Abstract

The super rogue wave dynamics in optical fibers are investigated within the framework of a generalized nonlinear Schrödinger equation containing group-velocity dispersion, Kerr and quintic nonlinearity, and self-steepening effect. In terms of the explicit rogue wave solutions up to the third order, we show that, for a rogue wave solution of order n, it can be shaped up as a single super rogue wave state with its peak amplitude 2n+1 times the background level, which results from the superposition of n(n+1)/2 Peregrine solitons. Particularly, we demonstrate that these super rogue waves involve a frequency chirp that is also localized in both time and space. The robustness of the super chirped rogue waves against white-noise perturbations as well as the possibility of generating them in a turbulent field is numerically confirmed, which anticipates their accessibility to experimental observation.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (2)

S. Chen, Y. Ye, J. M. Soto-Crespo, Ph. Grelu, and F. Baronio, “Peregrine solitons beyond the threefold limit and their two-soliton interactions,” Phys. Rev. Lett. 121, 104101 (2018).
[Crossref] [PubMed]

F. Baronio, B. Frisquet, S. Chen, G. Millot, S. Wabnitz, and B. Kibler, “Observation of a group of dark rogue waves in a telecommunication optical fiber,” Phys. Rev. A 97, 013852 (2018).
[Crossref]

2017 (8)

A. Ankiewicz and N. Akhmediev, “Multi-rogue waves and triangular numbers,” Rom. Rep. Phys. 69, 104 (2017).

F. Baronio, “Akhemdiev breathers and Peregrine solitary waves in a quadratic medium,” Opt. Lett. 42(9), 1756–1759 (2017).
[Crossref] [PubMed]

F. Baronio, S. Chen, and D. Mihalache, “Two-color walking Peregrine solitary waves,” Opt. Lett. 42(18), 3514–3517 (2017).
[Crossref] [PubMed]

S. Chen, Y. Ye, F. Baronio, Y. Liu, X.-M. Cai, and Ph. Grelu, “Optical Peregrine rogue waves of self-induced transparency in a resonant erbium-doped fiber,” Opt. Express 25(24), 29687–29698 (2017).
[Crossref] [PubMed]

S. Chen, F. Baronio, J. M. Soto-Crespo, Ph. Grelu, and D. Mihalache, “Versatile rogue waves in scalar, vector, and multidimensional nonlinear systems,” J. Phys. A Math. Theor. 50, 463001 (2017).
[Crossref]

D. Mihalache, “Multidimensional localized structures in optical and matter-wave media: A topical survey of recent literature,” Rom. Rep. Phys. 69, 403 (2017).

A. Safari, R. Fickler, M. J. Padgett, and R. W. Boyd, “Generation of caustics and rogue waves from nonlinear instability,” Phys. Rev. Lett. 119, 203901 (2017).
[Crossref] [PubMed]

A. Tikan, C. Billet, G. El, A. Tovbis, M. Bertola, T. Sylvestre, F. Gustave, S. Randoux, G. Genty, P. Suret, and J. M. Dudley, “Universality of the Peregrine soliton in the focusing dynamics of the cubic nonlinear Schrödinger equation,” Phys. Rev. Lett. 119, 033901 (2017).
[Crossref]

2016 (9)

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]

P. Suret, R. E. 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] [PubMed]

C. Bao, J. A. Jaramillo-Villegas, Y. Xuan, D. E. Leaird, M. Qi, and A. M. Weiner, “Observation of Fermi-Pasta-Ulam recurrence induced by breather solitons in an optical microresonator,” Phys. Rev. Lett. 117, 163901 (2016).
[Crossref] [PubMed]

J. M. Soto-Crespo, N. Devine, and N. Akhmediev, “Integrable turbulence and rogue waves: breathers or solitons?” Phys. Rev. Lett. 116, 103901 (2016).
[Crossref] [PubMed]

B. A. Malomed, “Multidimensional solitons: Well-established results and novel findings,” Eur. Phys. J. Spec. Top. 225, 2507–2532 (2016).
[Crossref]

S. Chen, X.-M. Cai, Ph. Grelu, J. M. Soto-Crespo, S. Wabnitz, and F. Baronio, “Complementary optical rogue waves in parametric three-wave mixing,” Opt. Express 24(6), 5886–5895 (2016).
[Crossref] [PubMed]

