Abstract

The (2+1)-dimensional coupled nonlinear Schrödinger equation with distributed coefficients in a graded-index grating waveguide is investigated, and an exact two-breather solution for certain functional relations is obtained. From this solution, the superposed Kuznetsov-Ma (KM) solitons can be constructed. The explicit functions that describe the evolution of the peak, width, center, and phase are found exactly, from which one knows that diffraction and chirp factors play important roles in the evolutional characteristics, such as phase, center and widths, while the gain/loss parameter only affects the evolution of their peaks. Moreover, we can change the propagation type of the superposed KM solitons by adjusting the relation between the maximum effective propagation distance Zm and the periodic locations Zij based on the maximum amplitude of the superposed KM solitons. The controllability for the type of excitation, such as partial excitation, maintenance, and postponement of the superposed KM solitons, is exhibited.

© 2013 Optical Society of America

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    [CrossRef]

2013

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]

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]

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15, 060201 (2013).
[CrossRef]

S. L. Xu, N. Z. Petrović, and M. R. Belić, “Vortex solitons in the (2+1)-dimensional nonlinear Schrödinger equation with variable diffraction and nonlinearity coefficients,” Phys. Scr. 87, 045401 (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]

S. L. Xu, M. R. Belić, and W. P. Zhong, “Three-dimensional spatiotemporal vector solitary waves in coupled nonlinear Schrödinger equations with variable coefficients,” J. Opt. Soc. Am. B 30, 113–122 (2013).
[CrossRef]

W. P. Zhong, M. R. Belić, and T. W. Huang, “Solitary waves in the nonlinear Schrödinger equation with spatially modulated Bessel nonlinearity,” J. Opt. Soc. Am. B 30, 1276–1283 (2013).
[CrossRef]

2012

W. P. Zhong, M. R. Belić, and T. W. Huang, “Two-dimensional accessible solitons in PT-symmetric potentials,” Nonlinear Dyn. 70, 2027–2034 (2012).
[CrossRef]

Y. J. He and D. Mihalache, “Soliton dynamics induced by periodic spatially inhomogeneous losses in optical media described by the complex Ginzburg-Landau model,” J. Opt. Soc. Am. B 29, 2554–2558 (2012).
[CrossRef]

L. C. Zhao and J. Liu, “Localized nonlinear waves in a two-mode nonlinear fiber,” J. Opt. Soc. Am. B 29, 3119–3127 (2012).
[CrossRef]

C. Q. Dai, Y. Y. Wang, Q. Tian, and J. F. Zhang, “The management and containment of self-similar rogue waves in the inhomogeneous nonlinear Schrödinger equation,” Ann. Phys. 327, 512–521 (2012).
[CrossRef]

C. N. Kumar, R. Gupta, A. Goyal, S. Loomba, T. S. Raju, and P. K. Panigrahi, “Controlled giant rogue waves in nonlinear fiber optics,” Phys. Rev. A 86, 025802 (2012).
[CrossRef]

C. Q. Dai, G. Q. Zhou, and J. F. Zhang, “Controllable optical rogue waves in the femtosecond regime,” Phys. Rev. E 85, 016603 (2012).
[CrossRef]

C. Q. Dai, Q. Tian, and S. Q. Zhu, “Controllable behaviours of rogue wave triplets in the nonautonomous nonlinear and dispersive system,” J. Phys. B 45, 085401 (2012).
[CrossRef]

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits,” Phys. Rev. E 85, 066601 (2012).
[CrossRef]

B. Kibler, J. Fatome, C. Finot, G. Millot, G. Genty, B. Wetzel, N. Akhmediev, F. Dias, and J. Dudley, “Observation of Kuznetsov-Ma soliton dynamics in optical fibre,” Sci. Rep. 2, 463 (2012).
[CrossRef]

2011

2010

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

C. Q. Dai, S. Q. Zhu, L. L. Wang, and J. F. Zhang, “Exact spatial similaritons for the generalized (2+1)-dimensional nonlinear Schrödinger equation with distributed coefficients,” Europhys. Lett. 92, 24005 (2010).
[CrossRef]

Z. Y. Yang, L. C. Zhao, T. Zhang, Y. H. Li, and R. H. Yue, “Snakelike nonautonomous solitons in a graded-index grating waveguid,” Phys. Rev. A 81, 043826 (2010).
[CrossRef]

