Abstract

The 𝒫𝒯 -symmetric and 𝒫𝒯 -antisymmetric Akhmediev breather (AB) and Kuznetsov-Ma (KM) soliton train solutions of a (2+1)-dimensional variable-coefficient coupled nonlinear Schrödinger equation in 𝒫𝒯 -symmetric coupled waveguides with gain and loss are derived via the Darboux transformation method. From these analytical solutions, we investigate the controllable behaviors of AB and KM soliton trains in a diffraction decreasing system with exponential profile. By adjusting the relation between the maximum Zm of effective propagation distance and the peak locations Zi of AB and KM soliton trains, we can control the restraint, maintenance and postpone excitations of AB and KM soliton trains.

© 2014 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Superposed Kuznetsov-Ma solitons in a two-dimensional graded-index grating waveguide

Chao-Qing Dai and Hai-Ping Zhu
J. Opt. Soc. Am. B 30(12) 3291-3297 (2013)

References

  • View by:
  • |
  • |
  • |

  1. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
    [Crossref]
  2. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
    [Crossref] [PubMed]
  3. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
    [Crossref] [PubMed]
  4. C. Q. Dai, X. G. Wang, and G. Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014).
    [Crossref]
  5. Y. V. Bludov, C. Hang, G. X. Huang, and V. V. Konotop, “PT-symmetric coupler with a coupling defect: soliton interaction with exceptional point,” Opt. Lett. 39, 3382–3385, 2014.
    [Crossref] [PubMed]
  6. D. H. Peregrine, “Water waves, nonlinear Schrödinger equations and their solutions,” J. Aust. Math. Soc. Ser. B 25, 16–43 (1983).
    [Crossref]
  7. N. Akhmediev and V.I. Korneev, ”Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. 69, 1089–1093 (1986).
    [Crossref]
  8. E. A. Kuznetsov, “Solitons in a parametrically unstable plasma,” Dokl. Akad. Nauk SSSR 236, 575–577 (1977).
  9. 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]
  10. 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]
  11. H. P. Zhu, Z. H. Pan, and J. P. Fang, “Controllability for two-Kuznetsov-Ma solitons in a (2 + 1)-dimensional graded-index grating waveguide,” Eur. Phys. J. D 68, 69–76 (2014).
    [Crossref]
  12. Yu V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, ”Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides,” J. Opt. 15, 064010 (2013).
    [Crossref]
  13. Y. Chen, A. W. Snyder, and D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
    [Crossref]
  14. F. Abdullaeev, Theory of Solitons in Inhomogeneous Media, (Wiley, 1994).
  15. D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, “Second-order nonlinear Schrodinger equation breather solutions in the degenerate and rogue wave limits,” Phys. Rev. E 85, 066601 (2012).
    [Crossref]
  16. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005).
    [Crossref]

2014 (3)

H. P. Zhu, Z. H. Pan, and J. P. Fang, “Controllability for two-Kuznetsov-Ma solitons in a (2 + 1)-dimensional graded-index grating waveguide,” Eur. Phys. J. D 68, 69–76 (2014).
[Crossref]

C. Q. Dai, X. G. Wang, and G. Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014).
[Crossref]

Y. V. Bludov, C. Hang, G. X. Huang, and V. V. Konotop, “PT-symmetric coupler with a coupling defect: soliton interaction with exceptional point,” Opt. Lett. 39, 3382–3385, 2014.
[Crossref] [PubMed]

2013 (1)

Yu V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, ”Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides,” J. Opt. 15, 064010 (2013).
[Crossref]

2012 (2)

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

2009 (1)

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]

2008 (1)

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

2007 (1)

2005 (1)

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005).
[Crossref]

1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

1992 (1)

Y. Chen, A. W. Snyder, and D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[Crossref]

1986 (1)

N. Akhmediev and V.I. Korneev, ”Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. 69, 1089–1093 (1986).
[Crossref]

1983 (1)

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

1977 (1)

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

Abdullaeev, F.

