Abstract

Annularly and radially phase-modulated spatiotemporal necklace-shaped patterns (SNPs) in the complex Ginzburg–Landau (CGL) and complex Swift–Hohenberg (CSH) equations are theoretically studied. It is shown that the annularly phase-modulated SNPs, with a small initial radius of the necklace and modulation parameters, can evolve into stable fundamental or vortex solitons. To the radially phase-modulated SNPs, the modulated “beads” on the necklace rapidly vanish under strong dissipation in transmission, which may have potential application for optical switching in signal processing. A prediction that the SNPs with large initial radii keep necklace-ring shapes upon propagation is demonstrated by use of balance equations for energy and momentum. Differences between both models for the evolution of solitons are revealed.

© 2009 OSA

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    [CrossRef] [PubMed]
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    [CrossRef]
  26. M. Tlidi and P. Mandel, “Three-dimensional optical crystals and localized structures in cavity second harmonic generation,” Phys. Rev. Lett. 83(24), 4995–4998 (1999).
    [CrossRef]
  27. M. Tlidi, “Three-dimensional crystals and localized structures in diffractive and dispersive nonlinear ring cavities,” J. Opt. B Quantum Semiclassical Opt. 2(3), 438–442 (2000).
    [CrossRef]
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    [CrossRef]
  29. N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Stable soliton pairs in optical transmission lines and fiber lasers,” J. Opt. Soc. Am. B 15(2), 515–523 (1998).
    [CrossRef]

2009 (1)

2007 (3)

Y. J. He, H. Z. Wang, and B. A. Malomed, “Fusion of necklace-ring patterns into vortex and fundamental solitons in dissipative media,” Opt. Express 15(26), 17502–17508 (2007).
[CrossRef] [PubMed]

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(4), 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(3), 033811 (2007).
[CrossRef]

2006 (4)

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 Stat. Nonlin. Soft Matter Phys. 74(1 Pt 2), 016611 (2006).
[CrossRef] [PubMed]

J. Burke and E. Knobloch, “Localized states in the generalized Swift-Hohenberg equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(5 Pt 2), 056211 (2006).
[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(7), 073904 (2006).
[CrossRef] [PubMed]

J. M. Soto-Crespo, P. Grelu, and N. Akhmediev, “Optical bullets and ‘rockets’ in nonlinear dissipative systems and their transformations and interactions,” Opt. Express 14(9), 4013–4025 (2006).
[CrossRef] [PubMed]

2005 (1)

N. Akhmediev and A. Ankiewicz, “Dissipative solitons in the complex Ginzburg–Landau and Swift–Hohenberg equations,” Lect. Notes Phys. 661, 1–17 (2005).
[CrossRef]

2004 (1)

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, “Soliton clusters in three-dimensional media with competing cubic and quintic nonlinearities,” J. Opt. B Quantum Semiclassical Opt. 6(5), S333–S340 (2004).
[CrossRef]

2003 (2)

L.-C. Crasovan, Y. V. Kartashov, D. Mihalache, L. Torner, and V. M. Pérez-García, “Soliton “molecules”: robust clusters of spatiotemporal optical solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4 Pt 2), 046610 (2003).
[CrossRef] [PubMed]

K.-I. Maruno, A. Ankiewicz, and N. Akhmediev, “Exact soliton solutions of the one-dimensional complex Swift–Hohenberg equation,” Physica D 176(1-2), 44–66 (2003).
[CrossRef]

2002 (5)

J. M. Soto-Crespo and N. Akhmediev, “Composite solitons and two-pulse generation in passively mode-locked lasers modeled by the complex quintic Swift-Hohenberg equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(6 Pt 2), 066610 (2002).
[CrossRef]

J. Buceta, K. Lindenberg, and J. M. R. Parrondo, “Stationary and oscillatory spatial patterns induced by global periodic switching,” Phys. Rev. Lett. 88(2), 024103 (2002).
[CrossRef] [PubMed]

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys. 74(1), 99–143 (2002).
[CrossRef]

A. G. Vladimirov, J. M. McSloy, D. V. Skryabin, and W. J. Firth, “Two-dimensional clusters of solitary structures in driven optical cavities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(44 Pt 2B), 046606 (2002).
[CrossRef] [PubMed]

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(4), 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 Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 63(1), 016605 (2001).
[CrossRef]

2000 (1)

