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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 15 — Jul. 20, 2009
  • pp: 12203–12209
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Annularly and radially phase-modulated spatiotemporal necklace-ring patterns in the Ginzburg–Landau and Swift–Hohenberg equations

Bin Liu, Ying-Ji He, Zhi-Ren Qiu, and He-Zhou Wang  »View Author Affiliations


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

The complex Ginzburg–Landau (CGL) equation is an important model that occurs in many areas such as in superconductivity and superfluidity, fluid dynamics, reaction-diffusion phenomena, nonlinear optics, Bose–Einstein condensates, and quantum field theories [1

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

,2

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

]. Many dynamical behaviors are achieved in such a model, such as the formation of periodic patterns and dissipative solitons [1

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

,2

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

]. Complex stable patterns have also been investigated in dissipative models based on the CGL equation with cubic-quintic (CQ) nonlinearity, including stable vortices [3

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

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]

], stable soliton clusters [7

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

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]

], fusions of 2D necklace-ring patterns [9

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]

], stable spatiotemporal solitons [10

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

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]

], and stable spatiotemporal necklace-ring solitons [12

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]

].

Another dissipative system described by the complex Swift–Hohenberg (CSH) equation is derived by adding the four-order diffusion term to the CGL model [13

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

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]

]. The CSH model possesses stronger friction force than the CGL mode due to the presence of the higher-order term, which leads to some differences between them in optics. The CSH model also has been widely used to study various localized states, including the formation of complex patterns [16

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]

] and localized foundational patterns [13

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

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

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

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]

]. Some states of this type have been computed in this equation with quadratic and cubic nonlinearities [20

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

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]

]. The CGL and CSH equations are generic ones describing systems near subcritical bifurcations [22

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

].

Spatiotemporal solitons in optical media have attracted much attention [23

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

]. A spatiotemporal soliton is referred to as a “light bullet” localized in all spatial dimensions and in the time dimension. Recently stable spatiotemporal soliton clusters have been reported in Hamiltonian nonlinear systems [24

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

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]

] and in the Swift–Hohenberg model [26

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

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]

]. The generation of a “light bullet” is of importance in soliton-based communication systems, where each soliton represents a bit of information.

In this work we study spatiotemporal necklace-shaped patterns (SNPs) with annular or radial phase modulation in the 3D CGL and 3D CSH equations. We demonstrate that SNPs with annular phase modulation can fuse into stable fundamental or vortex solitons in both models when the initial radius of the necklace is smaller than a critical value, which is similar to the fusion of 2D necklace-shaped patterns into stable fundamental or vortex solitons in a CGL Eq. (9). We predict that SNPs keep the shape of a necklace by the balance equations of both energy and momentum when their radii exceed a critical value. We find that it is easier to implement the above fusions of solitons in the CSH model than in the CGL model due to the effect of the higher-order diffusion term. When a radial phase modulation is added to SNPs, the modulated “bead” will move towards or off the center of the necklace and rapidly vanish due to strong dissipation.

2. The model

We consider the (3 + 1)D CQ complex GL equation in a general form [4

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

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]

]:
iuZ+iδu+(1/2iβ)(uXX+uYY+uTT)+(1iε)|u|2u(νiμ)|u|4u=0,
(1)
where ν is the quintic self-defocusing coefficient, δ is the linear loss (δ<0) or gain (δ>0) coefficient, μ<0 is the quintic-loss parameter, ε>0 is the cubic-gain coefficient, andβ>0 accounts for effective diffusion (viscosity). The temporal viscosity of Eq. (1) is βuTT, accounting for the spectral filtering in optics, while the spatial-diffusion term β(uXX+uYY) is a feature known in specific models of laser cavities [17

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

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]

].

Following [28

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]

], the initial necklace-shaped pattern with amplitude A, mean radius R 0, and width w can be taken (in polar coordinates r and θ) as
u(Z=0,r,θ)=Asech[(rR0)/w]cos(Nθ)exp(iMθ+iLr2).
(3)
Here, integer N determines the number of elements (“beads”) in the ring structure, which is 2N, and M and L are coefficients of annular and radial phase modulation, respectively.

