, and B. Jalali, “Photonic time stretch and its application to analog-to-digital conversion,” IEEE Trans. Microw. Theory Tech. 47, 1309–1314 (1999).
[Crossref]

Int. J. Nonlinear Opt. Phys. Mater. (1)

A. Ankiewicz, “Simplified description of soliton perturbation and interaction using averaged complex potentials,” Int. J. Nonlinear Opt. Phys. Mater. 4, 857–870 (1995).
[Crossref]

J. Exp. Theor. Phys. (1)

A. I. Maimistov, “Evolution of solitary waves which are approximately solitons of a nonlinear Schrödinger equation,” J. Exp. Theor. Phys. 77, 727 (1993) [Zh. Eksp. Teor. Fiz. 104, 3620 (1993), in Russian].

J. Fluid Mech. (1)

V. V. Voronovich, V. I. Shrira, and G. Thomas, “Can bottom friction suppress ‘freak wave’ formation?” J. Fluid Mech. 604, 263–296 (2008).
[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. (2)

C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue wave generation in multiple-pulsing mode-locked fiber laser,” J. Opt. 15, 064005 (2013).
[Crossref]

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

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

Nat. Commun. (2)

P. Suret, R. El Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, “Single-shot observation of optical rogue waves in integrable turbulence using time microscopy,” Nat. Commun. 7, 13136 (2016).
[Crossref]

M. Närhi, B. Wetzel, C. Billet, S. Toenger, T. Sylvestre, J.-M. Merolla, R. Morandotti, F. Dias, G. Genty, and J. M. Dudley, “Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability,” Nat. Commun. 7, 13675 (2016).
[Crossref]

Nat. Photonics (1)

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8, 755–764 (2014).
[Crossref]

Nature (1)

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

Nonlinear Dyn. (3)

C.-Q. Dai, J. Liu, Y. Fan, and D.-G. Yu, “Two-dimensional localized Peregrine solution and breather excited in a variable-coefficient nonlinear Schrödinger equation with partial nonlocality,” Nonlinear Dyn. 88, 1373–1383 (2017)..
[Crossref]

Y.-Y. Wang, C.-Q. Dai, G.-Q. Zhou, Y. Fan, and L. Chen, “Rogue wave and combined breather with repeatedly excited behaviors in the dispersion/diffraction decreasing medium,” Nonlinear Dyn. 87, 67–73 (2017).
[Crossref]

J.-T. Li, X.-T. Zhang, M. Meng, Q.-T. Liu, Y.-Y. Wang, and C.-Q. Dai, “Control and management of the combined Peregrine soliton and Akhmediev breathers in PT -symmetric coupled waveguides,” Nonlinear Dyn. 84, 473–479 (2016).
[Crossref]

Opt. Commun. (1)

N. Akhmediev, W. Krolikowski, and A. J. Lowery, “Influence of the Raman-effect on solitons in optical fibers,” Opt. Commun. 131, 260–266 (1996).
[Crossref]

Opt. Fiber Technol. (1)

A. Ankiewicz, N. Akhmediev, and N. Devine, “Dissipative solitons with a Lagrangian approach (Invited paper),” Opt. Fiber Technol. 13, 91–97 (2007).
[Crossref]

Opt. Lett. (5)

Optica (1)

Phys. Lett. A (4)

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, J. M. Soto-Crespo, and A. Ankiewicz, “Extreme waves that appear from nowhere: on the nature of rogue waves,” Phys. Lett. A 373, 2137–2145 (2009).
[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]

M. Lakshmanan, K. Porsezian, and M. Daniel, “Effect of discreteness on the continuum limit of the Heisenberg spin chain,” Phys. Lett. A 133, 483–488 (1988).
[Crossref]

Phys. Rep. (1)

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528, 47–89 (2013).
[Crossref]

Phys. Rev. A (3)

A. Zaviyalov, O. Egorov, R. Iliew, and F. Lederer, “Rogue waves in mode-locked fiber lasers,” Phys. Rev. A 85, 013828 (2012).
[Crossref]

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[Crossref]