S. Chen, J. M. Soto-Crespo, F. Baronio, Ph. Grelu, and D. Mihalache, “Rogue-wave bullets in a composite (2+1)D nonlinear medium,” Opt. Express 24(14), 15251–15260 (2016).
[Crossref] [PubMed]

B. Frisquet, B. Kibler, Ph. Morin, F. Baronio, M. Conforti, G. Millot, and S. Wabnitz, “Optical dark rogue wave,” Sci. Rep. 6, 20785 (2016).
[Crossref] [PubMed]

S. Chen, F. Baronio, J. M. Soto-Crespo, Y. Liu, and Ph. Grelu, “Chirped Peregrine solitons in a class of cubic-quintic nonlinear Schrödinger equations,” Phys. Rev. E 93, 062202 (2016).
[Crossref]

2015 (5)

F. Baronio, S. Chen, Ph. Grelu, S. Wabnitz, and M. Conforti, “Baseband modulation instability as the origin of rogue waves,” Phys. Rev. A 91, 033804 (2015).
[Crossref]

D. Pierangeli, F. Di Mei, C. Conti, A. J. Agranat, and E. DelRe, “Spatial rogue waves in photorefractive ferroelectrics,” Phys. Rev. Lett. 115, 093901 (2015).
[Crossref] [PubMed]

M. Leonetti and C. Conti, “Observation of three dimensional optical rogue waves through obstacles,” Appl. Phys. Lett. 106, 254103 (2015).
[Crossref]

J. He, S. Xu, and Y. Cheng, “The rational solutions of the mixed nonlinear Schrödinger equation,” AIP Adv. 5, 017105 (2015).
[Crossref]

S. Chen and D. Mihalache, “Vector rogue waves in the Manakov system: diversity and compossibility,” J. Phys. A Math. Theor. 48, 215202 (2015).
[Crossref]

2014 (5)

S. Chen and L.-Y. Song, “Peregrine solitons and algebraic soliton pairs in Kerr media considering space-time correction,” Phys. Lett. A 378, 1228–1232 (2014).
[Crossref]

F. Baronio, M. Conforti, A. Degasperis, S. Lombardo, M. Onorato, and S. Wabnitz, “Vector rogue waves and baseband modulation instability in the defocusing regime,” Phys. Rev. Lett. 113, 034101 (2014).
[Crossref] [PubMed]

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

H. N. Chan, K. W. Chow, D. J. Kedziora, R. H. J. Grimshaw, and E. Ding, “Rogue wave modes for a derivative nonlinear Schrödinger model,” Phys. Rev. E 89, 032914 (2014).
[Crossref]

S. Chen, J. M. Soto-Crespo, and Ph. Grelu, “Dark three-sister rogue waves in normally dispersive optical fibers with random birefringence,” Opt. Express 22(22), 27632–27642 (2014).
[Crossref] [PubMed]

2013 (4)

S. Birkholz, E. T. J. Nibbering, C. Brée, S. Skupin, A. Demircan, G. Genty, and G. Steinmeyer, “Spatiotemporal rogue events in optical multiple filamentation,” Phys. Rev. Lett. 111, 243903 (2013).
[Crossref]

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

S. Chen, “Twisted rogue-wave pairs in the Sasa-Satsuma equation,” Phys. Rev. E 88, 023202 (2013).
[Crossref]

Zhaqilao, “On Nth-order rogue wave solution to the generalized nonlinear Schrödinger equation,” Phys. Lett. A 377, 855–859 (2013).
[Crossref]

2012 (4)

S. Xu and J. He, “The rogue wave and breather solution of the Gerdjikov-Ivanov equation,” J. Math. Phys. 53, 063507 (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] [PubMed]

Z. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. 75, 086401 (2012).
[Crossref] [PubMed]

A. Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev, “Super rogue waves: observation of a higher-order breather in water waves,” Phys. Rev. X 2, 011015 (2012).

