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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 5 — Mar. 11, 2013
  • pp: 5561–5566
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Continuous emission of fundamental solitons from vortices in dissipative media by a radial-azimuthal potential

Bin Liu, Xing-Dao He, and Shu-Jing Li  »View Author Affiliations


Optics Express, Vol. 21, Issue 5, pp. 5561-5566 (2013)
http://dx.doi.org/10.1364/OE.21.005561


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Abstract

We report novel dynamical regimes of dissipative vortices supported by a radial-azimuthal potential (RAP) in the 2D complex Ginzburg-Landau (CGL) equation with the cubic-quintic nonlinearity. First, the stable solutions of vortices with intrinsic vorticity S = 1 and 2 are obtained in the CGL equation without potential. The RAP is a model of an active optical medium with respective expanding anti-waveguiding structures with m (integer) annularly periodic modulation. If the potential is strong enough, m jets fundamental of solitons are continuously emitted from the vortices. The influence of m, diffusivity term (viscosity) β, and cubic-gain coefficient ε on the dynamic region is studied. For a weak potential, the shape of vortices are stretched into the polygon, such as square for m = 4. But for a stronger potential, the vortices will be broke into m fundamental solitons.

© 2013 OSA

1. Introduction

An optical vortex soliton is a self-localized nonlinear wave, who has a point (“singularity”) of zero intensity, and with a phase that twists around that point, with a total phase accumulation of 2πS for a closed circuit around the singularity [1

1. C. López-Mariscal and J. C. Gutiérrez-Vega, “In your phase: all about optical vortices,” Opt. Photonics News 20(5), 10–13 (2009).

]. The quantity S is an integer number known as vorticity or topological charge of the solution.

Complex Ginzburg-Landau (CGL) equations are well known as basic models of the pattern formation in various nonlinear dissipative media [2

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

-4

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

]. The CGL equation with the cubic-quintic (CQ) nonlinearity has been widely used in nonlinear dissipative optics, due to the clear physical meaning of all its terms in any particular application. Among the important application are passively mode-locked laser systems and optical transmission lines [5

5. 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), 515–522 (1998). [CrossRef]

]. The reports in this model have been focused on complex stable patterns [6

6. L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg-Landau equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(1 Pt 2), 016605 (2001). [PubMed]

-10

10. 10D. 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).

] and interactions of localized pulses [11

11. D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77(3), 033817 (2008). [CrossRef]

-13

13. B. Liu, X. D. He, and S. J. Li, “Phase controlling of collisions between solitons in the two-dimensional complex Ginzburg-Landau equation without viscosity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(5), 056607 (2011). [CrossRef] [PubMed]

]. In the framework of the CQ CGL equations in two dimensions, stable solitary vortices alias vortex solitons, with topological charge_(vorticity)_S = 1 and 2, were constructed in works [6

6. L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg-Landau equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(1 Pt 2), 016605 (2001). [PubMed]

], and their 3D counterparts for S = 1, 2, and 3 have been obtained in Refs. [14

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

, 15

15. 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).

]. Recently, adding external potentials in CGL models has been a theme of extensive studies, expanding the already wide spectrum of relevant application [16

16. H. Leblond, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the Ginzburg-Landau model of two-dimensional lasing medium with a transverse grating,” Phys. Rev. A 80(3), 033835 (2009). [CrossRef]

-27

27. D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, “Solitons in a medium with linear dissipation and localized gain,” Opt. Lett. 36(7), 1200–1202 (2011). [CrossRef] [PubMed]

]. Desirable patterns of the refractive-index modulation in materials described by CGL equation, which may induce the effective potentials, can be achieved by means of various techniques, such as optics induction [28

28. C. Cleff, B. Gütlich, and C. Denz, “Gradient Induced Motion Control of Drifting Solitary Structures in a Nonlinear Optical Single Feedback Experiment,” Phys. Rev. Lett. 100(23), 233902 (2008). [CrossRef] [PubMed]

] and writing patterns by streams of ultrashort laser pulses [29

29. A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica,” Opt. Express 14(13), 6055–6062 (2006). [CrossRef] [PubMed]

].