M. Tlidi, “Three-dimensional crystals and localized structures in diffractive and dispersive nonlinear ring cavities,” J. Opt. B Quantum Semiclassical Opt. 2(3), 438–442 (2000).
[CrossRef]

1999 (2)

M. Tlidi and P. Mandel, “Three-dimensional optical crystals and localized structures in cavity second harmonic generation,” Phys. Rev. Lett. 83(24), 4995–4998 (1999).
[CrossRef]

V. J. Sánchez-Morcillo and K. Staliunas, “Stability of localized structures in the Swift-Hohenberg equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(55 Pt B), 6153–6156 (1999).
[CrossRef]

1998 (2)

M. Soljačić, S. Sears, and M. Segev, “Self-trapping of ‘necklace’ beams in self-focusing Kerr media,” Phys. Rev. Lett. 81(22), 4851–4854 (1998).
[CrossRef]

N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Stable soliton pairs in optical transmission lines and fiber lasers,” J. Opt. Soc. Am. B 15(2), 515–523 (1998).
[CrossRef]

1995 (1)

M. F. Hilali, S. Métens, P. Borckmans, and G. Dewel, “Pattern selection in the generalized Swift-Hohenberg model,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 51(3), 2046–2052 (1995).
[CrossRef] [PubMed]

1994 (2)

J. Lega, J. V. Moloney, and A. C. Newell, “Swift-Hohenberg equation for lasers,” Phys. Rev. Lett. 73(22), 2978–2981 (1994).
[CrossRef] [PubMed]

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73(5), 640–643 (1994).
[CrossRef] [PubMed]

Akhmediev, N.

J. M. Soto-Crespo, N. Akhmediev, C. Mejia-Cortés, and N. Devine, “Dissipative ring solitons with vorticity,” Opt. Express 17(6), 4236–4250 (2009).
[CrossRef] [PubMed]

J. M. Soto-Crespo, P. Grelu, and N. Akhmediev, “Optical bullets and ‘rockets’ in nonlinear dissipative systems and their transformations and interactions,” Opt. Express 14(9), 4013–4025 (2006).
[CrossRef] [PubMed]

N. Akhmediev and A. Ankiewicz, “Dissipative solitons in the complex Ginzburg–Landau and Swift–Hohenberg equations,” Lect. Notes Phys. 661, 1–17 (2005).
[CrossRef]

K.-I. Maruno, A. Ankiewicz, and N. Akhmediev, “Exact soliton solutions of the one-dimensional complex Swift–Hohenberg equation,” Physica D 176(1-2), 44–66 (2003).
[CrossRef]

J. M. Soto-Crespo and N. Akhmediev, “Composite solitons and two-pulse generation in passively mode-locked lasers modeled by the complex quintic Swift-Hohenberg equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(6 Pt 2), 066610 (2002).
[CrossRef]

Akhmediev, N. N.

Ankiewicz, A.

N. Akhmediev and A. Ankiewicz, “Dissipative solitons in the complex Ginzburg–Landau and Swift–Hohenberg equations,” Lect. Notes Phys. 661, 1–17 (2005).
[CrossRef]

K.-I. Maruno, A. Ankiewicz, and N. Akhmediev, “Exact soliton solutions of the one-dimensional complex Swift–Hohenberg equation,” Physica D 176(1-2), 44–66 (2003).
[CrossRef]

N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Stable soliton pairs in optical transmission lines and fiber lasers,” J. Opt. Soc. Am. B 15(2), 515–523 (1998).
[CrossRef]

Aranson, I. S.

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys. 74(1), 99–143 (2002).
[CrossRef]

Borckmans, P.

M. F. Hilali, S. Métens, P. Borckmans, and G. Dewel, “Pattern selection in the generalized Swift-Hohenberg model,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 51(3), 2046–2052 (1995).
[CrossRef] [PubMed]

Buceta, J.

J. Buceta, K. Lindenberg, and J. M. R. Parrondo, “Stationary and oscillatory spatial patterns induced by global periodic switching,” Phys. Rev. Lett. 88(2), 024103 (2002).
[CrossRef] [PubMed]

Burke, J.