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

2009

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

2007

A. Alexandrescu, G. D. Montesinos, and V. M. Pérez-García, “Stabilization of high-order solutions of the cubic nonlinear Schrödinger equation,” Phys. Rev. E 75, 046609 (2007).
[CrossRef]

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

U. Bortolozzo, A. Montina, F. T. Arecchi, J. P. Huignard, and S. Residori, “Spatiotemporal pulses in a liquid crystal optical oscillator,” Phys. Rev. Lett. 99, 023901 (2007).
[CrossRef]

2006

M. Centurion, M. A. Porter, P. G. Kevrekidis, and D. Psaltis, “Nonlinearity management in optics: experiment, theory, and simulation,” Phys. Rev. Lett. 97, 033903 (2006).
[CrossRef]

2004

R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, “Exact multisoliton solutions of the higher-order nonlinear Schrödinger equation with variable coefficients,” Phys. Rev. E 70, 066603 (2004).
[CrossRef]

2003

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef]

2002

2000

1999

K. B. Dysthe and K. Trulsen, “Note on breather type solutions of the NLS as models for freak-waves,” Phys. Scr. T82, 48–52 (1999).
[CrossRef]

1996

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

1994

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620–1622 (1994).
[CrossRef]

X. D. Cao and D. D. Meyerhofer, “Soliton collisions in optical birefringent fibers,” J. Opt. Soc. Am. B 11, 380–385 (1994).
[CrossRef]

1985

1979

J. C. Campbell, “Tapered waveguides for guided wave optics,” Appl. Opt. 18, 900–902 (1979).
[CrossRef]

Y. C. Ma, “The perturbed plane-wave solution of the cubic Schrödinger equation,” Stud. Appl. Math. 60, 43–58 (1979).

1977

E. A. Kuznetsov, “Solitons in a parametrically unstable plasma,” Dokl. Akad. Nauk SSSR 236, 575–577 (1977).

Abebe, M.

Akhmediev, N.

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]

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]

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15, 060201 (2013).
[CrossRef]

B. Kibler, J. Fatome, C. Finot, G. Millot, G. Genty, B. Wetzel, N. Akhmediev, F. Dias, and J. Dudley, “Observation of Kuznetsov-Ma soliton dynamics in optical fibre,” Sci. Rep. 2, 463 (2012).
[CrossRef]

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits,” Phys. Rev. E 85, 066601 (2012).
[CrossRef]

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

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]

Alexandrescu, A.

A. Alexandrescu, G. D. Montesinos, and V. M. Pérez-García, “Stabilization of high-order solutions of the cubic nonlinear Schrödinger equation,” Phys. Rev. E 75, 046609 (2007).
[CrossRef]

Ankiewicz, A.

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]

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, “Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits,” Phys. Rev. E 85, 066601 (2012).
[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]

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]

U. Bortolozzo, A. Montina, F. T. Arecchi, J. P. Huignard, and S. Residori, “Spatiotemporal pulses in a liquid crystal optical oscillator,” Phys. Rev. Lett. 99, 023901 (2007).
[CrossRef]

Belic, M. R.

S. L. Xu, N. Z. Petrović, and M. R. Belić, “Vortex solitons in the (2+1)-dimensional nonlinear Schrödinger equation with variable diffraction and nonlinearity coefficients,” Phys. Scr. 87, 045401 (2013).
[CrossRef]

S. L. Xu, M. R. Belić, and W. P. Zhong, “Three-dimensional spatiotemporal vector solitary waves in coupled nonlinear Schrödinger equations with variable coefficients,” J. Opt. Soc. Am. B 30, 113–122 (2013).
[CrossRef]

W. P. Zhong, M. R. Belić, and T. W. Huang, “Solitary waves in the nonlinear Schrödinger equation with spatially modulated Bessel nonlinearity,” J. Opt. Soc. Am. B 30, 1276–1283 (2013).
[CrossRef]

W. P. Zhong, M. R. Belić, and T. W. Huang, “Two-dimensional accessible solitons in PT-symmetric potentials,” Nonlinear Dyn. 70, 2027–2034 (2012).
[CrossRef]

Berge, L.

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]

U. Bortolozzo, A. Montina, F. T. Arecchi, J. P. Huignard, and S. Residori, “Spatiotemporal pulses in a liquid crystal optical oscillator,” Phys. Rev. Lett. 99, 023901 (2007).
[CrossRef]

Burns, W. K.

Campbell, J. C.

Cao, X. D.

Centurion, M.

