Abstract

We investigate dark rogue wave dynamics in normally dispersive birefringent optical fibers, based on the exact rational solutions of the coupled nonlinear Schrödinger equations. Analytical solutions are derived up to the second order via a nonrecursive Darboux transformation method. Vector dark “three-sister” rogue waves as well as their existence conditions are demonstrated. The robustness against small perturbations is numerically confirmed in spite of the onset of modulational instability, offering the possibility to observe such extreme events in normal optical fibers with random birefringence, or in other Manakov-type vector nonlinear media.

© 2014 Optical Society of America

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References

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  5. R. Driben and I. Babushkin, “Accelerated rogue waves generated by soliton fusion at the advanced stage of supercontinuum formation in photonic-crystal fibers,” Opt. Lett. 37, 5157–5159 (2012).
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2014 (5)

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]

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]

S. Chen, Ph. Grelu, and J. M. Soto-Crespo, “Dark- and bright-rogue-wave solutions for media with long-wave–short-wave resonance,” Phys. Rev. E 89, 011201(R) (2014).
[Crossref]

S. Chen, J. M. Soto-Crespo, and Ph. Grelu, “Coexisting rogue waves within the (2+1)-component long-wave–short-wave resonance,” Phys. Rev. E 90, 033203 (2014).
[Crossref]

L. Ling, B. Guo, and L.-C. Zhao, “High-order rogue waves in vector nonlinear Schrödinger equations,” Phys. Rev. E 89, 041201(R) (2014).
[Crossref]

2013 (6)

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Classifying the hierarchy of nonlinear-Schrödinger-equation rogue-wave solutions,” Phys. Rev. E 88, 013207 (2013).
[Crossref]

A. Chabchoub and N. Akhmediev, “Observation of rogue wave triplets in water waves,” Phys. Lett. A 377, 2590–2593 (2013).
[Crossref]

A. Degasperis and S. Lombardo, “Rational solitons of wave resonant-interaction models,” Phys. Rev. E 88, 052914 (2013).
[Crossref]

F. Baronio, M. Conforti, A. Degasperis, and S. Lombardo, “Rogue waves emerging from the resonant interaction of three waves,” Phys. Rev. Lett. 111, 114101 (2013).
[Crossref] [PubMed]

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]

2012 (6)

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]

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).

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]

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Triangular rogue wave cascades,” Phys. Rev. E 86, 056602 (2012).
[Crossref]

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

R. Driben and I. Babushkin, “Accelerated rogue waves generated by soliton fusion at the advanced stage of supercontinuum formation in photonic-crystal fibers,” Opt. Lett. 37, 5157–5159 (2012).
[Crossref] [PubMed]

2011 (4)

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

H. Bailung, S. K. Sharma, and Y. Nakamura, “Observation of Peregrine solitons in a multicomponent plasma with negative ions,” Phys. Rev. Lett. 107, 255005 (2011).
[Crossref]

A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, “Rogue wave observation in a water wave tank,” Phys. Rev. Lett. 106, 204502 (2011).
[Crossref] [PubMed]

2010 (1)

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

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

2008 (1)

2007 (1)

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

1998 (2)

D. Mihalache, D. Mazilu, and L. Torner, “Stability of walking vector solitons,” Phys. Rev. Lett. 81, 4353–4356 (1998).
[Crossref]

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: Physics and applications,” Phys. Rep. 298, 81–197 (1998).
[Crossref]

1997 (3)

A. P. Sheppard and Yu. S. Kivshar, “Polarized dark solitons in isotropic Kerr media,” Phys. Rev. E 55, 4773–4782 (1997).
[Crossref]

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15, 1735–1746 (1997).
[Crossref]

Z. Chen, M. Segev, T. Coskun, D. N. Christodoulides, and Yu. S. Kivshar, “Coupled photorefractive spatial-soliton pairs,” J. Opt. Soc. Am. B 14, 3066–3077 (1997).
[Crossref]

1996 (2)

J. U. Kang, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, “Observation of Manakov spatial solitons in AlGaAs planar waveguides,” Phys. Rev. Lett. 76, 3699–3702 (1996).
[Crossref] [PubMed]

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
[Crossref]

1993 (1)

1991 (1)

1989 (1)

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
[Crossref]

1987 (1)

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
[Crossref]

1983 (1)

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

Aitchison, J. S.