J. Burke and E. Knobloch, “Localized states in the generalized Swift-Hohenberg equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(5 Pt 2), 056211 (2006).
[CrossRef] [PubMed]

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(7), 073904 (2006).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, “Soliton clusters in three-dimensional media with competing cubic and quintic nonlinearities,” J. Opt. B Quantum Semiclassical Opt. 6(5), S333–S340 (2004).
[CrossRef]

L.-C. Crasovan, Y. V. Kartashov, D. Mihalache, L. Torner, and V. M. Pérez-García, “Soliton “molecules”: robust clusters of spatiotemporal optical solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4 Pt 2), 046610 (2003).
[CrossRef] [PubMed]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg–Landau equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 63(1), 016605 (2001).
[CrossRef]

Devine, N.

Dewel, G.

M. F. Hilali, S. Métens, P. Borckmans, and G. Dewel, “Pattern selection in the generalized Swift-Hohenberg model,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 51(3), 2046–2052 (1995).
[CrossRef] [PubMed]

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 Stat. Nonlin. Soft Matter Phys. 74(1 Pt 2), 016611 (2006).
[CrossRef] [PubMed]

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 Stat. Nonlin. Soft Matter Phys. 74(1 Pt 2), 016611 (2006).
[CrossRef] [PubMed]

Firth, W. J.

A. G. Vladimirov, J. M. McSloy, D. V. Skryabin, and W. J. Firth, “Two-dimensional clusters of solitary structures in driven optical cavities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(44 Pt 2B), 046606 (2002).
[CrossRef] [PubMed]

Grelu, P.

He, Y. J.

Y. J. He, H. Z. Wang, and B. A. Malomed, “Fusion of necklace-ring patterns into vortex and fundamental solitons in dissipative media,” Opt. Express 15(26), 17502–17508 (2007).
[CrossRef] [PubMed]

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 Stat. Nonlin. Soft Matter Phys. 74(1 Pt 2), 016611 (2006).
[CrossRef] [PubMed]

Hilali, M. F.

M. F. Hilali, S. Métens, P. Borckmans, and G. Dewel, “Pattern selection in the generalized Swift-Hohenberg model,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 51(3), 2046–2052 (1995).
[CrossRef] [PubMed]

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(7), 073904 (2006).
[CrossRef] [PubMed]

L.-C. Crasovan, Y. V. Kartashov, D. Mihalache, L. Torner, and V. M. Pérez-García, “Soliton “molecules”: robust clusters of spatiotemporal optical solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4 Pt 2), 046610 (2003).
[CrossRef] [PubMed]

Knobloch, E.

J. Burke and E. Knobloch, “Localized states in the generalized Swift-Hohenberg equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(5 Pt 2), 056211 (2006).
[CrossRef] [PubMed]

Kramer, L.

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys. 74(1), 99–143 (2002).
[CrossRef]

Leblond, H.

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(4), 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(3), 033811 (2007).
[CrossRef]

Lederer, F.

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(3), 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(4), 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(7), 073904 (2006).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, “Soliton clusters in three-dimensional media with competing cubic and quintic nonlinearities,” J. Opt. B Quantum Semiclassical Opt. 6(5), S333–S340 (2004).
[CrossRef]

Lefever, R.

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73(5), 640–643 (1994).
[CrossRef] [PubMed]

Lega, J.

J. Lega, J. V. Moloney, and A. C. Newell, “Swift-Hohenberg equation for lasers,” Phys. Rev. Lett. 73(22), 2978–2981 (1994).
[CrossRef] [PubMed]

Lindenberg, K.

J. Buceta, K. Lindenberg, and J. M. R. Parrondo, “Stationary and oscillatory spatial patterns induced by global periodic switching,” Phys. Rev. Lett. 88(2), 024103 (2002).
[CrossRef] [PubMed]

Malomed, B. A.

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(3), 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(4), 045803 (2007).
[CrossRef]

Y. J. He, H. Z. Wang, and B. A. Malomed, “Fusion of necklace-ring patterns into vortex and fundamental solitons in dissipative media,” Opt. Express 15(26), 17502–17508 (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(7), 073904 (2006).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, “Soliton clusters in three-dimensional media with competing cubic and quintic nonlinearities,” J. Opt. B Quantum Semiclassical Opt. 6(5), S333–S340 (2004).
[CrossRef]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg–Landau equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 63(1), 016605 (2001).
[CrossRef]

Mandel, P.

M. Tlidi and P. Mandel, “Three-dimensional optical crystals and localized structures in cavity second harmonic generation,” Phys. Rev. Lett. 83(24), 4995–4998 (1999).
[CrossRef]

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73(5), 640–643 (1994).
[CrossRef] [PubMed]

Maruno, K.-I.

