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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 16 — Aug. 2, 2010
  • pp: 17053–17058
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Stability conditions for moving dissipative solitons in one- and multidimensional systems with a linear potential

Wei- Ling Zhu and Ying-Ji He  »View Author Affiliations


Optics Express, Vol. 18, Issue 16, pp. 17053-17058 (2010)
http://dx.doi.org/10.1364/OE.18.017053


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Abstract

We analyze stability of moving dissipative solitons in the one-, two, and three-dimensional cubic-quintic complex Ginzburg-Landau equations in the presence of a linear potential (linear refractive index modulation). The expressions of stability conditions and propagation trajectory of solitons are derived by means of a generalized variational approximation. Predictions of the variational analysis are fully confirmed by direct numerical simulations. The results have potential applications to using spatial dissipative solitons in optics as individually addressable and shift registers of the all-optical data processing systems.

© 2010 OSA

1. Introduction

2. The model

We consider the CQ CGL equation of the general form which describes the propagation of an electromagnetic field with local amplitude u in an optical medium [19

19. N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: from stationary light bullets to pulsating complexes,” Chaos 17(3), 037112 (2007). [CrossRef] [PubMed]

,23

23. D. Mihalache and D. Mazilu, “Ginzburg-Landau spatiotemporal dissipative optical solitons,” Rom. Rep. Phys. 60, 749–761 (2008).

,24

24. V. Skarka and N. B. Aleksić, “Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimensional complex cubic-quintic Ginzburg-Landau equations,” Phys. Rev. Lett. 96(1), 013903 (2006). [CrossRef] [PubMed]

]:
iuz+(1/2)Δu+|u|2u+ν|u|4u=iR[u]αxu,
(1)
where Δ=2/x2+2/y2+2/t2 is the transverse Laplacian (x and y are the transverse coordinates, and t is the temporal variable, while z is the propagation distance). Further, ν accounts for the quintic self-defocusing, and R[u]=δu+βΔu+ε|u|2u+μ|u|4u, where δ is the coefficient of the linear loss (δ <0) or gain (δ >0), μ is the quintic-loss parameter, ε is the cubic-gain coefficient, and β denotes effective diffusion (viscosity). The last term in Eq. (1) is the LRIM, i.e., an effective linear potential that may be implemented via the local modulation of the refractive index in the transverse plane. The LRIM has been proposed as a tool for the controls of soliton dynamics in Kerr medium [25

25. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton percolation in random optical lattices,” Opt. Express 15(19), 12409–12417 (2007). [CrossRef] [PubMed]

] and soliton interaction in dissipative system [26

26. 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]

].

3. The stability of dissipative solitons within the framework of the VA

To analyze the stability of the moving dissipative solitons with a linear potential, we use the following isotropic ansatz in the general 3D case (cf. Refs [24

24. V. Skarka and N. B. Aleksić, “Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimensional complex cubic-quintic Ginzburg-Landau equations,” Phys. Rev. Lett. 96(1), 013903 (2006). [CrossRef] [PubMed]

,26

26. 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]

].):
u=A(z)exp{[xq(z)]2+y2+t22w2(z)+ic(z)[(xq(z))2+y2+t2]+iL(z)[xq(z)]+iψ(z)},
(2)
and its straightforward reductions in the 2D and 1D cases. Here, A, w, c, and ψ represent the amplitude, width, wave front curvature (alias chirp), and overall phase, respectively. Peak position q(z) and conjugate momentum L(z) account for the motion of the soliton along the direction of x, which results from the combination of the LRIM potential and friction force induced by the transverse-viscosity term in Eq. (1).

Firstly, we use the generalized VA to solve the CGL equation, as proposed in Refs [24

24. V. Skarka and N. B. Aleksić, “Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimensional complex cubic-quintic Ginzburg-Landau equations,” Phys. Rev. Lett. 96(1), 013903 (2006). [CrossRef] [PubMed]

,27

27. A. Kamagate, Ph. Grelu, P. Tchofo-Dinda, J. M. Soto-Crespo, and N. Akhmediev, “Stationary and pulsating dissipative light bullets from a collective variable approach,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(2), 026609 (2009). [CrossRef] [PubMed]