2011 (1)

A. Ankiewicz, D. J. Kedziora, and N. Akhmediev, “Rogue wave triplets,” Phys. Lett. A 375, 2782–2785 (2011).
[Crossref]

2010 (3)

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

M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, “Optical Cherenkov radiation in ultrafast cascaded second-harmonic generation,” Phys. Rev. A 82, 063806 (2010).
[Crossref]

V. I. Shrira and V. V. Geogjaev, “What makes the Peregrine soliton so special as a prototype of freak waves?” J. Eng. Math. 67, 11–22 (2010).
[Crossref]

2009 (3)

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, A. Ankiewicz, and J. M. Soto-Crespo, “Rogue waves and rational solutions of the nonlinear Schrödinger equation,” Phys. Rev. E 80, 026601 (2009).
[Crossref]

S. Chen and J. M. Dudley, “Spatiotemporal nonlinear optical self-similarity in three dimensions,” Phys. Rev. Lett. 102, 233903 (2009).
[Crossref] [PubMed]

2007 (2)

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

J. Moses, B. A. Malomed, and F. W. Wise, “Self-steepening of ultrashort optical pulses without self-phase-modulation,” Phys. Rev. A 76, 021802 (2007).
[Crossref]

2006 (2)

2005 (2)

S. Chen and L. Yi, “Chirped self-similar solutions of a generalized nonlinear Schrödinger equation model,” Phys. Rev. E 71, 016606 (2005).
[Crossref]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. Soc. Am. B 7, R53–R72 (2005).
[Crossref]

2002 (1)

Z. Li, L. Li, H. Tian, G. Zhou, and K. H. Spatschek, “Chirped femtosecond solitonlike laser pulse form with self-frequency shift,” Phys. Rev. Lett. 89, 263901 (2002).
[Crossref] [PubMed]

1997 (1)

1996 (1)

1987 (2)

P. A. Clarkson and C. M. Cosgrove, “Painlevé analysis of the nonlinear Schrödinger family of equations,” J. Phys. A Math. Gen. 20, 2003–2024 (1987).
[Crossref]

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. QE-23, 510–524 (1987).
[Crossref]

1984 (1)

A. Kundu, “Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger type equations,” J. Math. Phys. 25, 3433–3438 (1984).
[Crossref]

1983 (2)

V. S. Gerdjikov and M. I. Ivanov, “The quadratic bundle of general form and the nonlinear evolution equations. Hierarchies of Hamiltonian structures,” Bulg. J. Phys. 10, 130–143 (1983).

D. H. Peregrine, “Water waves, nonlinear Schödinger equations and their solutions,” J. Aust. Math. Soc. B Appl. Math. 25, 16–43 (1983).
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H. N. Chan, K. W. Chow, D. J. Kedziora, R. H. J. Grimshaw, and E. Ding, “Rogue wave modes for a derivative nonlinear Schrödinger model,” Phys. Rev. E 89, 032914 (2014).
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F. Baronio, B. Frisquet, S. Chen, G. Millot, S. Wabnitz, and B. Kibler, “Observation of a group of dark rogue waves in a telecommunication optical fiber,” Phys. Rev. A 97, 013852 (2018).
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[Crossref]