K.-I. Maruno, A. Ankiewicz, and N. Akhmediev, “Exact soliton solutions of the one-dimensional complex Swift–Hohenberg equation,” Physica D 176(1-2), 44–66 (2003).
[CrossRef]

Mazilu, D.

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(4), 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(3), 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(7), 073904 (2006).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, “Soliton clusters in three-dimensional media with competing cubic and quintic nonlinearities,” J. Opt. B Quantum Semiclassical Opt. 6(5), S333–S340 (2004).
[CrossRef]

McSloy, J. M.

A. G. Vladimirov, J. M. McSloy, D. V. Skryabin, and W. J. Firth, “Two-dimensional clusters of solitary structures in driven optical cavities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(44 Pt 2B), 046606 (2002).
[CrossRef] [PubMed]

Mejia-Cortés, C.

Métens, S.

M. F. Hilali, S. Métens, P. Borckmans, and G. Dewel, “Pattern selection in the generalized Swift-Hohenberg model,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 51(3), 2046–2052 (1995).
[CrossRef] [PubMed]

Mihalache, 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(3), 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(4), 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(7), 073904 (2006).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, “Soliton clusters in three-dimensional media with competing cubic and quintic nonlinearities,” J. Opt. B Quantum Semiclassical Opt. 6(5), S333–S340 (2004).
[CrossRef]

L.-C. Crasovan, Y. V. Kartashov, D. Mihalache, L. Torner, and V. M. Pérez-García, “Soliton “molecules”: robust clusters of spatiotemporal optical solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4 Pt 2), 046610 (2003).
[CrossRef] [PubMed]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg–Landau equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 63(1), 016605 (2001).
[CrossRef]

Moloney, J. V.

J. Lega, J. V. Moloney, and A. C. Newell, “Swift-Hohenberg equation for lasers,” Phys. Rev. Lett. 73(22), 2978–2981 (1994).
[CrossRef] [PubMed]

Newell, A. C.

J. Lega, J. V. Moloney, and A. C. Newell, “Swift-Hohenberg equation for lasers,” Phys. Rev. Lett. 73(22), 2978–2981 (1994).
[CrossRef] [PubMed]

Parrondo, J. M. R.

J. Buceta, K. Lindenberg, and J. M. R. Parrondo, “Stationary and oscillatory spatial patterns induced by global periodic switching,” Phys. Rev. Lett. 88(2), 024103 (2002).
[CrossRef] [PubMed]

Pérez-García, V. M.

L.-C. Crasovan, Y. V. Kartashov, D. Mihalache, L. Torner, and V. M. Pérez-García, “Soliton “molecules”: robust clusters of spatiotemporal optical solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4 Pt 2), 046610 (2003).
[CrossRef] [PubMed]

Sánchez-Morcillo, V. J.

V. J. Sánchez-Morcillo and K. Staliunas, “Stability of localized structures in the Swift-Hohenberg equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(55 Pt B), 6153–6156 (1999).
[CrossRef]

Sears, S.

M. Soljačić, S. Sears, and M. Segev, “Self-trapping of ‘necklace’ beams in self-focusing Kerr media,” Phys. Rev. Lett. 81(22), 4851–4854 (1998).
[CrossRef]

Segev, M.

M. Soljačić, S. Sears, and M. Segev, “Self-trapping of ‘necklace’ beams in self-focusing Kerr media,” Phys. Rev. Lett. 81(22), 4851–4854 (1998).
[CrossRef]

Skryabin, D. V.

A. G. Vladimirov, J. M. McSloy, D. V. Skryabin, and W. J. Firth, “Two-dimensional clusters of solitary structures in driven optical cavities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(44 Pt 2B), 046606 (2002).
[CrossRef] [PubMed]

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(4), 044101 (2002).
[CrossRef] [PubMed]

Soljacic, M.

M. Soljačić, S. Sears, and M. Segev, “Self-trapping of ‘necklace’ beams in self-focusing Kerr media,” Phys. Rev. Lett. 81(22), 4851–4854 (1998).
[CrossRef]

Soto-Crespo, J. M.

Staliunas, K.

V. J. Sánchez-Morcillo and K. Staliunas, “Stability of localized structures in the Swift-Hohenberg equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(55 Pt B), 6153–6156 (1999).
[CrossRef]

Tlidi, M.