W. Chang, J. M. Soto-Crespo, A. Ankiewicz, and N. Akhmediev, “Dissipative soliton resonances in anomalous dispersion regime,” Phys. Rev. A 79, 033840 (2009).
[Crossref]

Phys. Rev. E (7)

A. Ankiewicz, Yan Wang, S. Wabnitz, and N. Akhmediev, “Extended nonlinear Schrödinger equation with higher-order odd and even terms and its rogue wave solutions,” Phys. Rev. E 89, 012907 (2014).
[Crossref]

A. Ankiewicz and N. Akhmediev, “Rogue wave solutions for the infinite integrable nonlinear Schrödinger equation hierarchy,” Phys. Rev. E 96, 012219 (2017).
[Crossref]

M. Facão, M. I. Carvalho, and D. F. Parker, “Soliton self-frequency shift: Self-similar solutions and their stability,” Phys. Rev. E 81, 046604 (2010).
[Crossref]

A. Ankiewicz, D. J. Kedziora, A. Chowdury, U. Bandelow, and N. Akhmediev, “Infinite hierarchy of nonlinear Schrödinger equations and their solutions,” Phys. Rev. E 93, 012206 (2016).
[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]

W. Chang, J. M. Soto-Crespo, P. Vouzas, and N. Akhmediev, “Extreme soliton pulsations in dissipative systems,” Phys. Rev. E 92, 022926 (2015).
[Crossref]

J. M. Soto-Crespo, Ph. Grelu, and N. Akhmediev, “Dissipative rogue waves: extreme pulses generated by passively mode-locked lasers,” Phys. Rev. E 84, 016604 (2011).
[Crossref]

Phys. Rev. Lett. (3)

C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
[Crossref]

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, “Non-Gaussian statistics and extreme waves in a nonlinear optical cavity,” Phys. Rev. Lett. 103, 173901 (2009).
[Crossref]

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[Crossref]

Sci. Rep. (1)

P.-H. Hanzard, M. Talbi, D. Mallek, A. Kellou, H. Leblond, F. Sanchez, T. Godin, and A. Hideur, “Brillouin scattering-induced rogue waves in self-pulsing fiber lasers,” Sci. Rep. 78, 45868 (2017).
[Crossref]

Other (4)

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995).

N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chap. & Hall, 1997).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill book co., 1953), pp. 276, 314.