F. Abdullaeev, Theory of Solitons in Inhomogeneous Media, (Wiley, 1994).

Akhmediev, N.

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

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

N. Akhmediev and V.I. Korneev, ”Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. 69, 1089–1093 (1986).
[Crossref]

Ankiewicz, A.

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

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

Bender, C. M.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Bludov, Y. V.

Bludov, Yu V.

Yu V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, ”Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides,” J. Opt. 15, 064010 (2013).
[Crossref]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Chen, Y.

Y. Chen, A. W. Snyder, and D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[Crossref]

Christodoulides, D. N.

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

Dai, C. Q.

C. Q. Dai, X. G. Wang, and G. Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014).
[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]

Driben, R.

Yu V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, ”Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides,” J. Opt. 15, 064010 (2013).
[Crossref]

El-Ganainy, R.

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

Fang, J. P.

H. P. Zhu, Z. H. Pan, and J. P. Fang, “Controllability for two-Kuznetsov-Ma solitons in a (2 + 1)-dimensional graded-index grating waveguide,” Eur. Phys. J. D 68, 69–76 (2014).
[Crossref]

Hang, C.

Harvey, J. D.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005).
[Crossref]

Huang, G. X.

Kedziora, D. J.

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

Konotop, V. V.

Y. V. Bludov, C. Hang, G. X. Huang, and V. V. Konotop, “PT-symmetric coupler with a coupling defect: soliton interaction with exceptional point,” Opt. Lett. 39, 3382–3385, 2014.
[Crossref] [PubMed]

Yu V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, ”Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides,” J. Opt. 15, 064010 (2013).
[Crossref]

Korneev, V.I.

N. Akhmediev and V.I. Korneev, ”Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. 69, 1089–1093 (1986).
[Crossref]

Kruglov, V. I.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005).
[Crossref]

Kuznetsov, E. A.

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

Makris, K. G.

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

Malomed, B. A.

Yu V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, ”Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides,” J. Opt. 15, 064010 (2013).
[Crossref]

Musslimani, Z. H.

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

Pan, Z. H.

H. P. Zhu, Z. H. Pan, and J. P. Fang, “Controllability for two-Kuznetsov-Ma solitons in a (2 + 1)-dimensional graded-index grating waveguide,” Eur. Phys. J. D 68, 69–76 (2014).
[Crossref]

Payne, D. N.

Y. Chen, A. W. Snyder, and D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[Crossref]

Peacock, A. C.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005).
[Crossref]

Peregrine, D. H.

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

Snyder, A. W.

Y. Chen, A. W. Snyder, and D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[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]

Wang, X. G.

C. Q. Dai, X. G. Wang, and G. Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014).
[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]

Zhou, G. Q.

C. Q. Dai, X. G. Wang, and G. Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014).
[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]

Zhu, H. P.

H. P. Zhu, Z. H. Pan, and J. P. Fang, “Controllability for two-Kuznetsov-Ma solitons in a (2 + 1)-dimensional graded-index grating waveguide,” Eur. Phys. J. D 68, 69–76 (2014).
[Crossref]

Dokl. Akad. Nauk SSSR (1)

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

Eur. Phys. J. D (1)

H. P. Zhu, Z. H. Pan, and J. P. Fang, “Controllability for two-Kuznetsov-Ma solitons in a (2 + 1)-dimensional graded-index grating waveguide,” Eur. Phys. J. D 68, 69–76 (2014).
[Crossref]

IEEE J. Quantum Electron. (1)

Y. Chen, A. W. Snyder, and D. N. Payne, “Twin core nonlinear couplers with gain and loss,” IEEE J. Quantum Electron. 28, 239–245 (1992).
[Crossref]

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

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

J. Opt. (1)

Yu V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, ”Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides,” J. Opt. 15, 064010 (2013).
[Crossref]

Opt. Lett. (2)

Phys. Lett. A (1)

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]

Phys. Rev. A (1)