M. Centurion, M. A. Porter, P. G. Kevrekidis, and D. Psaltis, “Nonlinearity management in optics: experiment, theory, and simulation,” Phys. Rev. Lett. 97, 033903 (2006).
[CrossRef]

Chowdhury, M. A.

Christiansen, P. L.

Dai, C. Q.

C. Q. Dai, G. Q. Zhou, and J. F. Zhang, “Controllable optical rogue waves in the femtosecond regime,” Phys. Rev. E 85, 016603 (2012).
[CrossRef]

C. Q. Dai, Q. Tian, and S. Q. Zhu, “Controllable behaviours of rogue wave triplets in the nonautonomous nonlinear and dispersive system,” J. Phys. B 45, 085401 (2012).
[CrossRef]

C. Q. Dai, Y. Y. Wang, Q. Tian, and J. F. Zhang, “The management and containment of self-similar rogue waves in the inhomogeneous nonlinear Schrödinger equation,” Ann. Phys. 327, 512–521 (2012).
[CrossRef]

C. Q. Dai, S. Q. Zhu, L. L. Wang, and J. F. Zhang, “Exact spatial similaritons for the generalized (2+1)-dimensional nonlinear Schrödinger equation with distributed coefficients,” Europhys. Lett. 92, 24005 (2010).
[CrossRef]

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

Dias, F.

B. Kibler, J. Fatome, C. Finot, G. Millot, G. Genty, B. Wetzel, N. Akhmediev, F. Dias, and J. Dudley, “Observation of Kuznetsov-Ma soliton dynamics in optical fibre,” Sci. Rep. 2, 463 (2012).
[CrossRef]

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

Dudley, J.

B. Kibler, J. Fatome, C. Finot, G. Millot, G. Genty, B. Wetzel, N. Akhmediev, F. Dias, and J. Dudley, “Observation of Kuznetsov-Ma soliton dynamics in optical fibre,” Sci. Rep. 2, 463 (2012).
[CrossRef]

Dudley, J. M.

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15, 060201 (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).
[CrossRef]

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]

Dysthe, K. B.

K. B. Dysthe and K. Trulsen, “Note on breather type solutions of the NLS as models for freak-waves,” Phys. Scr. T82, 48–52 (1999).
[CrossRef]

Eggleton, B. J.

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]

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620–1622 (1994).
[CrossRef]

Fatome, J.

B. Kibler, J. Fatome, C. Finot, G. Millot, G. Genty, B. Wetzel, N. Akhmediev, F. Dias, and J. Dudley, “Observation of Kuznetsov-Ma soliton dynamics in optical fibre,” Sci. Rep. 2, 463 (2012).
[CrossRef]

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

Finot, C.

B. Kibler, J. Fatome, C. Finot, G. Millot, G. Genty, B. Wetzel, N. Akhmediev, F. Dias, and J. Dudley, “Observation of Kuznetsov-Ma soliton dynamics in optical fibre,” Sci. Rep. 2, 463 (2012).
[CrossRef]

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

Gaididei, Y. B.

Genty, G.

B. Kibler, J. Fatome, C. Finot, G. Millot, G. Genty, B. Wetzel, N. Akhmediev, F. Dias, and J. Dudley, “Observation of Kuznetsov-Ma soliton dynamics in optical fibre,” Sci. Rep. 2, 463 (2012).
[CrossRef]

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

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]

Goyal, A.

C. N. Kumar, R. Gupta, A. Goyal, S. Loomba, T. S. Raju, and P. K. Panigrahi, “Controlled giant rogue waves in nonlinear fiber optics,” Phys. Rev. A 86, 025802 (2012).
[CrossRef]

Gupta, R.

C. N. Kumar, R. Gupta, A. Goyal, S. Loomba, T. S. Raju, and P. K. Panigrahi, “Controlled giant rogue waves in nonlinear fiber optics,” Phys. Rev. A 86, 025802 (2012).
[CrossRef]

Hao, R. Y.

R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, “Exact multisoliton solutions of the higher-order nonlinear Schrödinger equation with variable coefficients,” Phys. Rev. E 70, 066603 (2004).
[CrossRef]

Harvey, J. D.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef]

Haus, H. A.

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

He, Y. J.

Huang, T. W.