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F. Baronio, S. Chen, and D. Mihalache, “Two-color walking Peregrine solitary waves,” Opt. Lett. 42(18), 3514–3517 (2017).
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S. Chen, J. M. Soto-Crespo, F. Baronio, Ph. Grelu, and D. Mihalache, “Rogue-wave bullets in a composite (2+1)D nonlinear medium,” Opt. Express 24(14), 15251–15260 (2016).
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F. Baronio, S. Chen, Ph. Grelu, S. Wabnitz, and M. Conforti, “Baseband modulation instability as the origin of rogue waves,” Phys. Rev. A 91, 033804 (2015).
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F. Baronio, M. Conforti, A. Degasperis, S. Lombardo, M. Onorato, and S. Wabnitz, “Vector rogue waves and baseband modulation instability in the defocusing regime,” Phys. Rev. Lett. 113, 034101 (2014).
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D. Pierangeli, F. Di Mei, C. Conti, A. J. Agranat, and E. DelRe, “Spatial rogue waves in photorefractive ferroelectrics,” Phys. Rev. Lett. 115, 093901 (2015).
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F. Baronio, M. Conforti, A. Degasperis, S. Lombardo, M. Onorato, and S. Wabnitz, “Vector rogue waves and baseband modulation instability in the defocusing regime,” Phys. Rev. Lett. 113, 034101 (2014).
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D. Pierangeli, F. Di Mei, C. Conti, A. J. Agranat, and E. DelRe, “Spatial rogue waves in photorefractive ferroelectrics,” Phys. Rev. Lett. 115, 093901 (2015).
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S. Birkholz, E. T. J. Nibbering, C. Brée, S. Skupin, A. Demircan, G. Genty, and G. Steinmeyer, “Spatiotemporal rogue events in optical multiple filamentation,” Phys. Rev. Lett. 111, 243903 (2013).
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B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
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H. N. Chan, K. W. Chow, D. J. Kedziora, R. H. J. Grimshaw, and E. Ding, “Rogue wave modes for a derivative nonlinear Schrödinger model,” Phys. Rev. E 89, 032914 (2014).
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A. Tikan, C. Billet, G. El, A. Tovbis, M. Bertola, T. Sylvestre, F. Gustave, S. Randoux, G. Genty, P. Suret, and J. M. Dudley, “Universality of the Peregrine soliton in the focusing dynamics of the cubic nonlinear Schrödinger equation,” Phys. Rev. Lett. 119, 033901 (2017).
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J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8, 755–764 (2014).
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J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8, 755–764 (2014).
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P. Suret, R. E. 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).
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B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
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A. Safari, R. Fickler, M. J. Padgett, and R. W. Boyd, “Generation of caustics and rogue waves from nonlinear instability,” Phys. Rev. Lett. 119, 203901 (2017).
[Crossref] [PubMed]

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B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys. 6, 790–795 (2010).
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F. Baronio, B. Frisquet, S. Chen, G. Millot, S. Wabnitz, and B. Kibler, “Observation of a group of dark rogue waves in a telecommunication optical fiber,” Phys. Rev. A 97, 013852 (2018).
[Crossref]

B. Frisquet, B. Kibler, Ph. Morin, F. Baronio, M. Conforti, G. Millot, and S. Wabnitz, “Optical dark rogue wave,” Sci. Rep. 6, 20785 (2016).
[Crossref] [PubMed]

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A. Tikan, C. Billet, G. El, A. Tovbis, M. Bertola, T. Sylvestre, F. Gustave, S. Randoux, G. Genty, P. Suret, and J. M. Dudley, “Universality of the Peregrine soliton in the focusing dynamics of the cubic nonlinear Schrödinger equation,” Phys. Rev. Lett. 119, 033901 (2017).
[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]

S. Birkholz, E. T. J. Nibbering, C. Brée, S. Skupin, A. Demircan, G. Genty, and G. Steinmeyer, “Spatiotemporal rogue events in optical multiple filamentation,” Phys. Rev. Lett. 111, 243903 (2013).
[Crossref]

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

S. Chen, Y. Ye, J. M. Soto-Crespo, Ph. Grelu, and F. Baronio, “Peregrine solitons beyond the threefold limit and their two-soliton interactions,” Phys. Rev. Lett. 121, 104101 (2018).
[Crossref] [PubMed]

S. Chen, F. Baronio, J. M. Soto-Crespo, Ph. Grelu, and D. Mihalache, “Versatile rogue waves in scalar, vector, and multidimensional nonlinear systems,” J. Phys. A Math. Theor. 50, 463001 (2017).
[Crossref]

S. Chen, Y. Ye, F. Baronio, Y. Liu, X.-M. Cai, and Ph. Grelu, “Optical Peregrine rogue waves of self-induced transparency in a resonant erbium-doped fiber,” Opt. Express 25(24), 29687–29698 (2017).
[Crossref] [PubMed]

S. Chen, J. M. Soto-Crespo, F. Baronio, Ph. Grelu, and D. Mihalache, “Rogue-wave bullets in a composite (2+1)D nonlinear medium,” Opt. Express 24(14), 15251–15260 (2016).
[Crossref] [PubMed]