N. Akhmediev and A. Ankiewicz, “Dissipative solitons in the complex Ginzburg-Landau and Swift-Hohenberg equations,” in Dissipative Solitons (Springer, 2005).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1.
Fig. 1. Plot of the relative soliton offset, t 0 / τ R , showing the effect of Raman delay. We compare the numerically found result (blue curve) with the analytic result of Eq. (29) (red curve). Here, delay is given by τ R = 0.02 and width w 0 = 0.4 .
Fig. 2.
Fig. 2. Plot of the soliton, Eq. (23), under the effect of Raman delay, using Eqs. (24)–(30). Here, delay is given by τ R = 0.02 and width w 0 = 0.4 . At the ends ( x = ± 3 ), the t offset is about 1.8.
Fig. 3.
Fig. 3. Plot of function r ( x ) for various c values. The blue curves are for c < 0 . This solution corresponds to the expansion in Eq. (37) with c = 0.04 (top blue curve) and c = 0.02 (next blue curve). The green curve, namely r ( x ) = z 2 = 1 + 4 x 2 , is for c = 0 . The two red curves are plots of Eqs. (35) and (36). The upper red curve is for c = 0.02 . The lower red curve is for c = 0.04 . In each case, the expansion, Eq. (37), is superimposed on the exact result. The curves are indistinguishable.
Fig. 4.
Fig. 4. Plot of function t 0 ( x ) / τ R , from Eq. (46) for various c : c = 0.04 (blue curve), c = 0.4 (magenta), c = 0.2 (red), and c = 0.1 (green).
Fig. 5.
Fig. 5. Plot of functions t 0 ( x ) , Eq. (46), r ( x ) , using the results of Section 6, including Eq. (38), and θ ( x ) , using Eq. (53), together with the same functions found numerically by solving Eqs. (33)–(41). Here, c = 0.04 , and the delay is given by τ R = 0.005 . The two upper curves give r ( x ) , while the two lower positive ones give t 0 ( x ) and the two negative ones show θ ( x ) . In each case, the exact numeric solution and the approximation for it are very close.
Fig. 6.
Fig. 6. Plot of the rogue wave, Eq. (7), using Eqs. (46)–(53) for delay given by τ R = 0.005 . Here c = 0.04 .
Fig. 7.
Fig. 7. Contour plot of the rogue wave, Eq. (7), using Eqs. (46)–(53) for delay given by τ R = 0.005 . Here c = 0.04 and the propagation direction, x , is the horizontal axis, while the transverse axis, t , is vertical.
Fig. 8.
Fig. 8. Plot of functions t 0 ( x ) / τ R (even function of x with t 0 ( 0 ) c 0 = 300 here), t 0 ( x ) / τ R (odd function of x ) and t 0 ( x ) / τ R (even function of x with t 0 ( 0 ) = 0 ) from original Eqs. (46), (45), and (48), respectively, together with a comparison with the Gaussian-based approximations for them, given by Eqs. (49), (50), and (51), all for c = 0.1 . Here, c 0 = 2 c ( 6 + 66 c + 215 c 2 ) .
Fig. 9.
Fig. 9. Plot of function t 0 ( x ) / τ R , from numerics (blue curve) for c = 0.1 . The magenta curve is from the analytic prediction of Eq. (52). It is in fairly good agreement with the numeric curve. Thus, the reduced model has provided a reasonably good description of the rogue wave dynamics.
Fig. 10.
Fig. 10. Function r 1 ( x ) found from numerical solution of the PDE. Three values of c , from (blue curve) 0.004 to (green curve) 0.006 are used, and the initial value is r ( 0 ) = 1 2 c . The analytic form, r 1 ( x ) = ( r ( x ) 1 4 x 2 ) / c 2 ( 1 + 4 x 2 ) 3 / 2 , shown by the red curve is close to the numerical results.
Fig. 11.
Fig. 11. Numerical PDE solution giving θ ( x ) / c . Various values of c are used, and the initial value is r ( 0 ) = 1 2 c . The analytic result is plotted as the magenta line, θ ( x ) / c = ( 35 / 2 ) arcsinh ( 2 x ) 35 x for x small, using Eq. (53). The numerically found curve for θ ( x ) / c is quite close to 35 x over this range, 0 < x < 1 .

Equations (87)