W. P. Zhong, M. R. Belić, and T. W. Huang, “Solitary waves in the nonlinear Schrödinger equation with spatially modulated Bessel nonlinearity,” J. Opt. Soc. Am. B 30, 1276–1283 (2013).
[CrossRef]

W. P. Zhong, M. R. Belić, and T. W. Huang, “Two-dimensional accessible solitons in PT-symmetric potentials,” Nonlinear Dyn. 70, 2027–2034 (2012).
[CrossRef]

Huignard, J. P.

U. Bortolozzo, A. Montina, F. T. Arecchi, J. P. Huignard, and S. Residori, “Spatiotemporal pulses in a liquid crystal optical oscillator,” Phys. Rev. Lett. 99, 023901 (2007).
[CrossRef]

Jalali, B.

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

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, “Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits,” Phys. Rev. E 85, 066601 (2012).
[CrossRef]

Kevrekidis, P. G.

M. Centurion, M. A. Porter, P. G. Kevrekidis, and D. Psaltis, “Nonlinearity management in optics: experiment, theory, and simulation,” Phys. Rev. Lett. 97, 033903 (2006).
[CrossRef]

Kibler, B.

B. Kibler, J. Fatome, C. Finot, G. Millot, G. Genty, B. Wetzel, N. Akhmediev, F. Dias, and J. Dudley, “Observation of Kuznetsov-Ma soliton dynamics in optical fibre,” Sci. Rep. 2, 463 (2012).
[CrossRef]

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

Koonath, P.

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

Krug, P. A.

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620–1622 (1994).
[CrossRef]

Kruglov, V. I.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef]

Kumar, C. N.

C. N. Kumar, R. Gupta, A. Goyal, S. Loomba, T. S. Raju, and P. K. Panigrahi, “Controlled giant rogue waves in nonlinear fiber optics,” Phys. Rev. A 86, 025802 (2012).
[CrossRef]

Kuznetsov, E. A.

E. A. Kuznetsov, “Solitons in a parametrically unstable plasma,” Dokl. Akad. Nauk SSSR 236, 575–577 (1977).

Li, L.

R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, “Exact multisoliton solutions of the higher-order nonlinear Schrödinger equation with variable coefficients,” Phys. Rev. E 70, 066603 (2004).
[CrossRef]

Li, Y. H.

Z. Y. Yang, L. C. Zhao, T. Zhang, Y. H. Li, and R. H. Yue, “Snakelike nonautonomous solitons in a graded-index grating waveguid,” Phys. Rev. A 81, 043826 (2010).
[CrossRef]

Li, Z. H.

R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, “Exact multisoliton solutions of the higher-order nonlinear Schrödinger equation with variable coefficients,” Phys. Rev. E 70, 066603 (2004).
[CrossRef]

Liu, J.

Loomba, S.

C. N. Kumar, R. Gupta, A. Goyal, S. Loomba, T. S. Raju, and P. K. Panigrahi, “Controlled giant rogue waves in nonlinear fiber optics,” Phys. Rev. A 86, 025802 (2012).
[CrossRef]

Ma, Y. C.

Y. C. Ma, “The perturbed plane-wave solution of the cubic Schrödinger equation,” Stud. Appl. Math. 60, 43–58 (1979).

Malomed, B. A.

Meyerhofer, D. D.

Mezentsev, V. K.

Mihalache, D.

Millot, G.

B. Kibler, J. Fatome, C. Finot, G. Millot, G. Genty, B. Wetzel, N. Akhmediev, F. Dias, and J. Dudley, “Observation of Kuznetsov-Ma soliton dynamics in optical fibre,” Sci. Rep. 2, 463 (2012).
[CrossRef]

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

Montesinos, G. D.

A. Alexandrescu, G. D. Montesinos, and V. M. Pérez-García, “Stabilization of high-order solutions of the cubic nonlinear Schrödinger equation,” Phys. Rev. E 75, 046609 (2007).
[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]

U. Bortolozzo, A. Montina, F. T. Arecchi, J. P. Huignard, and S. Residori, “Spatiotemporal pulses in a liquid crystal optical oscillator,” Phys. Rev. Lett. 99, 023901 (2007).
[CrossRef]

Onorato, M.

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]

Ouellette, F.

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620–1622 (1994).
[CrossRef]

Panigrahi, P. K.

C. N. Kumar, R. Gupta, A. Goyal, S. Loomba, T. S. Raju, and P. K. Panigrahi, “Controlled giant rogue waves in nonlinear fiber optics,” Phys. Rev. A 86, 025802 (2012).
[CrossRef]

Peacock, A. C.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef]

Pérez-García, V. M.