S. Chen, X.-M. Cai, Ph. Grelu, J. M. Soto-Crespo, S. Wabnitz, and F. Baronio, “Complementary optical rogue waves in parametric three-wave mixing,” Opt. Express 24(6), 5886–5895 (2016).
[Crossref] [PubMed]

S. Chen, F. Baronio, J. M. Soto-Crespo, Y. Liu, and Ph. Grelu, “Chirped Peregrine solitons in a class of cubic-quintic nonlinear Schrödinger equations,” Phys. Rev. E 93, 062202 (2016).
[Crossref]

F. Baronio, S. Chen, Ph. Grelu, S. Wabnitz, and M. Conforti, “Baseband modulation instability as the origin of rogue waves,” Phys. Rev. A 91, 033804 (2015).
[Crossref]

S. Chen, J. M. Soto-Crespo, and Ph. Grelu, “Dark three-sister rogue waves in normally dispersive optical fibers with random birefringence,” Opt. Express 22(22), 27632–27642 (2014).
[Crossref] [PubMed]

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] [PubMed]

Grimshaw, R. H. J.

H. N. Chan, K. W. Chow, D. J. Kedziora, R. H. J. Grimshaw, and E. Ding, “Rogue wave modes for a derivative nonlinear Schrödinger model,” Phys. Rev. E 89, 032914 (2014).
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Figures (6)

Fig. 1
Fig. 1 Surface (top) and contour (bottom) plots of the (a) first-order (fundamental), (b) second-order, and (c) third-order rogue waves in the self-focusing (or anomalous dispersion) regime, obtained with the same set of system parameters. (a) γ 2 = 1; (b) γ 2 = 1, γ 3 = 100; (c) γ 2 = 1, γ 5 = 2000. In each case, the other unshown γj will be set zero.
Fig. 2
Fig. 2 The preceding three low-order rogue wave states that have their respective maximum allowable amplitude factor (3, 5, 7) in the normal dispersion regime: (a) Peregrine soliton; (b) Super second-order rogue wave; (c) Super third-order rogue wave. We choose a = 1, σ = 1, γ = 1, μ = 3 / 2 and ω = 2 as the system parameters, and choose γ 2 = 1, γ 3 = 2 / 81 i 86 2 / 81, γ 5 = 359 / 1215 i 101 2 / 1215, and γ 1 = γ 4 = γ 6 = 0 as the specific structural parameters.
Fig. 3
Fig. 3 Super second-order rogue waves in the anomalous dispersion regime associated to (a) the GI equation, (b) the CLL-NLS equation, and (c) the KN-NLS equation, respectively. Left column: Amplitude |E|; Middle column: Phase Φ; Right column: Chirp δω.
Fig. 4
Fig. 4 Super second-order rogue wave solution of the KE equation in the anomalous dispersion regime, with the system parameters a = 1, σ = 1, γ = 0, μ = −1 and ω = 1.
Fig. 5
Fig. 5 Numerical simulations of the GI (see left column) and KE (see right column) super chirped second-order rogue waves, which are the same as in Figs. 3(a) and 4, but are now perturbed by white noises of ε = 0.01 and 0.02, respectively. (a), (d): Initial amplitude profiles (red line) as compared to the analytical ones (blue line); (b), (e): Numerical recurrence of rogue waves from the above initial conditions; (c), (f): the numerical amplitude profiles obtained at t = 0 (red line) compared with their analytical ones (blue line).
Fig. 6
Fig. 6 Numerical excitation of the GI super chirped rogue wave from a turbulent field under otherwise the same parameter condition as in Fig. 3(a). The panel (a) shows the maximum peak amplitude chosen from a very large t window for each specific value of z, and (b) displays the evolution of the field amplitude around z = 14 within a narrow temporal interval, where a typical super chirped rogue wave has been singled out by the black curve.