J. U. Kang, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, “Observation of Manakov spatial solitons in AlGaAs planar waveguides,” Phys. Rev. Lett. 76, 3699–3702 (1996).
[Crossref] [PubMed]

Akhmediev, N.

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Classifying the hierarchy of nonlinear-Schrödinger-equation rogue-wave solutions,” Phys. Rev. E 88, 013207 (2013).
[Crossref]

A. Chabchoub and N. Akhmediev, “Observation of rogue wave triplets in water waves,” Phys. Lett. A 377, 2590–2593 (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] [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).

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Triangular rogue wave cascades,” Phys. Rev. E 86, 056602 (2012).
[Crossref]

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

A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, “Rogue wave observation in a water wave tank,” Phys. Rev. Lett. 106, 204502 (2011).
[Crossref] [PubMed]

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]

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,” Nature Phys. 6, 790–795 (2010).
[Crossref]

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

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]

J. U. Kang, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, “Observation of Manakov spatial solitons in AlGaAs planar waveguides,” Phys. Rev. Lett. 76, 3699–3702 (1996).
[Crossref] [PubMed]

Ankiewicz, A.

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Classifying the hierarchy of nonlinear-Schrödinger-equation rogue-wave solutions,” Phys. Rev. E 88, 013207 (2013).
[Crossref]

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Triangular rogue wave cascades,” Phys. Rev. E 86, 056602 (2012).
[Crossref]

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

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]

Arecchi, F. T.

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]

Babushkin, I.

Bailung, H.

H. Bailung, S. K. Sharma, and Y. Nakamura, “Observation of Peregrine solitons in a multicomponent plasma with negative ions,” Phys. Rev. Lett. 107, 255005 (2011).
[Crossref]

Baronio, F.

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]

F. Baronio, M. Conforti, A. Degasperis, and S. Lombardo, “Rogue waves emerging from the resonant interaction of three waves,” Phys. Rev. Lett. 111, 114101 (2013).
[Crossref] [PubMed]

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]

Birkholz, S.

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]

Bortolozzo, U.

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]

Brée, C.

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]

Chabchoub, A.

A. Chabchoub and N. Akhmediev, “Observation of rogue wave triplets in water waves,” Phys. Lett. A 377, 2590–2593 (2013).
[Crossref]

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).

A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, “Rogue wave observation in a water wave tank,” Phys. Rev. Lett. 106, 204502 (2011).
[Crossref] [PubMed]

Chen, H. H.

Chen, S.

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]

S. Chen, Ph. Grelu, and J. M. Soto-Crespo, “Dark- and bright-rogue-wave solutions for media with long-wave–short-wave resonance,” Phys. Rev. E 89, 011201(R) (2014).
[Crossref]

S. Chen, J. M. Soto-Crespo, and Ph. Grelu, “Coexisting rogue waves within the (2+1)-component long-wave–short-wave resonance,” Phys. Rev. E 90, 033203 (2014).
[Crossref]

Chen, Z.

Christodoulides, D. N.

Conforti, M.

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]

F. Baronio, M. Conforti, A. Degasperis, and S. Lombardo, “Rogue waves emerging from the resonant interaction of three waves,” Phys. Rev. Lett. 111, 114101 (2013).
[Crossref] [PubMed]

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]

Coskun, T.

Degasperis, A.

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]

A. Degasperis and S. Lombardo, “Rational solitons of wave resonant-interaction models,” Phys. Rev. E 88, 052914 (2013).
[Crossref]

F. Baronio, M. Conforti, A. Degasperis, and S. Lombardo, “Rogue waves emerging from the resonant interaction of three waves,” Phys. Rev. Lett. 111, 114101 (2013).
[Crossref] [PubMed]

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.

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]

Dias, F.

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,” Nature Phys. 6, 790–795 (2010).
[Crossref]

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

Driben, R.

Dudley, J. M.

Eggleton, B. J.

Fatome, J.

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

Finot, C.

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

Genty, G.

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,” Nature Phys. 6, 790–795 (2010).
[Crossref]

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

J. M. Dudley, G. Genty, and B. J. Eggleton, “Harnessing and control of optical rogue waves in supercontinuum generation,” Opt. Express 16, 3644–3651 (2008).
[Crossref] [PubMed]

Grelu, Ph.

