Abstract

We numerically reveal the rich dynamics of a two-dimensional fundamental soliton coupled on the top of a sharp external potential in dissipative nonlinear media based on the cubic-quintic complex Ginzburg–Landau model. Here, we consider two kinds of radially symmetric potentials, namely, a tapered potential (TP) and a raised-cosine potential (RCP). It is found that if the sharpness and depth of the potential are large enough, the soliton can emit either one annular beam or a cluster of ring-like beams, all of which gradually expand upon propagation. By using the TP, one can get a nonstationary annular beam, while a single stationary annular beam can be achieved by using the RCP. The radius of the stationary annular beam is controllable by the modulation period of the potential. Other soliton dynamics, including soliton localization, soliton oscillation, lateral drift, soliton collapse, and soliton decay, are also revealed. The reported results provide what we believe to be a new method to generate annular beams in dissipative systems.

© 2010 Optical Society of America

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  1. I. S. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys. 74, 99–143 (2002).
    [CrossRef]
  2. N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer-Verlag, 2002).
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  4. B. A. Malomed, “Solitary pulses in linearly coupled Ginzburg–Landau equations,” Chaos 17, 037117 (2007).
    [CrossRef] [PubMed]
  5. N. Akhmediev, J. M. Soto-Crespo, and Ph. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsating complexes,” Chaos 17, 037112 (2007).
    [CrossRef] [PubMed]
  6. F. T. Arecchi, S. Boccaletti, and P. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
    [CrossRef]
  7. P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics,” J. Opt. B: Quantum Semiclassical Opt. 6, R60–R75 (2004).
    [CrossRef]
  8. N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Universal properties of self-organized localized structures,” Appl. Phys. B 81, 937–943 (2005).
    [CrossRef]
  9. N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190–1201 (1996).
    [CrossRef]
  10. W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
    [CrossRef]
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    [CrossRef]
  12. B. A. Malomed, “Bound solitons in the nonlinear Schrödinger–Ginzburg–Landau equation,” Phys. Rev. A 44, 6954–6957 (1991).
    [CrossRef] [PubMed]
  13. D. V. Skryabin and A. G. Vladimirov, “Vortex-induced rotation of clusters of localized states in the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 89, 044101 (2002).
    [CrossRef] [PubMed]
  14. Y. J. He, H. H. Fan, J. W. Dong, and H. Z. Wang, “Self-trapped spatiotemporal necklace-ring solitons in the Ginzburg–Landau equation,” Phys. Rev. E 74, 016611 (2006).
    [CrossRef]
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    [CrossRef] [PubMed]
  16. L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg–Landau equation,” Phys. Rev. E 63, 016605 (2001).
    [CrossRef]
  17. D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
    [CrossRef] [PubMed]
  18. D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. A 75, 033811 (2007).
    [CrossRef]
  19. D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability limits for three-dimensional vortex solitons in the Ginzburg–Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76, 045803 (2007).
    [CrossRef]
  20. Y. He, B. A. Malomed, D. Mihalache, F. Ye, and B. Hu, “Generation of arrays of spatiotemporal dissipative solitons by the phase modulation of a broad beam,” J. Opt. Soc. Am. B 27, 1266–1271 (2010).
    [CrossRef]
  21. H. Sakaguchi and B. A. Malomed, “Two-dimensional dissipative gap solitons,” Phys. Rev. E 80, 026606 (2009).
    [CrossRef]
  22. H. Leblond, B.A. Malomed, and D. Mihalache, “Stable vortex solitons in the Ginzburg–Landau model of a two-dimensional lasing medium with a transverse grating,” Phys. Rev. A 80, 033835 (2009).
    [CrossRef]
  23. Y.-J. He, B. A. Malomed, F. Ye, and B. Hu, “Dynamics of dissipative spatial solitons over a sharp potential,” J. Opt. Soc. Am. B 27, 1139–1142 (2010).
    [CrossRef]
  24. B. Liu, Y.-J. He, B. A. Malomed, X.-S. Wang, P. G. Kevrekidis, T.-B. Wang, F.-C. Leng, Z.-R. Qiu, and H.-Z. Wang, “Continuous generation of soliton patterns in two-dimensional dissipative media by razor, dagger, and needle potentials,” Opt. Lett. 35, 1974–1976 (2010).
    [CrossRef] [PubMed]
  25. J. M. Soto-Crespo, N. Akhmediev, C. Mejía-Cortés, and N. Devine, “Dissipative ring solitons with vorticity,” Opt. Express 17, 4236–4250 (2009).
    [CrossRef] [PubMed]
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    [CrossRef]