Equations (34)

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i E z + 1 2 E t t + σ | E | 2 E + i γ ( | E | 2 E ) t + i ( μ 2 γ ) ( | E | 2 ) t E + 1 2 ( μ γ ) ( μ 2 γ ) | E | 4 E = 0 ,
R t = UR ,     R z = VR ,
U = i ( λ σ ) σ 3 2 γ + λ Q + i ( μ 2 γ ) 2 σ 3 Q 2 , V = i ( λ σ ) 2 σ 3 4 γ 2 + λ 2 ( λ σ γ Q i λ σ 3 Q 2 + μ Q 3 + i σ 3 Q t ) + μ 2 γ 4 [ i ( 2 μ γ ) σ 3 Q 4 + ( Q Q t Q t Q ) ] ,
E 0 = a exp  [ i ( k z + ω t ) ] ,
k = η a 2 + 1 2 ( μ γ ) ( μ 2 γ ) a 4 ω 2 2 ,    ( here  η = σ γ ω ) .
R ( λ ) = G ( Γ 1 N 1 + Γ 2 N 2 ) ,
N j = [ 1 i [ a 2 γ ( μ 2 γ ) + 2 γ ϕ j + λ σ ] 2 a γ λ ] exp  [ i ( θ j z + ϕ j t ) ] ,
ϕ j = ω 2 ( 1 ) j 2 γ ( λ + a 2 γ 2 β 2 / a 2 ) 2 + 4 β 2 γ 2 ,
θ j = k 2 + ω 2 ϕ j 4 γ [ a 2 γ ( 2 μ γ ) + 2 ω γ + β 2 / a 2 λ ] ,
β = a η a 2 γ ( μ γ ) .
ϕ 0 = ω 2 ,    θ 0 = k 2 .
ω < σ / γ a 2 ( μ γ ) .
λ = λ 0 + χ ϵ 2 ,    χ = λ 0 λ 0 * = 4 i β γ ,
Γ 1 = 1 2 j = 1 n ( γ 2 j 1 + γ 2 j ϵ ) ϵ 2 ( j 1 ) ,   Γ 2 = 1 2 j = 1 n ( γ 2 j 1 γ 2 j ϵ ) ϵ 2 ( j 1 ) ,
Θ ( λ ) = Θ ( 0 ) + Θ ( 1 ) ϵ 2 + Θ ( 2 ) ϵ 4 + + O ( ϵ 2 n ) ,
E [ n ] = E 0 ( 1 i | E 0 | Y 1 ( M ) 1 Y 2 ) ( det ( M ) det ( M ) ) μ / γ ,
[ Y 1 Y 2 ] = [ Θ ( 0 ) , Θ ( 1 ) , Θ ( 2 ) , , Θ ( n 1 ) ] ,
Θ X Θ λ λ * = i j n M i j ϵ * 2 ( i 1 ) ϵ 2 ( j 1 ) + O ( | ϵ | 4 n ) ,     X = γ [ λ 0 0 λ * ] .
z 0 = a 2 γ 2 β ( a 4 γ 2 + β 2 ) Im ( γ 1 / γ 2 ) 2 β 2 ,    t 0 = z 0 ( a 2 μ + ω ) β 2 ( a 4 γ 2 + β 2 ) Re ( γ 1 / γ 2 ) 2 β ,
E [ 1 ] = E 0 [ 1 2 i ( γ τ + β 2 z / a 2 ) + 1 / a 2 M i N ] exp  ( i Φ ) ,
τ = t ( a 2 μ + ω ) z ,     M = ( β 2 a 2 + γ 2 a 2 ) ( τ 2 + β 2 z 2 ) + 1 4 a 2 , N = γ ( γ a 2 z τ ) ,    Φ = 2 ( μ γ 1 ) arctan   ( N M ) .
E [ 2 ] = E 0 ( 1 G + i H C i D ) exp  ( i Φ ) ,
Φ = 2 ( μ γ 1 ) arctan  ( D C ) .