In this paper, we introduced the 2D CGL models with an external radial-azimuthal potential. We consider the action of these potentials on stable dissipative vortices initially placed at the central position (apex of the respective potential). The extra force of potential will break the original dynamical balance of central vortex. A novel dynamic that the central vortex continuously emits m (annular periods) jets fundamental solitons along symmetry directions of the potential is observed. The regions of strength of potential by variety of m and diffusivity term, and cubic-gain coefficient are analyzed.

2. The model

We consider the following 2D CQ CGL equation in terms of nonlinear optics, as the evolution equation for the amplitude of electromagnetic wave in an active bulk optical medium [10

10. 10D. 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).

, 19

19. 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(12), 1974–1976 (2010). [CrossRef] [PubMed]

]
iuz+iδu+(1/2iβ)(uxx+uyy)+(1iε)|u|2u(νiμ)|u|4u=F(x,y)u,
(1)
where (x, y) are the transverse coordinates and z is the propagation distance. The coefficients of diffraction and cubic self-focusing nonlinearity are scaled respectively, to be 1/2 and 1. ν is the quintic self-defocusing coefficient, δ is the coefficient corresponding to the linear loss (δ > 0) or gain (δ < 0), m > 0 accounts for the quintic-loss parameter, and ε > 0 is the cubic-gain coefficient. β > 0 is the spatial-diffusion term, appears in a model of laser cavities, where it is generated by the interplay of the dephasing of the local polarization in the dielectric medium, cavity loss, and detuning between the cavity’s and atomic frequencies [30

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

]. Generic results may be adequately represented by setting δ = 0.5, m = 1, ε = 2.7, β = 0.5, and ν = 0.115, which corresponds to a physically realistic situation and, simultaneously, makes the evolution relatively fast, thus helping to elucidate its salient features [12

12. B. Liu, Y. J. He, Z. R. Qiu, and H. Z. Wang, “Annularly and radially phase-modulated spatiotemporal necklace-ring patterns in the Ginzburg-Landau and Swift-Hohenberg equations,” Opt. Express 17(15), 12203–12209 (2009). [CrossRef] [PubMed]

, 18

18. 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(19), 2976–2978 (2009). [CrossRef] [PubMed]

]. The last term on the right-hand side of Eq. (1) introduces the RAP in the transverse plane. The analytical form of F(x,y) is
F(x,y)=ar|cos(mθ/2)|,r=x2+y2,m2
(2)
where θ=arctanx/y is the angular coordinate, and a is the strength of potential. m stands for integer periods in angular coordinate.

The initial solutions of vortex in Eq. (1) are set as the following form:
u(z=0,x,y)=A|r|Sexp[(x2+y2)w2]exp(iSθ),
(3)
where A is the amplitude, w is the width. We distinguish the vortex solutions with topological charge S, also called angular momentum quantum number. The stable solutions of vortex with S = 1 and 2 were obtained in the numerical form by image-step propagation method without potential is shown in Figs. 1(a)
Fig. 1 (a) The profile of a stable vortex with S = 1 in (x, y)-plane, (c) is the phase; (b) The profile of a stable vortex with S = 2, (d) is the phase.
and 1(b), the vorticity S could been observed by phase [shown Figs. 1(c) and 1(d)]. The vortices with S > 2 have been reported that cannot steadily propagate in Ref. [6

6. L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg-Landau equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(1 Pt 2), 016605 (2001). [PubMed]

]. A series dynamics of the vortices with S = 1 and 2 are studied by introducing the external potential F(x,y).