2010 (3)

2009 (4)

J. M. Soto-Crespo, N. Akhmediev, C. Mejía-Cortés, and N. Devine, “Dissipative ring solitons with vorticity,” Opt. Express 17, 4236–4250 (2009).
[CrossRef] [PubMed]

H. Sakaguchi and B. A. Malomed, “Two-dimensional dissipative gap solitons,” Phys. Rev. E 80, 026606 (2009).
[CrossRef]

H. Leblond, B.A. Malomed, and D. Mihalache, “Stable vortex solitons in the Ginzburg–Landau model of a two-dimensional lasing medium with a transverse grating,” Phys. Rev. A 80, 033835 (2009).
[CrossRef]

Y. J. He, B. A. Malomed, D. Mihalache, B. Liu, H. C. Huang, H. Yang, and H. Z. Wang, “Bound states of one-, two-, and three-dimensional solitons in complex Ginzburg–Landau equations with a linear potential,” Opt. Lett. 34, 2976–2978 (2009).
[CrossRef] [PubMed]

2008 (1)

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[CrossRef]

2007 (4)

B. A. Malomed, “Solitary pulses in linearly coupled Ginzburg–Landau equations,” Chaos 17, 037117 (2007).
[CrossRef] [PubMed]

N. Akhmediev, J. M. Soto-Crespo, and Ph. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsating complexes,” Chaos 17, 037112 (2007).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. A 75, 033811 (2007).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability limits for three-dimensional vortex solitons in the Ginzburg–Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76, 045803 (2007).
[CrossRef]

2006 (2)

Y. J. He, H. H. Fan, J. W. Dong, and H. Z. Wang, “Self-trapped spatiotemporal necklace-ring solitons in the Ginzburg–Landau equation,” Phys. Rev. E 74, 016611 (2006).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

2005 (1)

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Universal properties of self-organized localized structures,” Appl. Phys. B 81, 937–943 (2005).
[CrossRef]

2004 (1)

P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics,” J. Opt. B: Quantum Semiclassical Opt. 6, R60–R75 (2004).
[CrossRef]

2003 (1)

2002 (2)

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys. 74, 99–143 (2002).
[CrossRef]

D. V. Skryabin and A. G. Vladimirov, “Vortex-induced rotation of clusters of localized states in the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 89, 044101 (2002).
[CrossRef] [PubMed]

2001 (1)

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg–Landau equation,” Phys. Rev. E 63, 016605 (2001).
[CrossRef]

1999 (1)

F. T. Arecchi, S. Boccaletti, and P. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

1996 (1)

N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190–1201 (1996).
[CrossRef]

1991 (1)

B. A. Malomed, “Bound solitons in the nonlinear Schrödinger–Ginzburg–Landau equation,” Phys. Rev. A 44, 6954–6957 (1991).
[CrossRef] [PubMed]

Afanasjev, V. V.

N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190–1201 (1996).
[CrossRef]

Akhmediev, N.

J. M. Soto-Crespo, N. Akhmediev, C. Mejía-Cortés, and N. Devine, “Dissipative ring solitons with vorticity,” Opt. Express 17, 4236–4250 (2009).
[CrossRef] [PubMed]

N. Akhmediev, J. M. Soto-Crespo, and Ph. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsating complexes,” Chaos 17, 037112 (2007).
[CrossRef] [PubMed]

N. Akhmediev and A. Ankiewicz, Dissipative Solitons, Vol. 661 of Lecture Notes in Physics (Springer, 2005).
[CrossRef]

Akhmediev, N. N.

N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190–1201 (1996).
[CrossRef]

Ankiewicz, A.

N. Akhmediev and A. Ankiewicz, Dissipative Solitons, Vol. 661 of Lecture Notes in Physics (Springer, 2005).
[CrossRef]

Aranson, I. S.

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys. 74, 99–143 (2002).
[CrossRef]

Arecchi, F. T.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

Boccaletti, S.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

Chong, A.

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[CrossRef]

Crasovan, L. -C.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg–Landau equation,” Phys. Rev. E 63, 016605 (2001).
[CrossRef]

Devine, N.

Dong, J. W.

Y. J. He, H. H. Fan, J. W. Dong, and H. Z. Wang, “Self-trapped spatiotemporal necklace-ring solitons in the Ginzburg–Landau equation,” Phys. Rev. E 74, 016611 (2006).
[CrossRef]

Fan, H. H.