C = 64 ( a 4 γ 2 + β 2 ) 3 ( β 2 z 2 + τ 2 ) 3 + ( a 4 γ 2 + β 2 ) [ 48 β 2 ( 4 a 8 γ 4 + 17 a 4 β 2 γ 2 + 9 β 4 ) z 4 + 384 a 6 β 2 γ 3 τ z 3 288 ( 2 a 4 γ 2 + β 2 ) ( a 4 γ 2 + β 2 ) τ 2 z 2 + 384 a 6 γ 3 τ 3 z 48 ( 3 a 4 γ 2 β 2 ) τ 4 ] + 36 ( 28 a 8 γ 4 + 35 a 4 β 2 γ 2 + 11 β 4 ) z 2 288 a 6 γ 3 τ z + 36 ( 7 a 4 γ 2 + 3 β 2 ) τ 2 + 9 , D = 192 a 2 γ ( a 4 γ 2 + β 2 ) 2 ( β 2 z 2 + τ 2 ) 2 ( a 2 γ z τ ) + 96 a 2 γ [ a 2 γ ( 6 a 8 γ 4 + 13 a 4 β 2 γ 2 + 9 β 4 ) z 3 + ( 6 a 8 γ 4 + 15 a 4 β 2 γ 2 + 3 β 4 ) τ z 2 + 3 a 2 γ ( a 4 γ 2 β 2 ) τ 2 z ( 3 a 4 γ 2 + β 2 ) τ 3 ] + 36 a 2 γ ( 11 a 2 γ z 3 τ ) , G = 192 ( a 4 γ 2 + β 2 ) 2 ( β 2 z 2 + τ 2 ) ( 4 a 2 γ τ z + 5 β 2 z 2 + τ 2 ) 1152 a 4 β 2 γ 2 z 2 + 288 ( 3 a 4 γ 2 + β 2 ) [ ( 2 a 4 γ 2 + 3 β 2 ) z 2 2 a 2 γ τ z + τ 2 ] 36 , H = 384 ( a 4 γ 2 + β 2 ) 2 ( β 2 z 2 + τ 2 ) 2 ( a 2 γ τ + β 2 z ) + 192 [ ( 4 a 8 γ 4 + 3 a 4 β 2 γ 2 + β 4 ) β 2 z 3 3 a 2 γ ( a 4 γ 2 + 3 β 2 ) ( 2 a 4 γ 2 + β 2 ) τ z 2 3 ( 2 a 8 γ 4 + a 4 β 2 γ 2 + β 4 ) τ 2 z + a 2 γ ( a 4 γ 2 β 2 ) τ 3 ] 72 [ ( 12 a 4 γ 2 + 5 β 2 ) z + a 2 γ τ ] .
δ ω = Φ t = 2 ( μ γ ) ( D C t C D t ) γ ( C 2 + D 2 ) ,
E [ 2 ] = E 0 ( 1 G + i H C ) exp  ( i Φ ) ,
C = 64 β 6 ( β 2 z 2 + τ 2 ) 3 + 48 β 4 ( 3 β 2 z 2 τ 2 ) 2 + 36 β 2 ( 11 β 2 z 2 + 3 τ 2 ) + 9 , G = 192 β 4 ( β 2 z 2 + τ 2 ) ( 5 β 2 z 2 + τ 2 ) + 288 β 2 ( 3 β 2 z 2 + τ 2 ) 36 , H = 384 β 6 ( β 2 z 2 + τ 2 ) 2 z + 24 β 2 z ( 8 β 4 z 2 24 β 2 τ 2 15 ) , Φ = μ a 2 β 2 ( ln   C ) t = 24 μ a 2 τ C [ 16 β 4 ( β 2 z 2 + τ 2 ) 2 8 β 2 ( 3 β 2 z 2 τ 2 ) + 9 ] .
δ ω = μ a 2 β 2 ( ln  C ) t t .
R 0 = γ 1 + 2 i γ 2 β ϑ , S 0 = γ 1 2 i γ 2 β ξ , R 1 = 2 γ 1 β 2 ϑ 2 + γ 2 ( 4 i 3 β 3 ϑ 3 + 5 i β ϑ + 4 β τ ) + γ 3 + 2 i γ 4 β ϑ , S 1 = 2 γ 1 β 2 ξ 2 γ 2 ( 4 i 3 β 3 ξ 3 + 5 i β ξ + 4 β τ i ρ ) γ 3 2 i γ 4 β ξ , R 2 = γ 1 g + γ 2 p 2 γ 3 β 2 ϑ 2 + γ 4 ( 4 i 3 β 3 ϑ 3 + 5 i β ϑ + 4 β τ ) + γ 5 + 2 i γ 6 β ϑ , S 2 = γ 1 h γ 2 q + 2 γ 3 β 2 ξ 2 γ 4 ( 4 i 3 β 3 ξ 3 + 5 i β ξ + 4 β τ i ρ ) γ 5 2 i γ 6 β ξ ,