A. Alexandrescu, G. D. Montesinos, and V. M. Pérez-García, “Stabilization of high-order solutions of the cubic nonlinear Schrödinger equation,” Phys. Rev. E 75, 046609 (2007).
[CrossRef]

Petrovic, N. Z.

S. L. Xu, N. Z. Petrović, and M. R. Belić, “Vortex solitons in the (2+1)-dimensional nonlinear Schrödinger equation with variable diffraction and nonlinearity coefficients,” Phys. Scr. 87, 045401 (2013).
[CrossRef]

Poladian, L.

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620–1622 (1994).
[CrossRef]

Porter, M. A.

M. Centurion, M. A. Porter, P. G. Kevrekidis, and D. Psaltis, “Nonlinearity management in optics: experiment, theory, and simulation,” Phys. Rev. Lett. 97, 033903 (2006).
[CrossRef]

Psaltis, D.

M. Centurion, M. A. Porter, P. G. Kevrekidis, and D. Psaltis, “Nonlinearity management in optics: experiment, theory, and simulation,” Phys. Rev. Lett. 97, 033903 (2006).
[CrossRef]

Raju, T. S.

C. N. Kumar, R. Gupta, A. Goyal, S. Loomba, T. S. Raju, and P. K. Panigrahi, “Controlled giant rogue waves in nonlinear fiber optics,” Phys. Rev. A 86, 025802 (2012).
[CrossRef]

Rasmussen, J. J.

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]

U. Bortolozzo, A. Montina, F. T. Arecchi, J. P. Huignard, and S. Residori, “Spatiotemporal pulses in a liquid crystal optical oscillator,” Phys. Rev. Lett. 99, 023901 (2007).
[CrossRef]

Ropers, C.

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

Solli, D. R.

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15, 060201 (2013).
[CrossRef]

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

Soto-Crespo, J. M.

A. Ankiewicz, J. M. Soto-Crespo, 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]

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]

Sulem, C.

C. Sulem and P. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (Springer-Verlag, 2000).

Sulem, P.

C. Sulem and P. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (Springer-Verlag, 2000).

Tian, Q.

C. Q. Dai, Y. Y. Wang, Q. Tian, and J. F. Zhang, “The management and containment of self-similar rogue waves in the inhomogeneous nonlinear Schrödinger equation,” Ann. Phys. 327, 512–521 (2012).
[CrossRef]

C. Q. Dai, Q. Tian, and S. Q. Zhu, “Controllable behaviours of rogue wave triplets in the nonautonomous nonlinear and dispersive system,” J. Phys. B 45, 085401 (2012).
[CrossRef]

Towers, I.

Trulsen, K.

K. B. Dysthe and K. Trulsen, “Note on breather type solutions of the NLS as models for freak-waves,” Phys. Scr. T82, 48–52 (1999).
[CrossRef]

Turitsyn, S. K.

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15, 060201 (2013).
[CrossRef]

Villarruel, C. A.

Wang, L. L.

C. Q. Dai, S. Q. Zhu, L. L. Wang, and J. F. Zhang, “Exact spatial similaritons for the generalized (2+1)-dimensional nonlinear Schrödinger equation with distributed coefficients,” Europhys. Lett. 92, 24005 (2010).
[CrossRef]

Wang, Y. Y.

C. Q. Dai, Y. Y. Wang, Q. Tian, and J. F. Zhang, “The management and containment of self-similar rogue waves in the inhomogeneous nonlinear Schrödinger equation,” Ann. Phys. 327, 512–521 (2012).
[CrossRef]

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

Wetzel, B.

B. Kibler, J. Fatome, C. Finot, G. Millot, G. Genty, B. Wetzel, N. Akhmediev, F. Dias, and J. Dudley, “Observation of Kuznetsov-Ma soliton dynamics in optical fibre,” Sci. Rep. 2, 463 (2012).
[CrossRef]

Wong, W. S.

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

Xu, S. L.

S. L. Xu, N. Z. Petrović, and M. R. Belić, “Vortex solitons in the (2+1)-dimensional nonlinear Schrödinger equation with variable diffraction and nonlinearity coefficients,” Phys. Scr. 87, 045401 (2013).
[CrossRef]

S. L. Xu, M. R. Belić, and W. P. Zhong, “Three-dimensional spatiotemporal vector solitary waves in coupled nonlinear Schrödinger equations with variable coefficients,” J. Opt. Soc. Am. B 30, 113–122 (2013).
[CrossRef]

Yang, Z. Y.