Y. J. He, H. H. Fan, J. W. Dong, and H. Z. Wang, “Self-trapped spatiotemporal necklace-ring solitons in the Ginzburg–Landau equation,” Phys. Rev. E 74, 016611 (2006).
[CrossRef]

Fedorov, S. V.

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Universal properties of self-organized localized structures,” Appl. Phys. B 81, 937–943 (2005).
[CrossRef]

Grelu, Ph.

N. Akhmediev, J. M. Soto-Crespo, and Ph. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsating complexes,” Chaos 17, 037112 (2007).
[CrossRef] [PubMed]

He, Y.

He, Y. J.

He, Y. -J.

Holzlöhner, R.

Hu, B.

Huang, H. C.

Kartashov, Y. V.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

Kevrekidis, P. G.

Kramer, L.

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys. 74, 99–143 (2002).
[CrossRef]

Leblond, H.

H. Leblond, B.A. Malomed, and D. Mihalache, “Stable vortex solitons in the Ginzburg–Landau model of a two-dimensional lasing medium with a transverse grating,” Phys. Rev. A 80, 033835 (2009).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. A 75, 033811 (2007).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability limits for three-dimensional vortex solitons in the Ginzburg–Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76, 045803 (2007).
[CrossRef]

Lederer, F.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability limits for three-dimensional vortex solitons in the Ginzburg–Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76, 045803 (2007).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. A 75, 033811 (2007).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

Leng, F. -C.

Liu, B.

Malomed, B. A.

B. Liu, Y.-J. He, B. A. Malomed, X.-S. Wang, P. G. Kevrekidis, T.-B. Wang, F.-C. Leng, Z.-R. Qiu, and H.-Z. Wang, “Continuous generation of soliton patterns in two-dimensional dissipative media by razor, dagger, and needle potentials,” Opt. Lett. 35, 1974–1976 (2010).
[CrossRef] [PubMed]

Y.-J. He, B. A. Malomed, F. Ye, and B. Hu, “Dynamics of dissipative spatial solitons over a sharp potential,” J. Opt. Soc. Am. B 27, 1139–1142 (2010).
[CrossRef]

Y. He, B. A. Malomed, D. Mihalache, F. Ye, and B. Hu, “Generation of arrays of spatiotemporal dissipative solitons by the phase modulation of a broad beam,” J. Opt. Soc. Am. B 27, 1266–1271 (2010).
[CrossRef]

H. Sakaguchi and B. A. Malomed, “Two-dimensional dissipative gap solitons,” Phys. Rev. E 80, 026606 (2009).
[CrossRef]

Y. J. He, B. A. Malomed, D. Mihalache, B. Liu, H. C. Huang, H. Yang, and H. Z. Wang, “Bound states of one-, two-, and three-dimensional solitons in complex Ginzburg–Landau equations with a linear potential,” Opt. Lett. 34, 2976–2978 (2009).
[CrossRef] [PubMed]

H. Leblond, B.A. Malomed, and D. Mihalache, “Stable vortex solitons in the Ginzburg–Landau model of a two-dimensional lasing medium with a transverse grating,” Phys. Rev. A 80, 033835 (2009).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. A 75, 033811 (2007).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability limits for three-dimensional vortex solitons in the Ginzburg–Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76, 045803 (2007).
[CrossRef]

B. A. Malomed, “Solitary pulses in linearly coupled Ginzburg–Landau equations,” Chaos 17, 037117 (2007).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg–Landau equation,” Phys. Rev. E 63, 016605 (2001).
[CrossRef]

B. A. Malomed, “Bound solitons in the nonlinear Schrödinger–Ginzburg–Landau equation,” Phys. Rev. A 44, 6954–6957 (1991).
[CrossRef] [PubMed]

B. A. Malomed, “Complex Ginzburg–Landau equation,” in Encyclopedia of Nonlinear Science, A.Scott, ed. (Routledge, 2005), pp. 157–160.

Mandel, P.

P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics,” J. Opt. B: Quantum Semiclassical Opt. 6, R60–R75 (2004).
[CrossRef]

Mazilu, D.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. A 75, 033811 (2007).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability limits for three-dimensional vortex solitons in the Ginzburg–Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76, 045803 (2007).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

Mejía-Cortés, C.

Menyuk, C. R.

Mihalache, D.