]. Using ansatz (2), after a straightforward algebra the following system of evolution equations for its parameters is derived in the D-dimensional case (D = 1, 2, 3):
dA/dz=[δ+2(2+D/2)(4+D)εA2+3(1+D/2)(3+D)μA4Dβw2DcβL2]A,
(3a)
dw/dz=[2(1+D/2)εA223(1+D/2)μA4+βw24βc2w2+2c]w,
(3b)
dc/dz=(2w4)12(1+D/2)A2w223(1+D/2)νA4w24βcw22c2,
(3c)
dL/dz=α2βLw2(1+4c2w4),
(3d)
dψ/dz=2DβcD(2w2)12(2+D/2)(4+D)A2+3(1+D/2)(3+D)νA4+αq+L2/24βcL2w2,
(3e)
dq/dz=L4βcLw2.
(3f)
Introducing small perturbations about a fixed point of Eqs. (3), A = A 0 + ∆α, w = w 0 + ∆w, c = c0 + ∆c, L = L0 + ∆L, we derive the respective linearized equations:
d(Δa)/dz=b1Δa+b2Δw+b3Δc+b4ΔL,
(4a)
d(Δw)/dz=b5Δa+b6Δr+b7Δc,
(4b)
d(Δc)/dz=b8Δa+b9Δr+b10Δc,
(4c)
d(ΔL)/dz=b11Δw+b12Δc+b13ΔL,
(4d)
where we defineb1=2(1+D/2)(4+D)εA02+43(1+D/2)(3+D)μA04, b2=2A0Dβw03, b3=DA0, b4=2βA0L0, b5=2D/2εA0w083(1+D/2)μA03w0, b6=(2βw02+8βc02w02), b7=2w08c0w03β, b8=A0(2D/2w02)18vA03(31+D/2w02)1, b9=2w05+8βc0w03+A02(2D/2w03)1+4νA04(3(1+D/2w03)1, b10=(4c0+4β/w02), b11=4βL0w0316βc02w0L0, b12=16βc0w02L0, b13=(2βw02+8βc02w02). Solutions to Eqs. (4) are looked for with the z-dependence in the form of exp(λz), where the eigenvalues are determined by the fourth-order equation with real coefficients,
|b1λb2b3b4b5b6λb70b8b9b10λ00b11b12b13λ|=0.
(5)
The fixed-point solution to Eqs. (4) is stable when the roots of Eq. (5) have negative or zero real parts. Rewriting the above equation in the form of λ4 + a 1λ3 + a 2λ2 + a 3λ + a 4 = 0, the stability condition amounts to the Routh-Hurwitz criterion, i.e., the coefficients a 1,2,3 must satisfy four inequalities:
a1>0,|a11a3a2|   >0,|a110a3a2a10a4a3|   >0,|a1100a3a2a110a4a3a2000a4|   >0,
(6)
where a1=(b1+b6+b10+b13), a2=b2b5b3b8b7b9+b6b10+b6b13+b10b13+b1(b6+b10+b13), a3=b1(b7b9b6b10)b4(b5b11+b8b12)b13[b1b6b7b9+b10(b1+b6)]+b3[b8(b6+b13)b5b9]+b2[b5(b10+b13)b7b8],
a4=b4[b7b8b11+b6b8b12+b5(b10b11b9b12)]+b13[b3(b5b9b6b8)+b2(b7b8b5b10)+b1(b6b10b7b9)].
Equation (3)d) relates the linear momentum L to the slope of the LRIM α, diffusivity β and parameters of the soliton, L0=αw02/[2β(1+4c02w04)]. Substituting L0=αw02/[2β(1+4c02w04)] into Eq. (3)f), one gets the expression of the peak position of soliton q(z):
q(z)=αw02(14βc0w02)/[2β(1+4c02w04)]z.
(7)
From Eq. (7), the peak position q(z) linearly changes with the propagation distance z when fixing the slope of the LRIM α and the parameters of soliton, which is verified by the numerical simulations below, see Figs. 2(a)
Fig. 2 Numerical simulation for moving 1D dissipative soliton. (a) Stable propagation of soliton with α = 0.1 and (b) soliton decay with α = 0.3.
, 3(a)
Fig. 3 Numerical simulation for moving 2D dissipative soliton in (a) Stable propagation of soliton with α = 0.2 and (b) soliton decay with α = 0.45.
, and 4(a)
Fig. 4 Numerical simulation for moving 3D dissipative soliton in (a) Stable propagation of soliton with α = 0.04 and (b) soliton decay with α = 0.1. In (a) and (b), the isosurfaces |u|=0.8.
. This is the consequence that the transverse push forces from the linear potential and the friction force induced by the effective diffusion (with β term in Eq. (1)) reach to balance. If without the effective diffusion, β = 0, from Eqs. (3d) and (3c), obviously, dL/dz = α and dq/dz = L, and thus q(z) ~αz2, which exhibits quadratic increase with z, similar to the moving soliton in the conservative system [25

25. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton percolation in random optical lattices,” Opt. Express 15(19), 12409–12417 (2007). [CrossRef] [PubMed]

]. Using Eqs. (3a)(3d), we find A0, w0, c0, and L0 for given α, and substitute the results into inequalities (6). Solving them in a numerical form, we thus identify the largest value of the slope, α = αcr, up to which (at α≤αcr) the solitons are stable.