3. Results and analysis

First, we consider the dynamic of the vortices with S = 1, whose center placed at the apex of the RAP. For appropriate strength a, simulations of Eq. (1) reveal several typical dynamical regimes. A novel dynamical regime that m streams of secondary pulses are continuously emitted from the vortices is observed, as shown in Figs. 2(a)
Fig. 2 (a), (b), (c), and (d): Isosurface plots of total intensity |u(x,y)|2, evolutions of the central vortex with S = 1 at (a = 0.3, m = 4), (a = 0.3, m = 5) (a = 0.08, m = 4), and (a = 0.6, m = 4). (e) Region of a for continuous emission of fundamental solitons by the variety of m. (f) Evolutions of energy at m = 4 with a = 0.5 (blue line) and 0.3 (red line).
and 2(b) at (m = 4, a = 0.3) and (m = 5, a = 0.3). Then, the emitted pulses self-trap into fundamental solitons which slide along slopes of the potential. By performing a large number of numerical simulation, we found that the corresponding region of a reduces with the growth of m [shown in Fig. 2(e)]. In addition, we have calculated the evolution of total amount of energy E that it carries,
E(z)=|u(x,y)|2dxdy,
(4)
as a function of propagation distance z. The evolutions of E [shown in Fig. 2(f)] reveal an obviously periodic emission. z1 and z2 stand for the propagation distance of one time emission at a = 0.5 and 0.3, respectively. The comparison of them demonstrates that the stronger potential provides for a higher emission rate.

For a weaker potential, jets of fundamental solitons are not emitted from the vortices. Instead, the circular vortex is gradually stretched into polygonal vortex, and the stretching force increases with a. The evolution of a quadrate vortex is shown in Fig. 2(c) at m = 4 and a = 0.12.

But for a stronger potential, the central vortices are broken into m fundamental solitons which slide along slopes of the potential, see a typical example in Fig. 2(d) with m = 4 and a = 0.6.

Next, we consider the potential act on the vortices with S = 2. The similarly dynamical regimes is observed for appropriate strength a. Figure 3(a)
Fig. 3 (a), (b), and (c) Isosurface plots of total intensity |u(x,y)|2, evolutions of the central vortex with S = 2 at (a = 0.12, m = 8), (a = 0.3, m = 8), and (a = 0.4, m = 8). (d) Region of a for continuous emission of fundamental solitons by the variety of m. (e) Evolutions of the energy E at m = 8 for a = 0.25 (blue line) and 0.3 (red line).
shows a typical example that 8 fundamental solitons are continuously emitted from the vortices (S = 2) at m = 8 and a = 0.25. The corresponding region of a also reduces with the growth of m [shown in Fig. 3(d)]. Figure 3(e) show the evolutions of total amount of energy E. By comparing z1 and z2 which stand for the propagation distance of one period at a = 0.3 and 0.25, it also reveal that the stronger potential provides for a higher emission rate. For a weaker potential, the circular vortex is also gradually stretch into a polygonal one whose size also increases with a. the evolution of an octagonal vortex is shown in Fig. 3(b) at m = 8 and a = 0.1. For a stronger potential, the central vortex is also broken into m fundamental solitons, as shown in Fig. 2(c) at m = 8 and a = 0.4.

The CQ CGL model is a dissipative system. The dissipative solitons or vortices should satisfy the balance between energy gain and loss. So we consider the influence of gain or loss coefficient on the novel dynamic regime of continuous emission of fundamental solitons from vortices. Figures 4(a)
Fig. 4 (a) and (b): Regions of a for continuous emission of fundamental solitons by the variety of ε at m = 4. (c) and (d): Regions of a for continuous emission by the variety of β at m = 4.
and 4(b) are show the regions of vortex dynamics with S = 1 and 2 by varying the cubic-gain coefficient ε. We can observed that the region of a obviously increase with the growth of ε. The region with S = 2 is larger than S = 1 under the same parameters. Furthermore, there is no appropriate strength a for attaining continuous emission, when ε<2.55 with S = 1 and ε<2.36 with S = 2. But the change of ε should be limited in a certain range. Otherwise the original vortex and generating fundamental soliton will become unstable.