Y. He, B. A. Malomed, D. Mihalache, F. Ye, and B. Hu, “Generation of arrays of spatiotemporal dissipative solitons by the phase modulation of a broad beam,” J. Opt. Soc. Am. B 27, 1266–1271 (2010).
[CrossRef]

Y. J. He, B. A. Malomed, D. Mihalache, B. Liu, H. C. Huang, H. Yang, and H. Z. Wang, “Bound states of one-, two-, and three-dimensional solitons in complex Ginzburg–Landau equations with a linear potential,” Opt. Lett. 34, 2976–2978 (2009).
[CrossRef] [PubMed]

H. Leblond, B.A. Malomed, and D. Mihalache, “Stable vortex solitons in the Ginzburg–Landau model of a two-dimensional lasing medium with a transverse grating,” Phys. Rev. A 80, 033835 (2009).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability limits for three-dimensional vortex solitons in the Ginzburg–Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76, 045803 (2007).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. A 75, 033811 (2007).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg–Landau equation,” Phys. Rev. E 63, 016605 (2001).
[CrossRef]

Qiu, Z. -R.

Ramazza, P.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

Renninger, W. H.

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[CrossRef]

Rosanov, N. N.

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Universal properties of self-organized localized structures,” Appl. Phys. B 81, 937–943 (2005).
[CrossRef]

N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer-Verlag, 2002).

Sakaguchi, H.

H. Sakaguchi and B. A. Malomed, “Two-dimensional dissipative gap solitons,” Phys. Rev. E 80, 026606 (2009).
[CrossRef]

Shatsev, A. N.

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Universal properties of self-organized localized structures,” Appl. Phys. B 81, 937–943 (2005).
[CrossRef]

Sinkin, O. V.

Skryabin, D. V.

D. V. Skryabin and A. G. Vladimirov, “Vortex-induced rotation of clusters of localized states in the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 89, 044101 (2002).
[CrossRef] [PubMed]

Soto-Crespo, J. M.

J. M. Soto-Crespo, N. Akhmediev, C. Mejía-Cortés, and N. Devine, “Dissipative ring solitons with vorticity,” Opt. Express 17, 4236–4250 (2009).
[CrossRef] [PubMed]

N. Akhmediev, J. M. Soto-Crespo, and Ph. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsating complexes,” Chaos 17, 037112 (2007).
[CrossRef] [PubMed]

N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190–1201 (1996).
[CrossRef]

Tlidi, M.

P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics,” J. Opt. B: Quantum Semiclassical Opt. 6, R60–R75 (2004).
[CrossRef]

Torner, L.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

Vladimirov, A. G.

D. V. Skryabin and A. G. Vladimirov, “Vortex-induced rotation of clusters of localized states in the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 89, 044101 (2002).
[CrossRef] [PubMed]

Wang, H. Z.

Wang, H. -Z.

Wang, T. -B.

Wang, X. -S.

Wise, F. W.

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[CrossRef]

Yang, H.

Ye, F.

Zweck, J.

Appl. Phys. B (1)

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Universal properties of self-organized localized structures,” Appl. Phys. B 81, 937–943 (2005).
[CrossRef]

Chaos (2)

B. A. Malomed, “Solitary pulses in linearly coupled Ginzburg–Landau equations,” Chaos 17, 037117 (2007).
[CrossRef] [PubMed]

N. Akhmediev, J. M. Soto-Crespo, and Ph. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsating complexes,” Chaos 17, 037112 (2007).
[CrossRef] [PubMed]

J. Lightwave Technol. (1)

J. Opt. B: Quantum Semiclassical Opt. (1)

P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics,” J. Opt. B: Quantum Semiclassical Opt. 6, R60–R75 (2004).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Express (1)

Opt. Lett. (2)

Phys. Rep. (1)

F. T. Arecchi, S. Boccaletti, and P. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

Phys. Rev. A (5)

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. A 75, 033811 (2007).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability limits for three-dimensional vortex solitons in the Ginzburg–Landau equation with the cubic-quintic nonlinearity,” Phys. Rev. A 76, 045803 (2007).
[CrossRef]

B. A. Malomed, “Bound solitons in the nonlinear Schrödinger–Ginzburg–Landau equation,” Phys. Rev. A 44, 6954–6957 (1991).
[CrossRef] [PubMed]

H. Leblond, B.A. Malomed, and D. Mihalache, “Stable vortex solitons in the Ginzburg–Landau model of a two-dimensional lasing medium with a transverse grating,” Phys. Rev. A 80, 033835 (2009).
[CrossRef]

Phys. Rev. E (4)

H. Sakaguchi and B. A. Malomed, “Two-dimensional dissipative gap solitons,” Phys. Rev. E 80, 026606 (2009).
[CrossRef]

Y. J. He, H. H. Fan, J. W. Dong, and H. Z. Wang, “Self-trapped spatiotemporal necklace-ring solitons in the Ginzburg–Landau equation,” Phys. Rev. E 74, 016611 (2006).
[CrossRef]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg–Landau equation,” Phys. Rev. E 63, 016605 (2001).
[CrossRef]

N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190–1201 (1996).
[CrossRef]

Phys. Rev. Lett. (2)

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
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D. V. Skryabin and A. G. Vladimirov, “Vortex-induced rotation of clusters of localized states in the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 89, 044101 (2002).
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Rev. Mod. Phys. (1)

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys. 74, 99–143 (2002).
[CrossRef]

Other (3)

N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer-Verlag, 2002).

