## Annularly and radially phase-modulated spatiotemporal necklace-ring patterns in the Ginzburg–Landau and Swift–Hohenberg equations

Optics Express, Vol. 17, Issue 15, pp. 12203-12209 (2009)

http://dx.doi.org/10.1364/OE.17.012203

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### 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

## 1. Introduction

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

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

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]

6. 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]

7. 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]

8. 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]

9. 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]

10. 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]

11. 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]

12. 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]

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

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

16. 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]

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

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

17. 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]

19. 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]

20. 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]

21. 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]

24. 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]

25. 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]

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]

## 2. The model

4. 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]

12. 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]

*ν*is the quintic self-defocusing coefficient,

*δ*is the linear loss (

17. 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]

18. 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]

28. 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]

*A*, mean radius

*R*

_{0}, and width

*w*can be taken (in polar coordinates

*r*and

*θ*) asHere, integer

*N*determines the number of elements (“beads”) in the ring structure, which is 2

*N*, and

*M*and

*L*are coefficients of annular and radial phase modulation, respectively.

4. 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]

5. 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]

11. 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]

25. 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]

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

*A*= 1.2 and

*w*= 2.5. The robustness of the SNPs is additionally tested in direct simulations of Eqs. (1) and (2) by multiplying Eq. (3) with [1 + ρ(

*X,Y,T*)], where ρ(

*X,Y,T*) is a Gaussian random function whose maximum is 10% of the soliton’s amplitude.

## 3. Results and analysis

*M*=

*N*the SNPs evolve into a fundamental soliton when the initial radius

*R*

_{0}of the necklace is smaller than a maximum value,

*R*

_{max}[Fig. 1(a) ]. From Eq. (3), the mean phase shift between adjacent beads is Δϕ = Δϕ

_{0}+ 2π

*M*/(2

*N*) = 0, where Δϕ

_{0}= -π corresponds to the out-of-phase difference between adjacent “beads” on the necklace with

*M*= 0. Therefore, individual elements in the array, being in-phase, attract each other, which lead to their fusion into a stable, fundamental soliton in the CGL and CSH models, corresponding to Figs. 1(b) and 1(c), respectively. It is also easy to understand the increase of

*R*

_{max}with

*N*[see Fig. 1(a)]. Indeed, the attraction between adjacent “beads” necessary for the fusion into the fundamental soliton is not too weak if the separation between them does not exceed a maximum value. The radius of the necklace grows linearly with

*N*, which shows the roughly linear form of dependence

*R*

_{max}(

*N*) in Fig. 1(a). From Figs. 1(b) and 1(c), the maximum value

*R*

_{max}in the CSH model is larger than that in the CGL model. This is because the effect of the higher-order diffusion in the CSH model produces stronger viscous forces between “beads,” which leads to easy fusion of the SNPs into a fundamental soliton with an even larger initial radius.

*R*

_{0}of the SNP is in a certain region [see Fig. 2(a) ], the SNP with annular phase modulation evolves into a stable vortex soliton, provided that

*M*=

*N*-1 or

*M*=

*N*+ 1 (Fig. 2). The initial configuration in Eq. (3) may be realized as a vorticity whose absolute topological charge is 1, hence the vorticity component that may survive in the course of the evolution. Indeed, the simulations confirm that the emerging vortex solitons feature precisely this value of the vorticity. Similarly, the SNP evolves into a stable vortex soliton whose absolute topological charge is 2, provided that

*M*=

*N*-2 or

*M*=

*N*+ 2 see (Fig. 3 ). The asymptotic form of the vortex at r→ 0 is

9. 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]

12. 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]

*R*

_{min}, the interaction between the “beads” in the necklace array becomes very weak, irrespective of its topological charge

*M*. As follows, we can predict the size of the

*R*

_{min}for the bound SNPs by the balance equations of both energy and momentum. The CGL and CSH equations have no known conserved quantities. Instead, the rate of change of energy and momentum with respect to

*z*are [29

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]

*u*is the stable soliton solution and

_{0}(X,Y,T)*x*

_{0}is the minimum separation of two solitons. Thus, the corresponding solutions must satisfy the set of two Eqs. (29):

*F*[

*u*] = 0 and

*J*[

*u*] = 0. The former equation indicates the necessary balance that must exist between loss and gain for any stationary solution, while the latter equation guarantees the balance between the transverse forces acting on solitons. The stationary solution is indicated by intersections

*A*(CSH model) and

*B*(CGL model) in Fig. 4(a) , respectively, corresponding to the minimum separations

*x*

_{0}= 10.7 and

*x*

_{0}= 11.6. This result can achieve the minimum radius of the SNP for keeping the necklace shape by an approximate relation:

*x*

_{0}≈π

*R*

_{min}/

*N*, which is in agreement with the simulation results as shown in Fig. 4(b).