In addition, the viscosity term in Eq. (1), ~β, plays an important role in maintaining the position of the central vortices. With little or zero viscosity, the vortices cannot steadily exist in the CQ CGL model. So, we also study the influence of β on the regime of continuous emission. The gray region in Figs. 4(c) and 4(d) show the regions of a depending on β with S = 1 and 2, respectively. There are just small changes with the variety of β.

4. Conclusions

We have introduced a radial-azimuthal potential into 2D CGL equation with the CQ nonlinearity. For an appropriate strength of potential, the setting gives rise to the continuous emission of arrays of fundamental solitons from the stable vortices with S = 1 and 2. The regions of a (strength of potential) decrease with the growth of annular period m or cubic-gain coefficient ε. But, the influence of variety of diffusivity term β on is not very obvious. The rate of emission increases with the strength of potential. If the potential is weak, stretch of the circular vortex into a polygonal is observed instead. For a stronger potential, the central vortex is broken into m fundamental solitons.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 61205119, 41066001 and 11104128) and the Doctor Startup Foundation of Nanchang Hangkong University (Grant No. EA201008231).

References and links

1.

C. López-Mariscal and J. C. Gutiérrez-Vega, “In your phase: all about optical vortices,” Opt. Photonics News 20(5), 10–13 (2009).

2.

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

3.

N. Rosanov, “Solitons in laser systems with absorption,” in Dissipative Solitons, N. Akhmediev and A. Ankievicz, eds. (Springer-Verlag, Berlin, 2005).

4.

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

5.

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), 515–522 (1998). [CrossRef]

6.

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg-Landau equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(1 Pt 2), 016605 (2001). [PubMed]

7.

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]

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.

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]

10.

10D. 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).

11.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77(3), 033817 (2008). [CrossRef]

12.

B. Liu, Y. J. He, Z. R. Qiu, and H. Z. Wang, “Annularly and radially phase-modulated spatiotemporal necklace-ring patterns in the Ginzburg-Landau and Swift-Hohenberg equations,” Opt. Express 17(15), 12203–12209 (2009). [CrossRef] [PubMed]

13.

B. Liu, X. D. He, and S. J. Li, “Phase controlling of collisions between solitons in the two-dimensional complex Ginzburg-Landau equation without viscosity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(5), 056607 (2011). [CrossRef] [PubMed]

14.

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

15.

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).

16.

H. Leblond, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the Ginzburg-Landau model of two-dimensional lasing medium with a transverse grating,” Phys. Rev. A 80(3), 033835 (2009). [CrossRef]

17.

H. Sakaguchi and B. A. Malomed, “Two-dimensional dissipative gap solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(2), 026606 (2009). [CrossRef] [PubMed]

18.

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(19), 2976–2978 (2009). [CrossRef] [PubMed]

19.

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(12), 1974–1976 (2010). [CrossRef] [PubMed]

20.

D. Mihalache, “Three-dimensional Ginzburg-Landau dissipative solitons supported by a two-dimensional transverse grating,” Proc. Romanian Acad. Ser. A 11(2), 142–147 (2010).

21.

B. Liu and X. D. He, “Continuous generation of “light bullets” in dissipative media by an annularly periodic potential,” Opt. Express 19(21), 20009–20014 (2011). [CrossRef] [PubMed]

22.

Y. J. He, B. A. Malomed, F. Ye, J. Dong, Z. Qiu, H. Z. Wang, and B. Hu, “Splitting broad beams into arrays of dissipative satial solitns by material and virtual gratings,” Phys. Scr. 82(6), 065404 (2010). [CrossRef]

23.

C. P. Yin, D. Mihalache, and Y. J. He, “Dynamics of two-dimensional dissipative spatial solitons interacting with an umbrella-shaped potential,” J. Opt. Soc. Am. B 28(2), 342–346 (2011).

24.