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

i ψ x + 1 2 ψ t t + | ψ | 2 ψ = R ,
R = τ R ψ ( | ψ | 2 ) t ν | ψ | 4 ψ ν 2 n | ψ | 2 n ψ + i γ 3 ψ t t t d 4 ψ t t t t d 6 ψ t t t t t t i s ( ψ | ψ | 2 ) t ,
L = L d d t ,
L d = i 2 ( ψ * ψ x ψ ψ * x ) + 1 2 | ψ t | 2 1 2 | ψ | 4 ,
L d ψ * + x L d ψ x * + t L d ψ t * 2 t 2 ( L d ψ t t * ) = R ,
d d x ( L c x j ) L c ( j ) = 2 Re ( R ψ * c ( j ) d t ) J [ c j ; R ]
i ψ x + 1 2 ψ t t + | ψ | 2 ψ = 0 ,
ψ = [ 4 1 + 2 i x c ( x ) + 4    t 2 1 ] exp [ i x ]
L = 2 π c ( x ) 7 / 2 [ x c ( x ) 2 c ( x ) 14 ( 4 x 2 + 1 ) c ( x ) + 2 c ( x ) 3 + 10 ( 4 x 2 + 1 ) 2 ] .
70 π ( 4 x 2 + 1 ) ( c ( x ) + 4 x 2 + 1 ) c ( x ) 9 / 2 = 0 ,
c ( x ) = 1 + 4 x 2 ,
ψ = [ 4 1 + 2 i x 1 + 4 x 2 + 4 t 2 1 ] exp [ i x ] .
ψ ( x , t ) = [ 4 1 + 2 i x D m ( x , t ) 1 ] e i F m ( x , t ) ,
D m ( x , t ) = r ( x ) + 4 [ t t 0 ( x ) ] 2 ,
F m ( x , t ) = x + a ( x ) [ t t 0 ( x ) ] θ ( x ) ,
L r 7 / 2 ( x ) 2 π = r 2 ( x ) [ 2 X a ( x ) t 0 ( x ) + X a 2 ( x ) + x r ( x ) 2 X θ ( x ) ] r 3 ( x ) [ 2 2 a ( x ) t 0 ( x ) + a 2 ( x ) 2 θ ( x ) ] + 14 X r ( x ) 10 X 2 ,
r ( x ) [ r ( x ) + 16 x ] 3 ( 4 x 2 + 1 ) r ( x ) = 0 ,
[ t 0 ( x ) ] 2 + 2 θ ( x ) = 0 .
ψ = [ 4 1 + 2 i x 1 + 4 x 2 + 4 ( t v x ) 2 1 ] e i [ v t + ( 1 v 2 2 ) x ] .
ψ ( x , t ) = [ 4 1 + 2 i B x D m ( x , t ) 1 ] e i F m ( x , t ) ,
L d = L d 1 + L d 2 + L d 3 ,
L d 1 = i 2 ( ψ * ψ x ψ ψ * x ) + a 1 a ( x ) t 0 ( x ) + h 1 θ ( x ) u 1 , L d 2 = 1 2 | ψ t | 2 ,
L d 3 = 1 2 ( | ψ | 4 1 ) .
( a 1 1 ) a ( x ) t 0 ( x ) + ( h 1 1 ) θ ( x ) u 1 + 1 ,
L r 7 / 2 ( x ) 2 π = 14 X r ( x ) 10 X 2 + r ( x ) 2 [ a 2 ( x ) X 2 a ( x ) X t 0 ( x ) 2 X θ ( x ) + B x r ( x ) + 4 B 4 ] r ( x ) 3 [ a 2 ( x ) + 2 B 2 a ( x ) t 0 ( x ) 2 θ ( x ) ] ,
S [ c ( j ) ] d d x ( L c x j ) L c ( j ) ,
r ( x ) 3 / 2 4 π [ d d x ( L a x ) L a ] = r ( x ) 3 / 2 4 π S [ a ] = ( r ( x ) X ) [ a ( x ) t 0 ( x ) ] .
r ( x ) 5 / 2 2 π [ d d x ( L θ x ) L θ ] = r ( x ) 5 / 2 2 π S [ θ ] = [ 3 X r ( x ) ] r ( x ) 16 B 2 x r ( x ) ,
r ( x ) 5 / 2 2 π [ d d x ( L t 0 ( x ) ) L t 0 ( x ) ] = r ( x ) 5 / 2 2 π S [ t 0 ] = 2 r ( x ) a ( x ) [ r ( x ) X ] a ( x ) { [ r ( x ) 3 X ] r ( x ) + 16 B 2 x r ( x ) } .