Z. Y. Yang, L. C. Zhao, T. Zhang, and R. H. Yue, “Bright chirp-free and chirped nonautonomous solitons under dispersion and nonlinearity management,” J. Opt. Soc. Am. B 28, 236–240 (2011).
[CrossRef]

Z. Y. Yang, L. C. Zhao, T. Zhang, Y. H. Li, and R. H. Yue, “Snakelike nonautonomous solitons in a graded-index grating waveguid,” Phys. Rev. A 81, 043826 (2010).
[CrossRef]

Yue, R. H.

Z. Y. Yang, L. C. Zhao, T. Zhang, and R. H. Yue, “Bright chirp-free and chirped nonautonomous solitons under dispersion and nonlinearity management,” J. Opt. Soc. Am. B 28, 236–240 (2011).
[CrossRef]

Z. Y. Yang, L. C. Zhao, T. Zhang, Y. H. Li, and R. H. Yue, “Snakelike nonautonomous solitons in a graded-index grating waveguid,” Phys. Rev. A 81, 043826 (2010).
[CrossRef]

Zhang, J. F.

C. Q. Dai, G. Q. Zhou, and J. F. Zhang, “Controllable optical rogue waves in the femtosecond regime,” Phys. Rev. E 85, 016603 (2012).
[CrossRef]

C. Q. Dai, Y. Y. Wang, Q. Tian, and J. F. Zhang, “The management and containment of self-similar rogue waves in the inhomogeneous nonlinear Schrödinger equation,” Ann. Phys. 327, 512–521 (2012).
[CrossRef]

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

C. Q. Dai, S. Q. Zhu, L. L. Wang, and J. F. Zhang, “Exact spatial similaritons for the generalized (2+1)-dimensional nonlinear Schrödinger equation with distributed coefficients,” Europhys. Lett. 92, 24005 (2010).
[CrossRef]

Zhang, T.

Z. Y. Yang, L. C. Zhao, T. Zhang, and R. H. Yue, “Bright chirp-free and chirped nonautonomous solitons under dispersion and nonlinearity management,” J. Opt. Soc. Am. B 28, 236–240 (2011).
[CrossRef]

Z. Y. Yang, L. C. Zhao, T. Zhang, Y. H. Li, and R. H. Yue, “Snakelike nonautonomous solitons in a graded-index grating waveguid,” Phys. Rev. A 81, 043826 (2010).
[CrossRef]

Zhao, L. C.

Zhong, W. P.

Zhou, G. Q.

C. Q. Dai, G. Q. Zhou, and J. F. Zhang, “Controllable optical rogue waves in the femtosecond regime,” Phys. Rev. E 85, 016603 (2012).
[CrossRef]

Zhou, G. S.

R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, “Exact multisoliton solutions of the higher-order nonlinear Schrödinger equation with variable coefficients,” Phys. Rev. E 70, 066603 (2004).
[CrossRef]

Zhu, S. Q.

C. Q. Dai, Q. Tian, and S. Q. Zhu, “Controllable behaviours of rogue wave triplets in the nonautonomous nonlinear and dispersive system,” J. Phys. B 45, 085401 (2012).
[CrossRef]

C. Q. Dai, S. Q. Zhu, L. L. Wang, and J. F. Zhang, “Exact spatial similaritons for the generalized (2+1)-dimensional nonlinear Schrödinger equation with distributed coefficients,” Europhys. Lett. 92, 24005 (2010).
[CrossRef]

Ann. Phys.

C. Q. Dai, Y. Y. Wang, Q. Tian, and J. F. Zhang, “The management and containment of self-similar rogue waves in the inhomogeneous nonlinear Schrödinger equation,” Ann. Phys. 327, 512–521 (2012).
[CrossRef]

Appl. Opt.

Dokl. Akad. Nauk SSSR

E. A. Kuznetsov, “Solitons in a parametrically unstable plasma,” Dokl. Akad. Nauk SSSR 236, 575–577 (1977).

Electron. Lett.

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620–1622 (1994).
[CrossRef]

Europhys. Lett.