M. Tlidi, “Three-dimensional crystals and localized structures in diffractive and dispersive nonlinear ring cavities,” J. Opt. B Quantum Semiclassical Opt. 2(3), 438–442 (2000).
[CrossRef]

M. Tlidi and P. Mandel, “Three-dimensional optical crystals and localized structures in cavity second harmonic generation,” Phys. Rev. Lett. 83(24), 4995–4998 (1999).
[CrossRef]

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73(5), 640–643 (1994).
[CrossRef] [PubMed]

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(7), 073904 (2006).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, “Soliton clusters in three-dimensional media with competing cubic and quintic nonlinearities,” J. Opt. B Quantum Semiclassical Opt. 6(5), S333–S340 (2004).
[CrossRef]

L.-C. Crasovan, Y. V. Kartashov, D. Mihalache, L. Torner, and V. M. Pérez-García, “Soliton “molecules”: robust clusters of spatiotemporal optical solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4 Pt 2), 046610 (2003).
[CrossRef] [PubMed]

Vladimirov, A. G.

A. G. Vladimirov, J. M. McSloy, D. V. Skryabin, and W. J. Firth, “Two-dimensional clusters of solitary structures in driven optical cavities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(44 Pt 2B), 046606 (2002).
[CrossRef] [PubMed]

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(4), 044101 (2002).
[CrossRef] [PubMed]

Wang, H. Z.

Y. J. He, H. Z. Wang, and B. A. Malomed, “Fusion of necklace-ring patterns into vortex and fundamental solitons in dissipative media,” Opt. Express 15(26), 17502–17508 (2007).
[CrossRef] [PubMed]

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 Stat. Nonlin. Soft Matter Phys. 74(1 Pt 2), 016611 (2006).
[CrossRef] [PubMed]

J. Opt. B Quantum Semiclassical Opt. (2)

M. Tlidi, “Three-dimensional crystals and localized structures in diffractive and dispersive nonlinear ring cavities,” J. Opt. B Quantum Semiclassical Opt. 2(3), 438–442 (2000).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, “Soliton clusters in three-dimensional media with competing cubic and quintic nonlinearities,” J. Opt. B Quantum Semiclassical Opt. 6(5), S333–S340 (2004).
[CrossRef]

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

Lect. Notes Phys. (1)

N. Akhmediev and A. Ankiewicz, “Dissipative solitons in the complex Ginzburg–Landau and Swift–Hohenberg equations,” Lect. Notes Phys. 661, 1–17 (2005).
[CrossRef]

Opt. Express (3)

Phys. Rev. A (2)

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(4), 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(3), 033811 (2007).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (5)

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 Stat. Nonlin. Soft Matter Phys. 74(1 Pt 2), 016611 (2006).
[CrossRef] [PubMed]

J. Burke and E. Knobloch, “Localized states in the generalized Swift-Hohenberg equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(5 Pt 2), 056211 (2006).
[CrossRef] [PubMed]

J. M. Soto-Crespo and N. Akhmediev, “Composite solitons and two-pulse generation in passively mode-locked lasers modeled by the complex quintic Swift-Hohenberg equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(6 Pt 2), 066610 (2002).
[CrossRef]

L.-C. Crasovan, Y. V. Kartashov, D. Mihalache, L. Torner, and V. M. Pérez-García, “Soliton “molecules”: robust clusters of spatiotemporal optical solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4 Pt 2), 046610 (2003).
[CrossRef] [PubMed]

A. G. Vladimirov, J. M. McSloy, D. V. Skryabin, and W. J. Firth, “Two-dimensional clusters of solitary structures in driven optical cavities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(44 Pt 2B), 046606 (2002).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (3)

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg–Landau equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 63(1), 016605 (2001).
[CrossRef]

M. F. Hilali, S. Métens, P. Borckmans, and G. Dewel, “Pattern selection in the generalized Swift-Hohenberg model,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 51(3), 2046–2052 (1995).
[CrossRef] [PubMed]

V. J. Sánchez-Morcillo and K. Staliunas, “Stability of localized structures in the Swift-Hohenberg equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(55 Pt B), 6153–6156 (1999).
[CrossRef]

Phys. Rev. Lett. (7)

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73(5), 640–643 (1994).
[CrossRef] [PubMed]

J. Lega, J. V. Moloney, and A. C. Newell, “Swift-Hohenberg equation for lasers,” Phys. Rev. Lett. 73(22), 2978–2981 (1994).
[CrossRef] [PubMed]