S. Chen, Ph. Grelu, and J. M. Soto-Crespo, “Dark- and bright-rogue-wave solutions for media with long-wave–short-wave resonance,” Phys. Rev. E 89, 011201(R) (2014).
[Crossref]

S. Chen, J. M. Soto-Crespo, and Ph. Grelu, “Coexisting rogue waves within the (2+1)-component long-wave–short-wave resonance,” Phys. Rev. E 90, 033203 (2014).
[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]

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]

Guo, B.

L. Ling, B. Guo, and L.-C. Zhao, “High-order rogue waves in vector nonlinear Schrödinger equations,” Phys. Rev. E 89, 041201(R) (2014).
[Crossref]

Hoffmann, N.

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).

Hoffmann, N. P.

A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, “Rogue wave observation in a water wave tank,” Phys. Rev. Lett. 106, 204502 (2011).
[Crossref] [PubMed]

Holden, T.

J. F. Wolff and T. Holden, Lake Superior Shipwrecks, (2nd ed.) (Lake Superior Port Cities, 1990).

Jalali, B.

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

Kang, J. U.

J. U. Kang, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, “Observation of Manakov spatial solitons in AlGaAs planar waveguides,” Phys. Rev. Lett. 76, 3699–3702 (1996).
[Crossref] [PubMed]

Kedziora, D. J.

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Classifying the hierarchy of nonlinear-Schrödinger-equation rogue-wave solutions,” Phys. Rev. E 88, 013207 (2013).
[Crossref]

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Triangular rogue wave cascades,” Phys. Rev. E 86, 056602 (2012).
[Crossref]

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

Kibler, B.

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

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

Kivshar, Yu. S.

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: Physics and applications,” Phys. Rep. 298, 81–197 (1998).
[Crossref]

Z. Chen, M. Segev, T. Coskun, D. N. Christodoulides, and Yu. S. Kivshar, “Coupled photorefractive spatial-soliton pairs,” J. Opt. Soc. Am. B 14, 3066–3077 (1997).
[Crossref]

A. P. Sheppard and Yu. S. Kivshar, “Polarized dark solitons in isotropic Kerr media,” Phys. Rev. E 55, 4773–4782 (1997).
[Crossref]

Yu. S. Kivshar and S. K. Turitsyn, “Vector dark solitons,” Opt. Lett. 18, 337–339 (1993).
[Crossref] [PubMed]

Koonath, P.

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

Lecaplain, C.

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]

Ling, L.

L. Ling, B. Guo, and L.-C. Zhao, “High-order rogue waves in vector nonlinear Schrödinger equations,” Phys. Rev. E 89, 041201(R) (2014).
[Crossref]

Lombardo, S.

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]

A. Degasperis and S. Lombardo, “Rational solitons of wave resonant-interaction models,” Phys. Rev. E 88, 052914 (2013).
[Crossref]

F. Baronio, M. Conforti, A. Degasperis, and S. Lombardo, “Rogue waves emerging from the resonant interaction of three waves,” Phys. Rev. Lett. 111, 114101 (2013).
[Crossref] [PubMed]

Luther-Davies, B.

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: Physics and applications,” Phys. Rep. 298, 81–197 (1998).
[Crossref]

Manakov, S. V.

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP38, 248–253 (1974) [Zh. Eksp. Teor. Fiz. 65, 505–516 (1973)].

Marcuse, D.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15, 1735–1746 (1997).
[Crossref]

Matveev, V. B.

V. B. Matveev and M. A. Salle, Darboux Transformation and Solitons (Springer, 1991).
[Crossref]

Mazilu, D.

D. Mihalache, D. Mazilu, and L. Torner, “Stability of walking vector solitons,” Phys. Rev. Lett. 81, 4353–4356 (1998).
[Crossref]

Menyuk, C. R.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15, 1735–1746 (1997).
[Crossref]

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
[Crossref]

P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt. Lett. 16, 1231–1233 (1991).
[Crossref] [PubMed]

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
[Crossref]

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
[Crossref]

Mihalache, D.

D. Mihalache, D. Mazilu, and L. Torner, “Stability of walking vector solitons,” Phys. Rev. Lett. 81, 4353–4356 (1998).
[Crossref]

Millot, G.

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

Montina, A.

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]

Nakamura, Y.