r ( x ) 9 / 2 π [ d d x ( L r x ) L r ] = r ( x ) 9 / 2 π S [ r ] = r ( x ) { r ( x ) { ( 3 X r ( x ) ) [ a 2 ( x ) 2 a ( x ) t 0 ( x ) 2 θ ( x ) ] + 12 ( B 1 ) } + 70 X } 70 X 2 .
J [ c j ; R ] = 2 Re ( R ψ * c ( j ) d t ) .
R = i α 3 ( ψ t t t + 6 | ψ | 2 ψ t ) R h .
r ( x ) 7 / 2 12 π α 3 J [ a ; R h ] = [ X r ( x ) ] × { r ( x ) [ ( a 2 ( x ) 2 ) r ( x ) + 4 ] 10 X } ,
r ( x ) 9 / 2 2 π α 3 a ( x ) J [ r ; R h ] = 210 X 2 + r ( x ) × { r ( x ) [ a 2 ( x ) ( r ( x ) 3 X ) + 6 ( 6 r ( x ) + 3 X ) ] 210 X } ,
v s = α 3 ( 1 k 2 2 ) v 1 k .
α 3 k 2 ( 2 c 2 + v 1 2 ) 2 α 3 ( v 1 6 ) + ( 2 c 1 + 1 ) k = 0 .
t 0 ( x ) = x [ 3 α 3 ( k 2 2 ) + k ] , θ ( x ) = 1 2 k 2 x ( 4 α 3 k + 1 ) .
i ψ x + 1 2 ψ t t + | ψ | 2 ψ = R [ ψ ( x , t ) ] ,
R [ ψ ( x , t ) ] = α 4 ( ψ t t t t + 6 ψ t 2 ψ * + 4 ψ | ψ t | 2 + 8 ψ t t | ψ | 2 + 2 ψ t t * ψ 2 + 6 ψ | ψ | 4 ) R l .
r ( x ) 7 / 2 16 π a ( x ) α 4 J [ a ; R l ] = [ r ( x ) X ] { r ( x ) × [ ( a 2 ( x ) 6 ) r ( x ) + 12 ] 30 X }
r ( x ) 13 / 2 2 π α 4 J [ r ; R l ] = 2772 X 3 + r ( x ) × { r ( x ) [ 12 a 2 ( x ) ( r ( x ) ( r ( x ) [ r ( x ) 3 X 6 ] + 35 X ) 35 X 2 ) + a 4 ( x ) r ( x ) 2 [ 3 X r ( x ) ] 2 r ( x ) ( 3 r ( x ) [ r ( x ) 3 ( X + 4 ) ] + 70 ( 3 X + 2 ) ) + 28 X ( 15 X + 91 ) ] 5040 X 2 } ,
t 0 ( x ) = v e x = k [ 4 α 4 ( k 2 6 ) 1 ] x , θ ( x ) = { 3 α 4 [ ( k 2 2 ) 2 6 ] k 2 2 } x .
ψ ( x , t ) = Q ( x ) 2 w ( x ) sech ( y 1 w ( x ) ) exp { i [ a ( x ) y 1 θ ( x ) ] } ,
Q ( x ) = 0 ,
Q ( x ) [ a ( x ) t 0 ( x ) ] = 0 ,
Q ( x ) 6 w 3 ( x ) [ w ( x ) Q ( x ) 2 ] ,
8 τ R 15 w 0 4 t 0 ( x ) = 0 ,
t 0 ( x ) = a ( x ) = 4 τ R Q ( x ) 15 w 0 3 x = 8 τ R 15 w 0 4 x ,
t 0 ( x ) = 4 τ R x 2 15 w 0 4 .
1 w 0 2 + t 0 ( x ) 2 + 2 θ ( x ) = 0 ,
ψ ( x , t ) = 1 w 0 sech ( t w 0 ) exp { i x 2 w 0 2 }
= Q 0 2 sech ( Q 0 t 2 ) e 1 8 i Q 0 2 x ,
r ( x ) [ r ( x ) + 16 x ] = 3 ( 4 x 2 + 1 ) r ( x ) .
[ 3 z 2 s 2 ( z ) ] s ( z ) = 2 z s ( z ) .
s ( z ) = 1 6 c ( F 1 / 3 + F 1 / 3 1 ) ,
F = 54 c 2 z 2 1 + 6 c z 3 27 c 2 z 2 1 ,
r ( z ) = z 2 ( 1 2 c z + 6 c 2 z 2 21 c 3 z 3 + 80 c 4 z 4 + ) .