4. Results and analysis

The generic case can be adequately represented by the choice of parameters δ = −0.5, β = 0.5, μ = −1, and for the 1D case: ε = 1.85 and ν = −0.115; for the 2D case: ε = 2.4 and ν = −0.01; and for the 3D case: ε = 2.5 and ν = −0.115. For all cases, the stable soliton solutions were obtained in the numerical form, see Figs. 1(a)
Fig. 1 Profiles of solitons for 1D, 2D, and 3D cases, respectively, in (a), (b), and (c).
, 1(b), and 1(c). The robustness of the stable solitons is additionally tested in direct simulations 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.

In 1D case, the largest slope of the linear potential at conditions (6), which was derived by means of the VA, admits a stable single-soliton solution at αcr ≈0.24. On the other hand, direct simulations of Eq. (1) yield αcr ≈0.23, also indicated in Ref [26

26. 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]

]. By using the same method, in 2D case, the largest slope of the linear potential is αcr ≈0.4, and direct simulations of Eq. (1) obtain αcr ≈0.41 [26

26. 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]

], while in 3D case, the largest slope of the linear potential is αcr ≈0.08, and direct simulations of Eq. (1) find αcr ≈0.082 [26

26. 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]

]. All these cases, the largest slope of the linear potential induced by numerical simulation demonstrates the accuracy provided by the VA. The numerical examples are shown in Figs. 2(a) and 2(b), Figs. 3(a) and 3(b), Figs. 4(a) and 4(b), respectively, for the 1D, 2D, and 3D cases. It is clearly demonstrated that soliton can maintain stable propagation when the slope of linear potential is smaller than the critical value [Figs. 2(a), 3(a), and 4(a)]. On contrary, for the slope of linear potential is larger than the critical value, soliton will disappear due to loss large energy [Figs. 2(b), 3(b), and 4(b)], because that stronger linear modulation pushes soliton to move faster, and the effective diffusion of model (1) leads to larger friction force when the soliton is drifted faster. In addition, the stable moving solitons exhibit linearly transverse drift, as above analytical result given by Eq. (7).

5. Conclusions

In this work, we have studied the stability of isolated dissipative solitons in the 1D, 2D, and 3D versions of the complex Ginzburg-Landau equation with cubic-quintic nonlinearity. A key element of the model is the linear potential term, which represents the linear refractive-index modulation effect in laser-cavity models. The existence and stability of single solitons was studied using the generalized version of the variational approximation based on the Gaussian ansatz. In particular, the largest value of the slope of the linear potential was identified, below which the solitons are stable. Predictions of the variational approximation are well corroborated by direct numerical results. The results suggest a simple method allowing one to stabilize and control one- and multidimensional moving dissipative solitons, with potential applications to the all-optical data processing systems.

Acknowledgment

This work was supported by Guangdong Province Natural Science Foundation of China (Grant No. 9451063301003516).

References and links

1.

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

2.

G. I. Stegeman, D. N. Christodoulides, and M. Segev, “Optical spatial solitons: historical perspectives,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1419–1427 (2000). [CrossRef]

3.

N. N. Akhmediev, and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, London, 1997).

4.

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B Quantum Semiclassical Opt. 7(5), R53–R72 (2005). [CrossRef]

5.

W. J. Firth, in Self-Organization in Optical Systems and Applications in Information Technology, edited by M. A. Vorontsov and W. B. Miller (Springer-Verlag, Berlin, 1995), p. 69.

6.

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286(5444), 1518–1523 (1999). [CrossRef] [PubMed]

7.

W. J. Firth and A. J. Scroggie, “Optical bullet holes: Robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76(10), 1623–1626 (1996). [CrossRef] [PubMed]

8.

L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, “Spatial solitons in semiconductor microcavities,” Phys. Rev. A 58(3), 2542–2559 (1998). [CrossRef]

9.

F. Pedaci, S. Barland, E. Caboche, P. Genevet, M. Giudici, J. R. Tredicce, T. Ackemann, A. J. Scroggie, W. J. Firth, G.-L. Oppo, G. Tissoni, and R. Jäger, “All-optical delay line using semiconductor cavity solitons,” Appl. Phys. Lett. 92(1), 011101–011103 (2008). [CrossRef]

10.

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]

11.

P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Bistable and addressable localized vortices in semiconductor lasers,” Phys. Rev. Lett. 104(22), 223902 (2010). [CrossRef] [PubMed]

12.

M. L. van Hecke, E. de Wit, and W. van Saarloos, “Coherent and incoherent drifting pulse dynamics in a complex Ginzburg-Landau equation,” Phys. Rev. Lett. 75(21), 3830–3833 (1995). [CrossRef] [PubMed]

13.