H. Bailung, S. K. Sharma, and Y. Nakamura, “Observation of Peregrine solitons in a multicomponent plasma with negative ions,” Phys. Rev. Lett. 107, 255005 (2011).
[Crossref]

Nibbering, E. T. J.

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]

Ohta, Y.

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

Onorato, M.

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]

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]

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).

Peregrine, D. H.

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

Residori, S.

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]

Ropers, C.

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

Salle, M. A.

V. B. Matveev and M. A. Salle, Darboux Transformation and Solitons (Springer, 1991).
[Crossref]

Segev, M.

Sharma, S. K.

H. Bailung, S. K. Sharma, and Y. Nakamura, “Observation of Peregrine solitons in a multicomponent plasma with negative ions,” Phys. Rev. Lett. 107, 255005 (2011).
[Crossref]

Sheppard, A. P.

A. P. Sheppard and Yu. S. Kivshar, “Polarized dark solitons in isotropic Kerr media,” Phys. Rev. E 55, 4773–4782 (1997).
[Crossref]

Skupin, S.

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]

Solli, D. R.

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

Song, L.-Y.

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]

Soto-Crespo, J. M.

S. Chen, Ph. Grelu, and J. M. Soto-Crespo, “Dark- and bright-rogue-wave solutions for media with long-wave–short-wave resonance,” Phys. Rev. E 89, 011201(R) (2014).
[Crossref]

S. Chen, J. M. Soto-Crespo, and Ph. Grelu, “Coexisting rogue waves within the (2+1)-component long-wave–short-wave resonance,” Phys. Rev. E 90, 033203 (2014).
[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]

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]

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]

Stegeman, G. I.

J. U. Kang, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, “Observation of Manakov spatial solitons in AlGaAs planar waveguides,” Phys. Rev. Lett. 76, 3699–3702 (1996).
[Crossref] [PubMed]

Steinmeyer, G.

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]

Taki, M.

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

Torner, L.

D. Mihalache, D. Mazilu, and L. Torner, “Stability of walking vector solitons,” Phys. Rev. Lett. 81, 4353–4356 (1998).
[Crossref]

Turitsyn, S. K.

Wabnitz, S.

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]

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]

Wai, P. K. A.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15, 1735–1746 (1997).
[Crossref]

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
[Crossref]

P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Stability of solitons in randomly varying birefringent fibers,” Opt. Lett. 16, 1231–1233 (1991).
[Crossref] [PubMed]

Wolff, J. F.

J. F. Wolff and T. Holden, Lake Superior Shipwrecks, (2nd ed.) (Lake Superior Port Cities, 1990).

Yang, J.

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

Zhao, L.-C.

L. Ling, B. Guo, and L.-C. Zhao, “High-order rogue waves in vector nonlinear Schrödinger equations,” Phys. Rev. E 89, 041201(R) (2014).
[Crossref]

IEEE J. Quantum Electron. (2)

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
[Crossref]

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
[Crossref]

J. Aust. Math. Soc. Series B, Appl. Math. (1)

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

J. Lightwave Technol. (2)

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
[Crossref]

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 15, 1735–1746 (1997).
[Crossref]

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

Nature (London) (1)

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

Nature Phys. (1)

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

Opt. Express (2)

Opt. Lett. (3)

Phys. Lett. A (4)

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]

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

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

A. Chabchoub and N. Akhmediev, “Observation of rogue wave triplets in water waves,” Phys. Lett. A 377, 2590–2593 (2013).
[Crossref]

Phys. Rep. (2)

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: Physics and applications,” Phys. Rep. 298, 81–197 (1998).
[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]

Phys. Rev. E (9)

A. P. Sheppard and Yu. S. Kivshar, “Polarized dark solitons in isotropic Kerr media,” Phys. Rev. E 55, 4773–4782 (1997).
[Crossref]

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Triangular rogue wave cascades,” Phys. Rev. E 86, 056602 (2012).
[Crossref]

A. Degasperis and S. Lombardo, “Rational solitons of wave resonant-interaction models,” Phys. Rev. E 88, 052914 (2013).
[Crossref]

S. Chen, Ph. Grelu, and J. M. Soto-Crespo, “Dark- and bright-rogue-wave solutions for media with long-wave–short-wave resonance,” Phys. Rev. E 89, 011201(R) (2014).
[Crossref]

S. Chen, J. M. Soto-Crespo, and Ph. Grelu, “Coexisting rogue waves within the (2+1)-component long-wave–short-wave resonance,” Phys. Rev. E 90, 033203 (2014).
[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]