*R*

_{min}on modulation number

*N*of initial pattern in Eq. (3) is approximately linear [see Fig. 5(a)]. Note that each individual element in the established SNPs observed in Fig. 5 features an isotropic (circular) shape, unlike the “beads” in the initial pattern. This is explained by the fact that each “bead” evolves into a fundamental soliton.

*R*

_{0}= 10 by setting

*M*= 0 in Eq. (3). For

*L*< 0, the modulated “bead” moves toward the center of necklace. And if

*L*is up to a certain value, the moving “bead” rapidly disappears (decays to zero) upon propagation as shown in Fig. 6 , which is the result of the strong dissipative property of the models. The strong dissipation is generated by the diffusion term in Eqs. (1) and (2) (the one proportional to

*β*and

*s*) in the necklace array. Naturally, the diffusive dissipation is stronger for a larger radial phase gradient. Note that the CGL and CSH equations, being dissipative ones, do not have any dynamical invariant, hence the total momentum is not conserved.

## 4. Conclusions

*R*

_{0}is not too large. The key parameters that determine the results of the evolution are modulation number

*N*and topological charge

*M*of the initial necklace of Eq. (3). These results are similar the 2D cases in a CGL Eq. (9). When radial phase modulation is added to the SNP, the modulated “beads” will move toward or off the center of the necklace and quickly vanish due to the strong dissipation by which some “beads” can be switching off upon propagation. This offers a potential application to optical switching in signal processing.

## Acknowledgments

## References and links

1. | I. S. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys. |

2. | B. A. Malomed, “Complex Ginzburg–Landau equation,” in |

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 |

4. | 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. |

5. | 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 |

6. | J. M. Soto-Crespo, N. Akhmediev, C. Mejia-Cortés, and N. Devine, “Dissipative ring solitons with vorticity,” Opt. Express |

7. | 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. |

8. | D. V. Skryabin and A. G. Vladimirov, “Vortex induced rotation of clusters of localized states in the complex Ginzburg-Landau equation,” Phys. Rev. Lett. |

9. | 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 |

10. | J. M. Soto-Crespo, P. Grelu, and N. Akhmediev, “Optical bullets and ‘rockets’ in nonlinear dissipative systems and their transformations and interactions,” Opt. Express |

11. | 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 |

12. | 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. |

13. | J. Lega, J. V. Moloney, and A. C. Newell, “Swift-Hohenberg equation for lasers,” Phys. Rev. Lett. |

14. | 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. |

15. | N. Akhmediev and A. Ankiewicz, “Dissipative solitons in the complex Ginzburg–Landau and Swift–Hohenberg equations,” Lect. Notes Phys. |

16. | J. Buceta, K. Lindenberg, and J. M. R. Parrondo, “Stationary and oscillatory spatial patterns induced by global periodic switching,” Phys. Rev. Lett. |

17. | 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 |

18. | K.-I. Maruno, A. Ankiewicz, and N. Akhmediev, “Exact soliton solutions of the one-dimensional complex Swift–Hohenberg equation,” Physica D |

19. | J. Burke and E. Knobloch, “Localized states in the generalized Swift-Hohenberg equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

20. | M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. |

21. | 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 |

22. | Y. Kuramoto, |

23. | Y. S. Kivshar, and G. P. Agrawal, |

24. | 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. |

25. | 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. |

26. | M. Tlidi and P. Mandel, “Three-dimensional optical crystals and localized structures in cavity second harmonic generation,” Phys. Rev. Lett. |

27. | M. Tlidi, “Three-dimensional crystals and localized structures in diffractive and dispersive nonlinear ring cavities,” J. Opt. B Quantum Semiclassical Opt. |

28. | M. Soljačić, S. Sears, and M. Segev, “Self-trapping of ‘necklace’ beams in self-focusing Kerr media,” Phys. Rev. Lett. |

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 |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: April 9, 2009

Revised Manuscript: May 10, 2009

Manuscript Accepted: May 13, 2009

Published: July 6, 2009

**Citation**

Bin Liu, Ying-Ji He, Zhi-Ren Qiu, and He-Zhou Wang, "Annularly and radially phase-modulated spatiotemporal necklace-ring patterns in
the Ginzburg–Landau and Swift–Hohenberg equations," Opt. Express **17**, 12203-12209 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-12203

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### References

- I. S. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys. 74(1), 99–143 (2002). [CrossRef]
- B. A. Malomed, “Complex Ginzburg–Landau equation,” in Encyclopedia of Nonlinear Science, A. Scott, ed. (Routledge, 2005), pp. 157–160.
- 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]
- 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, 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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. Lega, J. V. Moloney, and A. C. Newell, “Swift-Hohenberg equation for lasers,” Phys. Rev. Lett. 73(22), 2978–2981 (1994). [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]
- N. Akhmediev and A. Ankiewicz, “Dissipative solitons in the complex Ginzburg–Landau and Swift–Hohenberg equations,” Lect. Notes Phys. 661, 1–17 (2005). [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]
- 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]
- 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. 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]
- 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]
- 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]
- 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).
- 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]
- 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]
- 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]
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