J. Buceta, K. Lindenberg, and J. M. R. Parrondo, “Stationary and oscillatory spatial patterns induced by global periodic switching,” Phys. Rev. Lett. 88(2), 024103 (2002).
[CrossRef] [PubMed]

M. Soljačić, S. Sears, and M. Segev, “Self-trapping of ‘necklace’ beams in self-focusing Kerr media,” Phys. Rev. Lett. 81(22), 4851–4854 (1998).
[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(7), 073904 (2006).
[CrossRef] [PubMed]

M. Tlidi and P. Mandel, “Three-dimensional optical crystals and localized structures in cavity second harmonic generation,” Phys. Rev. Lett. 83(24), 4995–4998 (1999).
[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(4), 044101 (2002).
[CrossRef] [PubMed]

Physica D (1)

K.-I. Maruno, A. Ankiewicz, and N. Akhmediev, “Exact soliton solutions of the one-dimensional complex Swift–Hohenberg equation,” Physica D 176(1-2), 44–66 (2003).
[CrossRef]

Rev. Mod. Phys. (1)

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys. 74(1), 99–143 (2002).
[CrossRef]

Other (3)

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

Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, (Springer, 1984).

Y. S. Kivshar, and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

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Figures (6)

Fig. 1
Fig. 1

Fusion into a fundamental soliton of the necklace array whose initial radius is not larger than R max, and M = N. (a) The largest radius, admitting the fusion, versus N. Examples of the fusion of the SNPs are given at the propagation distance (from left to right) z = 0, z = 34, z = 102, and z = 170 with R0 = 5 and M = N = 3 for (b) CGL and (c) CSH models [the examples (including the figures below) are shown by means of the isosurface plots of the power |u(X,Y,T)|2 ].

Fig. 2
Fig. 2

Fusion into a stable vortex soliton with topological charge 1. (a) The region of the initial radius admitting the fusion versus N; evolutions of fusion of the SNPs plotted at z = 0, z = 34, z = 102, and z = 170 with R0 = 6 and M = 4 for (b) CGL and (c) CSH models.

Fig. 3
Fig. 3

Fusion into a stable vortex soliton with topological charge 2. (a) The regions of initial radius admitting the fusion versus N; evolutions of fusion of the SNPs given at z = 0, z = 34, z = 102, and z = 170 with R0 = 6 and M = 3 for (b) CGL and (c) CSH models.

Fig. 4
Fig. 4

(a) Intersections based on energy F[u] = 0 (solid curves) and momentum J[u] = 0 (dotted curve); A and B indicate the bound states as a function the separation x 0, respectively, corresponding to the CSH (blue curve) and CGL (red curve) models. Note that J[u] = 0 occurs in the CSH and CGL models for bound states. (b) Minimum radius of the necklace achieved by simulations admits the keeping of the necklace-ring shape.

Fig. 5
Fig. 5

The SNPs keep necklace-ring shapes. Evolutions of the SNPs given at z = 0, z = 25, and z = 50 with N = 6, and R0 = 13.6 for (a) CGL model and (b) R0 = 12.5 for CSH model.

Fig. 6
Fig. 6

Switching off a “bead” on the necklace by radially phase modulating the “bead” in (a) CGL and (b) CSH models. The evolutions are plotted at z = 0, z = 25, and z = 50. Left column is the phase maps, and the arrow indicates the radial phase modulation with L = −0.5.

Equations (6)

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

iuZ+iδu+(1/2iβ)(uXX+uYY+uTT)+(1iε)|u|2u(νiμ)|u|4u=0,
iuZ+iδu+(1/2iβ)(uXX+uYY+uTT)+(1iε)|u|2u(νiμ)|u|4uis(uXXXX+uYYYY+uTTTT)=0.
u(Z=0,r,θ)=Asech[(rR0)/w]cos(Nθ)exp(iMθ+iLr2).
dE[u]dz=2{δ|u|2+ε|u|4μ|u|6β(|uX|2+|uY|2+|uT|2)             +s/2[u*(uXXXX+uXXXX+uXXXX)+u(uXXXX*+uYYYY*+uTTTT*)]}dXdYdTF[u],
dM[u]dz=i{(δ+ε|u|2μ|u|4)(uxu*ux*u)+β(uxuxx*ux*uxx)+s(uxuxxxx*ux*uxxxx)]}dxJ[u].
u(X,Y,T)=u0(Xx0/2,Y,T)exp(iπ)+u0(X+x0/2,Y,T) ,

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