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]

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Classifying the hierarchy of nonlinear-Schrödinger-equation rogue-wave solutions,” Phys. Rev. E 88, 013207 (2013).
[Crossref]

L. Ling, B. Guo, and L.-C. Zhao, “High-order rogue waves in vector nonlinear Schrödinger equations,” Phys. Rev. E 89, 041201(R) (2014).
[Crossref]

Phys. Rev. Lett. (9)

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]

H. Bailung, S. K. Sharma, and Y. Nakamura, “Observation of Peregrine solitons in a multicomponent plasma with negative ions,” Phys. Rev. Lett. 107, 255005 (2011).
[Crossref]

D. Mihalache, D. Mazilu, and L. Torner, “Stability of walking vector solitons,” Phys. Rev. Lett. 81, 4353–4356 (1998).
[Crossref]

A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, “Rogue wave observation in a water wave tank,” Phys. Rev. Lett. 106, 204502 (2011).
[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]

F. Baronio, M. Conforti, A. Degasperis, and S. Lombardo, “Rogue waves emerging from the resonant interaction of three waves,” Phys. Rev. Lett. 111, 114101 (2013).
[Crossref] [PubMed]

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. U. Kang, G. I. Stegeman, J. S. Aitchison, and N. Akhmediev, “Observation of Manakov spatial solitons in AlGaAs planar waveguides,” Phys. Rev. Lett. 76, 3699–3702 (1996).
[Crossref] [PubMed]

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]

Phys. Rev. X (1)

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).

Proc. R. Soc. A (1)

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

Other (3)

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP38, 248–253 (1974) [Zh. Eksp. Teor. Fiz. 65, 505–516 (1973)].

J. F. Wolff and T. Holden, Lake Superior Shipwrecks, (2nd ed.) (Lake Superior Port Cities, 1990).

V. B. Matveev and M. A. Salle, Darboux Transformation and Solitons (Springer, 1991).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1 (a),(b) Dark-bright Peregrine structures and (c), (d) dark-bright rogue wave triplets formed at a1 = 2, a2 = 1, δ = 1, and κ = 0. The structural parameters intended for (c), (d) are given by γ1 = 6i, γ2 = 1, and γ3 = γ4 = 0.
Fig. 2
Fig. 2 Fundamental (first-order) rogue wave states formed at (a), (b) δ = 1.3 and (c), (d) δ = 3. The other initial parameters are given by a1 = a2 = a = 1 and κ = −1/2. The insets in the lower right corner show the contour distributions.
Fig. 3
Fig. 3 Second-order rogue wave states formed at δ = 3 with two different sets of structural parameters: (a), (b) dark composite structures (γ2 = 1, γ1 = γ3 = γ4 = 0); (c), (d) dark triplet (or three-sister) structures (γ1 = −3i, γ2 = 1, γ3 = γ4 = 0). The other initial plane-wave parameters are the same as in Fig. 2.
Fig. 4
Fig. 4 Numerical simulations of the vector dark three-sister rogue waves [using otherwise identical parameters as in Figs. 3(c) and 3(d)]: (a), (b) the unperturbed situation; (c), (d) perturbed by the initial white noise. Provided in (e) and (f) are the numerical results of the typical dark-bright triplets perturbed by the white noise of the same level.

Equations (23)