The generic case can be adequately represented for parameters δ = 0.5, β = 0.5, ν = 0.11, and m = 1 for both CGL and SCH models and ε = 2.52 (for CGL model) and ε = 2.47 (for CSH model), which corresponds to a physically realistic situation and, simultaneously, makes the evolution relatively fast, thus helping to elucidate its salient features [4

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

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

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

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]

]. The typical coefficient of the higher-order term is s = −0.1 [15

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]

,22

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

]. For these parameters, the amplitude and width of the individual 3D stable fundamental soliton are 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

When M0 and L=0, only the annular phase modulation is added to the SNPs. In this case, for 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)
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].
]. From Eq. (3), the mean phase shift between adjacent beads is Δϕ = Δϕ0 + 2πM/(2N) = 0, where Δϕ0 = -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.

When radius R 0 of the SNP is in a certain region [see Fig. 2(a)
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.
], 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
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.
). The asymptotic form of the vortex at r→ 0 is constrS (here S|MN|), which shows that topological charge 1 corresponds to the smaller radius of the inner hole in the vortex soliton than that of topological charge 2 [see Figs. 2(b), 2(c), 3(b), and 3(c)]. But, SNPs cannot evolve into stable vorticities with a larger topological charge than 2 because such vorticities were unstable for the CGL Eq. (4) as well as for the CSH equation in our simulations. The values of the radius of the SNPs in the CSH model are smaller than those in the CGL model from Figs. 2(a) and 3(a). This is the reason that if the radius of the SNP is too small, some solitons will fuse into a soliton as described in [9

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

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]

]. The viscidity of the CSH model is stronger than that of the CGL model due to its higher-order diffusion, which allows the SNP with a smaller radius to be transformed into the vortex soliton in the CSH mode.

If the initial radius exceeds a larger threshold value 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]

]:
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],
(4a)
dM[u]dz=i{(δ+ε|u|2μ|u|4)(uxu*ux*u)+β(uxuxx*ux*uxx)+s(uxuxxxx*ux*uxxxx)]}dxJ[u].
(4b)
By means of two-soliton perturbation theory, two solitons with out-of-phase difference are:
u(X,Y,T)=u0(Xx0/2,Y,T)exp(iπ)+u0(X+x0/2,Y,T),
(5)
where u0(X,Y,T) is the stable soliton solution and 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)
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.
, 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).

As a result, the pattern keeps its necklace-like structure and the initial radius, thus taking the form of a stable SNP (see 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.
). The dependence of the respective minimum radius 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.

Finally, we radially phase modulate the SNP with R 0 = 10 by setting L0 and 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
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.
, 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

We have studied SNPs with annular and radial phase modulating of necklaces in two kinds of models of dissipative optical media based on the 3D CGL and CSH equations with cubic-quintic nonlinearity. In the presence of annular phase modulation, the SNPs in both models can fuse into stable fundamental and vortice solitons provided that the initial radius of the ring 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

This work was supported by the National Natural Science Foundation of China (NNSFC) grants 10874250 and 10674183, National 973 Project of China grant 2004CB719804, Ph.D. Degrees Foundation of Ministry of Education of China grant 20060558068, and Introductory Programs for Science and Technology Development of Bureau of Science and Technology of Guangzhou Municipality grant 2005Z3-C7451. B. Liu and Y.-J. He equally contributed to this work.

References and links

1.

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

2.

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

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]

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]

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]

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. 66(6 Pt 2), 066610 (2002). [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]

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]

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]

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]

22.

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

23.

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

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]

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]

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]

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

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  2. B. A. Malomed, “Complex Ginzburg–Landau equation,” in Encyclopedia of Nonlinear Science, A. Scott, ed. (Routledge, 2005), pp. 157–160.
  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]
  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]
  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]
  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. 66(6 Pt 2), 066610 (2002). [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]
  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]
  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]
  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]
  22. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, (Springer, 1984).
  23. Y. S. Kivshar, and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).
  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]
  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]
  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]

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