Y. J. He, D. Mihalache, B. A. Malomed, Y. Qiu, Z. Chen, and Y. Li, “Generation of polygonal soliton clusters and fundamental solitons in dissipative systems by necklace-ring beams with radial-azimuthal phase modulation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(6), 066206 (2012). [CrossRef] [PubMed]

25.

D. Mihalache, D. Mazilu, V. Skarka, B. Malomed, H. Leblond, N. Aleksić, and F. Lederer, “Stable topological modes in two-dimensional Ginzburg-Landau models with trapping potentials,” Phys. Rev. A 82(2), 023813 (2010). [CrossRef]

26.

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, “Varieties of Stable Vortical Solitons in Ginzburg-Landau Media with Radially Inhomogeneous Losses,” Phys. Rev. Lett. 105(21), 213901 (2010). [CrossRef] [PubMed]

27.

D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, “Solitons in a medium with linear dissipation and localized gain,” Opt. Lett. 36(7), 1200–1202 (2011). [CrossRef] [PubMed]

28.

C. Cleff, B. Gütlich, and C. Denz, “Gradient Induced Motion Control of Drifting Solitary Structures in a Nonlinear Optical Single Feedback Experiment,” Phys. Rev. Lett. 100(23), 233902 (2008). [CrossRef] [PubMed]

29.

A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica,” Opt. Express 14(13), 6055–6062 (2006). [CrossRef] [PubMed]

30.

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

OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

ToC Category:
Nonlinear Optics

History
Original Manuscript: December 4, 2012
Revised Manuscript: January 14, 2013
Manuscript Accepted: January 14, 2013
Published: February 27, 2013

Citation
Bin Liu, Xing-Dao He, and Shu-Jing Li, "Continuous emission of fundamental solitons from vortices in dissipative media by a radial-azimuthal potential," Opt. Express 21, 5561-5566 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-5561


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References

  1. C. López-Mariscal and J. C. Gutiérrez-Vega, “In your phase: all about optical vortices,” Opt. Photonics News20(5), 10–13 (2009).
  2. I. S. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys.74(1), 99–143 (2002). [CrossRef]
  3. N. Rosanov, “Solitons in laser systems with absorption,” in Dissipative Solitons, N. Akhmediev and A. Ankievicz, eds. (Springer-Verlag, Berlin, 2005).
  4. B. A. Malomed, “Complex Ginzburg–Landau equation,” in Encyclopedia of Nonlinear Science, A. Scott, ed. (Routledge, New York, 2005), p. 157.
  5. N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Stable soliton pairs in optical transmission lines and fiber lasers,” J. Opt. Soc. Am. B15(2), 515–522 (1998). [CrossRef]
  6. L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg-Landau equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.63(1 Pt 2), 016605 (2001). [PubMed]
  7. J. M. Soto-Crespo, N. Akhmediev, C. Mejia-Cortés, and N. Devine, “Dissipative ring solitons with vorticity,” Opt. Express17(6), 4236–4250 (2009). [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. J. M. Soto-Crespo, P. Grelu, and N. Akhmediev, “Optical bullets and “rockets” in nonlinear dissipative systems and their transformations and interactions,” Opt. Express14(9), 4013–4025 (2006). [CrossRef] [PubMed]
  10. 10D. 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. A75, 033811 (2007).
  11. D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A77(3), 033817 (2008). [CrossRef]
  12. B. Liu, Y. J. He, Z. R. Qiu, and H. Z. Wang, “Annularly and radially phase-modulated spatiotemporal necklace-ring patterns in the Ginzburg-Landau and Swift-Hohenberg equations,” Opt. Express17(15), 12203–12209 (2009). [CrossRef] [PubMed]
  13. B. Liu, X. D. He, and S. J. Li, “Phase controlling of collisions between solitons in the two-dimensional complex Ginzburg-Landau equation without viscosity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.84(5), 056607 (2011). [CrossRef] [PubMed]
  14. D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stablevotex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. Lett.97, 073904 (2006).
  15. 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. A76, 045803 (2007).
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