N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(1), 1190–1201 (1996). [CrossRef] [PubMed]

14.

N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 79(21), 4047–4051 (1997). [CrossRef]

15.

J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, “Pulsating, creeping, and erupting solitons in dissipative systems,” Phys. Rev. Lett. 85(14), 2937–2940 (2000). [CrossRef] [PubMed]

16.

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

17.

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

18.

B. A. Malomed, “Solitary pulses in linearly coupled Ginzburg-Landau equations,” Chaos 17(3), 037117 (2007). [CrossRef] [PubMed]

19.

N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: from stationary light bullets to pulsating complexes,” Chaos 17(3), 037112 (2007). [CrossRef] [PubMed]

20.

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]

21.

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]

22.

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]

23.

D. Mihalache and D. Mazilu, “Ginzburg-Landau spatiotemporal dissipative optical solitons,” Rom. Rep. Phys. 60, 749–761 (2008).

24.

V. Skarka and N. B. Aleksić, “Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimensional complex cubic-quintic Ginzburg-Landau equations,” Phys. Rev. Lett. 96(1), 013903 (2006). [CrossRef] [PubMed]

25.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton percolation in random optical lattices,” Opt. Express 15(19), 12409–12417 (2007). [CrossRef] [PubMed]

26.

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]

27.

A. Kamagate, Ph. Grelu, P. Tchofo-Dinda, J. M. Soto-Crespo, and N. Akhmediev, “Stationary and pulsating dissipative light bullets from a collective variable approach,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(2), 026609 (2009). [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: June 10, 2010
Revised Manuscript: July 14, 2010
Manuscript Accepted: July 15, 2010
Published: July 27, 2010

Citation
Wei- Ling Zhu and Ying-Ji He, "Stability conditions for moving dissipative solitons in one- and multidimensional systems with a linear potential," Opt. Express 18, 17053-17058 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-17053


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References

  1. Y. S. Kivshar, and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).
  2. G. I. Stegeman, D. N. Christodoulides, and M. Segev, “Optical spatial solitons: historical perspectives,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1419–1427 (2000). [CrossRef]
  3. N. N. Akhmediev, and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, London, 1997).
  4. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B Quantum Semiclassical Opt. 7(5), R53–R72 (2005). [CrossRef]
  5. W. J. Firth, in Self-Organization in Optical Systems and Applications in Information Technology, edited by M. A. Vorontsov and W. B. Miller (Springer-Verlag, Berlin, 1995), p. 69.
  6. G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286(5444), 1518–1523 (1999). [CrossRef] [PubMed]
  7. W. J. Firth and A. J. Scroggie, “Optical bullet holes: Robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76(10), 1623–1626 (1996). [CrossRef] [PubMed]
  8. L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, “Spatial solitons in semiconductor microcavities,” Phys. Rev. A 58(3), 2542–2559 (1998). [CrossRef]
  9. F. Pedaci, S. Barland, E. Caboche, P. Genevet, M. Giudici, J. R. Tredicce, T. Ackemann, A. J. Scroggie, W. J. Firth, G.-L. Oppo, G. Tissoni, and R. Jäger, “All-optical delay line using semiconductor cavity solitons,” Appl. Phys. Lett. 92(1), 011101–011103 (2008). [CrossRef]
  10. 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]
  11. P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Bistable and addressable localized vortices in semiconductor lasers,” Phys. Rev. Lett. 104(22), 223902 (2010). [CrossRef] [PubMed]
  12. M. L. van Hecke, E. de Wit, and W. van Saarloos, “Coherent and incoherent drifting pulse dynamics in a complex Ginzburg-Landau equation,” Phys. Rev. Lett. 75(21), 3830–3833 (1995). [CrossRef] [PubMed]
  13. N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(1), 1190–1201 (1996). [CrossRef] [PubMed]
  14. N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 79(21), 4047–4051 (1997). [CrossRef]
  15. J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, “Pulsating, creeping, and erupting solitons in dissipative systems,” Phys. Rev. Lett. 85(14), 2937–2940 (2000). [CrossRef] [PubMed]
  16. I. S. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys. 74(1), 99–143 (2002). [CrossRef]
  17. B. A. Malomed, “Complex Ginzburg-Landau equation,” in: Encyclopedia of Nonlinear Science, A. Scott, ed., (Routledge, New York, 2005) pp. 157–160.
  18. B. A. Malomed, “Solitary pulses in linearly coupled Ginzburg-Landau equations,” Chaos 17(3), 037117 (2007). [CrossRef] [PubMed]
  19. N. Akhmediev, J. M. Soto-Crespo, and P. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: from stationary light bullets to pulsating complexes,” Chaos 17(3), 037112 (2007). [CrossRef] [PubMed]
  20. 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]
  21. 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]
  22. 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]
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