κ = γ a 2 + i β ,     ρ = 4 β κ + 4 i β 2 κ 2 + 8 β 3 3 κ 3 , τ = t ( a 2 μ + ω ) z ,     ϑ = i τ β z ,     ξ = ϑ + 1 / κ , g = 2 β 4 ϑ 4 3 10 β 2 ϑ 2 + 8 i β 2 ϑ τ , h = 2 β 4 ξ 4 3 10 β 2 ξ 2 + 8 i β 2 ξ τ + 2 ρ β ξ , p = 4 i β 5 ϑ 5 15 10 i β 3 ϑ 3 + 7 i β ϑ 4 8 β 3 ϑ 2 τ + 2 β τ , q = 4 i β 5 ξ 5 15 10 i β 3 ξ 3 + 7 i β ξ 4 8 β 3 ξ 2 τ + 2 β τ + 2 i ρ β 2 ξ 2 2 i β κ + 2 β 2 κ 2 12 i β 3 κ 3 16 β 4 κ 4 + 32 i β 5 5 κ 5 .
E [ 1 ] = E 0 ( 1 + i χ R 0 S 0 * γ a m 11 * ) ( m 11 * m 11 ) μ / γ ,
E [ 2 ] = E 0 { 1 + i χ [ R 0 ( S 0 * m 22 * S 1 * m 21 * ) + R 1 ( S 1 * m 11 * S 0 * m 12 * ) ] γ a ( m 11 * m 22 * m 12 * m 21 * ) } ( m 11 * m 22 * m 12 * m 21 * m 11 m 22 m 12 m 21 ) μ / γ ,
E [ 3 ] = E 0 ( 1 + i χ γ a [ R 0 , R 1 , R 2 ] [ m 11 * , m 21 * , m 31 * m 12 * , m 22 * , m 32 * m 13 * , m 23 * , m 33 * ] 1 [ S 0 * S 1 * S 2 * ] ) × ( ( | m 11 , m 12 , m 13 m 21 , m 22 , m 23 m 31 , m 32 , m 33 | * / | m 11 , m 12 , m 13 m 21 , m 22 , m 23 m 31 , m 32 , m 33 | ) μ / γ ,
m 11 = λ 0 | R 0 | 2 + λ 0 * | S 0 | 2 , m 12 = λ 0 R 0 * R 1 + λ 0 * S 0 * S 1 + χ | R 0 | 2 2 λ 0 m 11 , m 13 = λ 0 R 0 * R 2 + λ 0 * S 0 * S 2 + χ R 0 * 2 λ 0 ( R 1 χ R 0 4 λ 0 ) m 12 , m 21 = λ 0 R 0 R 1 * + λ 0 * S 0 S 1 * χ | S 0 | 2 2 λ 0 * m 11 , m 22 = λ 0 | R 1 | 2 + λ 0 * | S 1 | 2 + χ 2 ( R 0 R 1 * λ 0 S 0 * S 1 λ 0 * ) m 12 m 21 , m 23 = λ 0 R 1 * R 2 + λ 0 * S 1 * S 2 χ S 0 * S 2 2 λ 0 * + χ R 1 * 2 λ 0 ( R 1 χ R 0 4 λ 0 ) m 22 m 13 , m 31 = λ 0 R 0 R 2 * + λ 0 * S 0 S 2 * χ S 0 2 λ 0 * ( S 1 * + χ S 0 * 4 λ 0 * ) m 21 , m 32 = λ 0 R 1 R 2 * + λ 0 * S 1 S 2 * + χ R 0 R 2 * 2 λ 0 χ S 1 2 λ 0 * ( S 1 * + χ S 0 * 4 λ 0 * ) m 22 m 31 , m 33 = λ 0 | R 2 | 2 + λ 0 * | S 2 | 2 + χ R 2 * 2 λ 0 ( R 1 χ R 0 4 λ 0 ) χ S 2 2 λ 0 * ( S 1 * + χ S 0 * 4 λ 0 * ) m 23 m 32 .

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