C. Q. Dai, S. Q. Zhu, L. L. Wang, and J. F. Zhang, “Exact spatial similaritons for the generalized (2+1)-dimensional nonlinear Schrödinger equation with distributed coefficients,” Europhys. Lett. 92, 24005 (2010).
[CrossRef]

J. Opt.

N. Akhmediev, J. M. Dudley, D. R. Solli, and S. K. Turitsyn, “Recent progress in investigating optical rogue waves,” J. Opt. 15, 060201 (2013).
[CrossRef]

J. Opt. Soc. Am. B

X. D. Cao and D. D. Meyerhofer, “Soliton collisions in optical birefringent fibers,” J. Opt. Soc. Am. B 11, 380–385 (1994).
[CrossRef]

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

Z. Y. Yang, L. C. Zhao, T. Zhang, and R. H. Yue, “Bright chirp-free and chirped nonautonomous solitons under dispersion and nonlinearity management,” J. Opt. Soc. Am. B 28, 236–240 (2011).
[CrossRef]

Y. J. He and D. Mihalache, “Soliton dynamics induced by periodic spatially inhomogeneous losses in optical media described by the complex Ginzburg-Landau model,” J. Opt. Soc. Am. B 29, 2554–2558 (2012).
[CrossRef]

L. C. Zhao and J. Liu, “Localized nonlinear waves in a two-mode nonlinear fiber,” J. Opt. Soc. Am. B 29, 3119–3127 (2012).
[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]

S. L. Xu, M. R. Belić, and W. P. Zhong, “Three-dimensional spatiotemporal vector solitary waves in coupled nonlinear Schrödinger equations with variable coefficients,” J. Opt. Soc. Am. B 30, 113–122 (2013).
[CrossRef]

W. P. Zhong, M. R. Belić, and T. W. Huang, “Solitary waves in the nonlinear Schrödinger equation with spatially modulated Bessel nonlinearity,” J. Opt. Soc. Am. B 30, 1276–1283 (2013).
[CrossRef]

J. Phys. B

C. Q. Dai, Q. Tian, and S. Q. Zhu, “Controllable behaviours of rogue wave triplets in the nonautonomous nonlinear and dispersive system,” J. Phys. B 45, 085401 (2012).
[CrossRef]

Nat. Phys.

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

Nature

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

Nonlinear Dyn.

W. P. Zhong, M. R. Belić, and T. W. Huang, “Two-dimensional accessible solitons in PT-symmetric potentials,” Nonlinear Dyn. 70, 2027–2034 (2012).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett. A

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]

Phys. Rep.

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. A

C. N. Kumar, R. Gupta, A. Goyal, S. Loomba, T. S. Raju, and P. K. Panigrahi, “Controlled giant rogue waves in nonlinear fiber optics,” Phys. Rev. A 86, 025802 (2012).
[CrossRef]

Z. Y. Yang, L. C. Zhao, T. Zhang, Y. H. Li, and R. H. Yue, “Snakelike nonautonomous solitons in a graded-index grating waveguid,” Phys. Rev. A 81, 043826 (2010).
[CrossRef]

Phys. Rev. E

R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, “Exact multisoliton solutions of the higher-order nonlinear Schrödinger equation with variable coefficients,” Phys. Rev. E 70, 066603 (2004).
[CrossRef]

C. Q. Dai, G. Q. Zhou, and J. F. Zhang, “Controllable optical rogue waves in the femtosecond regime,” Phys. Rev. E 85, 016603 (2012).
[CrossRef]

D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Second-order nonlinear Schrödinger equation breather solutions in the degenerate and rogue wave limits,” Phys. Rev. E 85, 066601 (2012).
[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]

A. Alexandrescu, G. D. Montesinos, and V. M. Pérez-García, “Stabilization of high-order solutions of the cubic nonlinear Schrödinger equation,” Phys. Rev. E 75, 046609 (2007).
[CrossRef]

Phys. Rev. Lett.

M. Centurion, M. A. Porter, P. G. Kevrekidis, and D. Psaltis, “Nonlinearity management in optics: experiment, theory, and simulation,” Phys. Rev. Lett. 97, 033903 (2006).
[CrossRef]

U. Bortolozzo, A. Montina, F. T. Arecchi, J. P. Huignard, and S. Residori, “Spatiotemporal pulses in a liquid crystal optical oscillator,” Phys. Rev. Lett. 99, 023901 (2007).
[CrossRef]

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef]

Phys. Scr.

K. B. Dysthe and K. Trulsen, “Note on breather type solutions of the NLS as models for freak-waves,” Phys. Scr. T82, 48–52 (1999).
[CrossRef]

S. L. Xu, N. Z. Petrović, and M. R. Belić, “Vortex solitons in the (2+1)-dimensional nonlinear Schrödinger equation with variable diffraction and nonlinearity coefficients,” Phys. Scr. 87, 045401 (2013).
[CrossRef]

Rev. Mod. Phys.