r ( x ) = ( 1 + 4 x 2 ) [ 1 2 c 1 + 4 x 2 + 6 c 2 ( 1 + 4 x 2 ) 21 c 3 ( 1 + 4 x 2 ) 3 / 2 + 80 c 4 ( 1 + 4 x 2 ) 2 + ] .
r ( x ) 9 / 2 32 π J [ t 0 ; R r ] = τ R { r ( x ) [ 5 X r ( x ) ] 7 X 2 } ,
16 τ R { r ( x ) [ r ( x ) 5 X ] + 7 X 2 } + r ( x ) 2 { 2 r ( x ) ( a ( x ) [ r ( x ) X ] 8 x a ( x ) ) + a ( x ) [ 3 X r ( x ) ] r ( x ) } = 0 ,
r ( x ) { r ( x ) [ r ( x ) 3 X ] [ a 2 ( x ) 2 a ( x ) t 0 ( x ) 2 θ ( x ) ] 70 X } + 70 X 2 = 0
[ X r ( x ) ] [ a ( x ) t 0 ( x ) ] = 0 .
r ( x ) { r ( x ) [ t 0 ( x ) ( ( r ( x ) 3 X ) r ( x ) + 16 x r ( x ) ) + 2 r ( x ) ( r ( x ) X ) t 0 ( x ) + 16 τ R ] 80 τ R X } + 112 τ R X 2 = 0 ,
r ( x ) { r ( x ) [ 3 X r ( x ) ] [ t 0 ( x ) 2 + 2 θ ( x ) ] 70 X } + 70 X 2 = 0 .
r ( x ) = X [ 1 2 c X 1 / 2 + 6 c 2 X 21 c 3 X 3 / 2 + 80 c 4 X 2 + ] .
a ( x ) = t 0 ( x ) = 2 τ R x c X 3 / 2 [ 2 ( 8 x 2 + 3 ) + 33 c X 1 / 2 + 215 c 2 X ] + 33 τ R arctan ( 2 x ) .
t 0 ( x ) = τ R { 1 c ( 8 x 2 + 1 X 1 / 2 1 ) + 33 x arctan ( 2 x ) + 215 2 c [ X 1 / 2 1 ] } .
t 0 ( x ) τ R ( 215 c + 6 c + 66 ) x 2 ( 215 c + 10 c + 88 ) x 4 + .
t 0 ( x ) 6 τ R c x 2 ( 1 5 3 x 2 + ) .
a ( x ) τ R = t 0 ( x ) τ R = 2 c [ 6 X 5 / 2 + 66 c X 2 + 215 c 2 X 3 / 2 ] .
t 0 ( x ) τ R c 0 e 6 x 2 .
t 0 ( x ) τ R c 0 π 6 erf ( 6 x ) ,
t 0 ( x ) τ R c 0 2 π 6 [ x erf ( 6 x ) 1 e 6 x 2 6 x ] .
t 0 ( x ) τ R c 0 2 x 2 [ 1 x 2 + 6 x 4 5 ] .
t 0 ( x ) 125 τ R x 2
θ ( x ) = 35 c 2 arcsinh ( 2 x ) .
θ ( x ) = 35 c x ( 1 2 3 x 2 + ) ,
θ ( x ) = 35 c X ,
I = { r ( x ) + 4 [ t t 0 ( x ) ] 2 4 } 2 + 64 x 2 ( r ( x ) + 4 [ t t 0 ( x ) ] 2 ) 2 .
t 0 ( x ) 25 τ R x 2 ( 1 + 14.7 c ) ,
ϕ [ x , t = t 0 ( x ) ; c ] = ( 1 + 4 ( 1 + 2 i x ) c n r 1 ( x ) + 4 x 2 + 1 ) e i ( x + c n y ( x ) ) ,
ϕ [ x , t = 0 ] = ( 1 + 4 ( 1 + 2 i x ) 4 x 2 + 1 ) e i ( x ) .
Δ = c n e i x [ f ( x ) + i n ( x ) ] .
f ( x ) + i n ( x ) + 4 i r 1 ( x ) i 2 x + ( 1 2 i x ) ( 2 x 3 i ) y ( x ) ( 2 x + i ) 2 = 0 .
r 1 ( x ) = 1 12 ( 4 x 2 + 1 ) [ ( 3 4 x 2 ) f ( x ) + 8 x n ( x ) ] ,
y ( x ) = 1 3 [ 2 x f ( x ) n ( x ) ] .
y ( x ) θ ( x ) / ( c ) = 35 x + 2 ( 96 + 35 3 ) x 3 + .

Metrics