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i u ξ + 𝒟 2 u τ τ + ( | u | 2 + | v | 2 ) u = 0 ,
i v ξ + 𝒟 2 v τ τ + ( | v | 2 + | u | 2 ) v = 0 ,
R τ = ( λ E + Q ) R , R ξ = [ 3 2 λ 2 E 3 2 λ Q + i 2 σ 3 ( Q 2 Q τ ) ] R ,
Q = [ 0 u v u * 0 0 v * 0 0 ] .
u [ n ] = u 0 ( 1 + 3 i | u 0 | Y 1 M 1 Y 2 ) ,
v [ n ] = v 0 ( 1 + 3 i | v 0 | Y 1 M 1 Y 3 ) ,
u 0 ( ξ , τ ) = a 1 exp [ i ( ω 1 τ k 1 ξ ) ] ,
v 0 ( ξ , τ ) = a 2 exp [ i ( ω 2 τ k 2 ξ ) ] ,
[ Y 1 Y 2 Y 3 ] = [ Φ ( 0 ) , Φ ( 1 ) , Φ ( 2 ) , , Φ ( n 1 ) ] ,
Φ ( λ ) = Φ ( 0 ) + Φ ( 1 ) ε 2 + Φ ( 2 ) ε 4 + + Φ ( n 1 ) ε 2 ( n 1 ) + 𝒪 ( ε 2 n ) .
Φ σ 3 Φ λ λ * = i j n M i j ε * 2 ( i 1 ) ε 2 ( j 1 ) + 𝒪 ( | ε | 4 n ) .
| δ | < δ m = { 3 2 ( A 2 B 2 ) 1 / 3 [ ( A B ) 1 / 3 + ( A + B ) 1 / 3 ] + A } 1 / 2 ,
u [ 1 ] = u 0 [ 1 + 3 ( λ 0 λ 0 * ) θ 1 * ϑ / α 1 * | ϑ | 2 a 1 2 | θ 1 / α 1 | 2 a 2 2 | θ 2 / α 2 | 2 ] ,
v [ 1 ] = v 0 [ 1 + 3 ( λ 0 λ 0 * ) θ 2 * ϑ / α 2 * | ϑ | 2 a 1 2 | θ 1 / α 1 | 2 a 2 2 | θ 2 / α 2 | 2 ] ,
u [ 2 ] = u 0 { 1 + 3 i ( λ 0 λ 0 * ) [ R 1 * ( R 0 m 22 S 0 m 21 ) + S 1 * ( S 0 m 11 R 0 m 12 ) ] a 1 ( m 11 m 22 m 12 m 21 ) } ,
v [ 2 ] = v 0 { 1 + 3 i ( λ 0 λ 0 * ) [ R 2 * ( R 0 m 22 S 0 m 21 ) + S 2 * ( S 0 m 11 R 0 m 12 ) ] a 2 ( m 11 m 22 m 12 m 21 ) } ,
R 0 = 2 γ 1 2 i γ 2 ϕ ϑ , R j = 2 a j ( i γ 1 + γ 2 ϕ θ j ) / α j , S 0 = γ 1 p i γ 2 ϕ ( q ϑ 2 σ ξ ) + 2 γ 3 2 i γ 4 ϕ ϑ , S j = a j α j [ γ 1 ( i p 2 ϕ 2 θ j + 2 i σ α j ) + 2 i γ 3 + 2 γ 4 ϕ θ j ] + a j γ 2 ϕ α j [ q θ j 2 σ ( ξ + ϑ α j 2 i α j 2 ) + 2 ϕ 2 ( i α j ϑ 2 + 3 θ j ) 3 α j 2 ] , m 11 = | R 0 | 2 | R 1 | 2 | R 2 | 2 , m 12 = R 0 * S 0 R 1 * S 1 R 2 * S 2 m 11 m 21 * , m 22 = | S 0 | 2 | S 1 | 2 | S 2 | 2 m 12 m 21 .
λ 0 = 1 6 κ B + g 6 δ ± i 6 | δ | g 2 + c + 2 e / g ,
μ 0 = κ 3 + 1 6 δ [ ( B + g i δ | δ | g 2 + c + 2 e / g ) 2 + 3 δ 2 ( δ 2 4 A ) ] 1 / 2 ,
λ 0 = 1 6 κ + i 4 η i ( 4 A δ 2 ) 12 η , μ 0 = 1 3 κ + i 4 η + i ( 4 A δ 2 ) 12 η ,
η = ± [ 2 A ( A + 2 δ 2 ) 2 A δ 2 ] 1 / 2 , ( for | δ | < δ m = 2 A ) .
u [ 1 ] = u 0 { 1 + 8 i δ 2 ξ 2 i [ δ 2 ξ + δ ( 2 τ κ ξ ) 2 i ] ( 1 + 1 + 2 δ 2 / A ) δ 2 [ ( 2 τ κ ξ ) 2 + η 2 ξ 2 + 4 / η 2 ] } ,
v [ 1 ] = v 0 { 1 + 8 i δ 2 ξ 2 i [ δ 2 ξ δ ( 2 τ κ ξ ) 2 i ] ( 1 + 1 + 2 δ 2 / A ) δ 2 [ ( 2 τ κ ξ ) 2 + η 2 ξ 2 + 4 / η 2 ] } ,

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