C. Q. Dai, X. G. Wang, and G. Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014).
[Crossref]

Phys. Rev. E (3)

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 Schrodinger equation breather solutions in the degenerate and rogue wave limits,” Phys. Rev. E 85, 066601 (2012).
[Crossref]

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005).
[Crossref]

Phys. Rev. Lett. (2)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT-symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

Theor. Math. Phys. (1)

N. Akhmediev and V.I. Korneev, ”Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. 69, 1089–1093 (1986).
[Crossref]

Other (1)

F. Abdullaeev, Theory of Solitons in Inhomogeneous Media, (Wiley, 1994).

Cited By

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

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1 (a) and (b) KM soliton and AB trains for Eq. (3) in the ZX coordinates, respectively. Parameters are chosen as v0 = 0.1 with (a) Z0 = 5, κ = 0.6i and (b) Z0 = 8, κ = 0.6.
Fig. 2
Fig. 2 We take y = 2. The physical quantity I = |u| of controllable behaviors for KM soliton train: (a), (c) and (h) maintenance of the second-order PS, the first and third PS pairs, (b),(g) and (d),(i) restraint and postpone of the first and third PS pairs. Parameters are chosen as A0 = 0.5, a = 0.05, b = 0.06, β10 = 0.1, β20 = 0.12, k = 2.5, l = 3, γ = 0.2, c = 1, d = 1.2 with (a)–(d), (g)–(i) σ = 0.07, 0.032, 0.0315, 0.0295, 0.018, 0.0178, 0.0174, respectively. Results are similar for other values of y. (e) and (f) Magnitude and phase of ���� -symmetric and -antisymmetric solution corresponding to (c). (j)–(l) Evolutional plots corresponding to (a), (c) and (g) at z = 60, 180, 190, respectively.
Fig. 3
Fig. 3 We take y = 2. The physical quantity I = |u| of controllable behaviors for AB train: (a), (d) and (g) restraint of the front wing, the second-order PS and back wing, (b), (e) and (h) maintenance of the front wing, the second-order PS and back wing, (c), (f) and (i) postpone of the front wing, the second-order PS and back wing. Parameters are chosen as the same as those in Fig. 2 except for (a)–(i) σ = 0.5, 0.148, 0.11, 0.075, 0.0597, 0.054, 0.043, 0.0372, 0.035, respectively. Results are similar for other values of y. (j)–(l) Evolutional plots corresponding to (b), (d) and (i) at z = 60, 80, 150, respectively.

Equations (6)

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

i u z + 1 2 [ β 1 ( z ) u x x + β 2 ( z ) u y y ] + [ χ 1 ( z ) | u | 2 + χ ( z ) | v | 2 ] u = η ( z ) ( v + i γ u ) , i v z + 1 2 [ β 1 ( z ) u x x + β 2 ( z ) u y y ] + [ χ ( z ) | u | 2 + χ 1 ( z ) | v | 2 ] v = η ( z ) ( u + i γ v ) ,
v ( z , x , y ) = ± u ( z , x , y ) exp ( ± i θ ) , u ( z , x , y ) = A 0 α 1 ( z ) α 2 ( z ) U [ Z ( z ) , X ( z , x , y ) ] exp { i [ cos ( θ ) Ω ( z ) + ϕ ( z , x , y ) ] } ,
i U Z + 1 2 U X X + | U | 2 U = 0 ,
χ ( z ) + χ 1 ( z ) = 1 4 A 0 2 [ k 2 β 1 ( z ) α 1 ( z ) α 2 ( z ) + l 2 β 2 ( z ) α 2 ( z ) α 1 ( z ) ] ,
u = A 0 α 1 ( z ) α 2 ( z ) [ 1 + P + i Q R ] exp { i [ ( 1 v 0 2 ) ( Z Z 0 ) + v 0 X + cos ( θ ) Ω ( z ) + ϕ ] } ,
β 1 ( z ) = β 10 exp ( σ z ) , β 2 ( z ) = β 20 exp ( σ z ) ,

Metrics