B. A. Malomed, “Complex Ginzburg–Landau equation,” in Encyclopedia of Nonlinear Science, A.Scott, ed. (Routledge, 2005), pp. 157–160.

N. Akhmediev and A. Ankiewicz, Dissipative Solitons, Vol. 661 of Lecture Notes in Physics (Springer, 2005).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

External TPs, for p = 1 and (a) n = 0.6 , (b) n = 1 , and (c) n = 1.4 .

Fig. 2
Fig. 2

RCPs for p = 5 and (a) m = 3 and n = 2 , (b) m = 6 and n = 8 .

Fig. 3
Fig. 3

Various dynamical regimes of the soliton under the effect of the TP in the parameter plane ( n , p ) . Region A: soliton localization; region B: soliton lateral drift; region C: soliton collapse; region D: soliton continually emitting annular beams; region E: soliton emitting single annular beams; region F: soliton oscillation; region G: soliton decay.

Fig. 4
Fig. 4

Input soliton emits a single annular beam when n = 1 and p = 1 . (a) Evolution of the soliton into a single annular beam. (b) The radius of the annular beam increases with the growth of propagation distance. (c) The amplitude change of annular beam.

Fig. 5
Fig. 5

Input soliton emits a single annular beam when soliton deviates from radial symmetry of the potential in the presence of 10% noise. (a) Deviation of soliton from radial symmetry of the potential Δ r /(soliton width) (i.e., ratio of deviation of soliton to soliton width) versus the modulation depth p. (b) Example of the evolution of soliton with large deviation Δ r / ( soliton   width ) = 20 % , into a single annular beam, for p = 2 and n = 0.6 . The transverse box is ( 45 , 45 ) × ( 45 , 45 ) .

Fig. 6
Fig. 6

Soliton emits a cluster of annular beams when n = 1 and p = 0.6 . (a) Evolution of the soliton into annular beams. (b) The radius changes of the generated annular beams are indicated by curves 1–6, which correspond to the annular beams 1–6 shown in (a).

Fig. 7
Fig. 7

Same as Fig. 5, but soliton emits a cluster of annular beams. (a) Deviation of soliton from radial symmetry of the potential Δ r /(soliton width) versus the modulation depth p. (b) Example of the evolution of soliton with large deviation Δ r / ( soliton   width ) = 7 % , into a cluster of annular beams, for p = 0.8 and n = 1 .

Fig. 8
Fig. 8

(a) Soliton localization when n = 1.4 and p = 0.01 . (b) Soliton lateral drift when n = 1 and p = 0.2 . (c) Soliton oscillation when n = 1.4 and p = 0.5 . (d) Soliton collapse when n = 0.6 and p = 0.78 . (e) Soliton decay when n = 1 and p = 1.3 .

Fig. 9
Fig. 9

Annular soliton induced by using the RCP with p = 5 . (a) The region in plane ( m , n ) above the curve is the domain where annular soliton is generated. (b) Input fundamental soliton evolves into a small annular soliton when m = 3 and n = 8 . (c) Input soliton evolves into a large annular soliton when m = 6 and n = 8 . (d) Soliton oscillation when m = 6 and n = 4 .

Fig. 10
Fig. 10

Same as Fig. 5, but in the case of RCP with p = 5 . (a) Deviation of soliton from radial symmetry of the potential Δ r /(soliton width) versus the index n. (b) Example of the evolution of soliton with large deviation Δ r / ( soliton   width ) = 15 % , into a single annular beam, for m = 6 and n = 8 . The transverse box is ( 10 , 10 ) × ( 10 , 10 ) .

Equations (4)

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i u z + ( 1 / 2 ) Δ u + | u | 2 u + ν | u | 4 u = i R [ u ] + V u ,
V ( x , y ) = p r n ,
V ( x , y ) = p [ cos ( r / m ) ] n ,
u = A   exp [ x 2 + y 2 2 w x , y 2 ] ,

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