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

Sci. Rep.

B. Kibler, J. Fatome, C. Finot, G. Millot, G. Genty, B. Wetzel, N. Akhmediev, F. Dias, and J. Dudley, “Observation of Kuznetsov-Ma soliton dynamics in optical fibre,” Sci. Rep. 2, 463 (2012).
[CrossRef]

Stud. Appl. Math.

Y. C. Ma, “The perturbed plane-wave solution of the cubic Schrödinger equation,” Stud. Appl. Math. 60, 43–58 (1979).

Other

C. Sulem and P. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (Springer-Verlag, 2000).

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Figures (5)

Fig. 1.
Fig. 1.

Long-periodic grating structure.

Fig. 2.
Fig. 2.

(a) and (c) Two KM solitons with different position in X direction. Nonlinear superposition of KM soliton built from (b) first-order RW pairs and second-order RWs and (d) quin-RW clusters for Eq. (8) in the ZX coordinates. The parameters are chosen as n1=1.25i, n2=1.14i, v=0.2 with (a) Z0=Z0=8, X0=X0=5, (b) Z0=Z0=X0=X0=8, (c) Z0=8, Z0=4, X0=X0=5, and (d) Z0=8, Z0=4, X0=X0=8.

Fig. 3.
Fig. 3.

We take y=2. The total intensity I=|u1|2+|u2|2 of the superposed KM I soliton is shown: (a), (d), and (g) three kinds of partial excitation of the superposed KM I soliton; (b), (e), and (h) three kinds of maintenance of the superposed KM I soliton; (c), (f), and (i) three kinds of postponement of the superposed KM I soliton. Parameters are chosen as γ=0.005, ρ0=0.05, l=0.2, ω=0.25, H0=0.2, p=1, q=2,Cp,0=0.1, β0=0.02, c=2, c1=c2=1 with (a)–(i) σ=0.3, 0.2, 0.16, 0.095, 0.082, 0.07, 0.06, 0.055, 0.051, respectively. Results are similar for other values of y.

Fig. 4.
Fig. 4.

We take y=2. (a) and (d) Two kinds of partial excitation of the superposed KM II soliton, (b) and (e) two kinds of maintenance of the superposed KM II soliton, (c) and (f) two kinds of postponement of the superposed KM II soliton. Parameters are chosen as chosen as that in Fig. 3 except for (a)–(i) σ=0.115, 0.1, 0.095, 0.086, 0.075, 0.068, respectively. Results are similar for other values of y.

Fig. 5.
Fig. 5.

Numerical rerun of KM I and KM II solitons in Figs. 3(b), 3(f), 4(a), and 4(f). Only the dependence on x is shown. An added 5% white noise is added to the initial values. The parameters are the same as those in the corresponding analytical plots.

Equations (10)

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izu1+12β(z)Δu1+χ(z)(c1|u1|2+c|u2|2)u1+lcos(ωz)(x+y)u1+12ν(z)(x2+y2)u1=iγ(z)u1,izu2+12β(z)Δu2+χ(z)(c2|u2|2+c|u1|2)u2+lcos(ωz)(x+y)u2+12ν(z)(x2+y2)u2=iγ(z)u2,
u1=ρ0(cc2c2c1c2)12exp[Γ(z)]D2(z)U(Z,X)exp[iϕ(z,x,y)],u2=ρ0(cc1c2c1c2)12exp[Γ(z)]D2(z)U(Z,X)exp[iϕ(z,x,y)],
Z=0z(p2+q2)Ω(s)ds,
X=pD2(z)x+qD2(z)y0z(p+q)Ω(s)H(s)ds,
ϕ=Cp(z)(x2+y2)+H(z)D2(z)(x+y)140zΩ(s)H2(s)ds
2Cp,z+4βCp2ν=0,
χ(z)=p2+q2ρ02β(z)exp[2Γ(z)],
iUZ+12UXX+|U|2U=0.
u1=ρ0(cc2c2c1c2)12exp[Γ(z)]D2(z)[1+G+iHF]exp(iΦ),u2=ρ0(cc1c2c1c2)12exp[Γ(z)]D2(z)[1+G+iHF]exp(iΦ),
β(z)=β0exp(σz)

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