## Continuous generation of “light bullets” in dissipative media by an annularly periodic potential |

Optics Express, Vol. 19, Issue 21, pp. 20009-20014 (2011)

http://dx.doi.org/10.1364/OE.19.020009

Acrobat PDF (1112 KB)

### Abstract

We report novel dynamical regimes of “light bullets” supported by an annularly periodic potential in the three-dimensional (3D) complex Ginzburg-Landau equation with the cubic-quintic nonlinearity. This is a model of an active optical medium with respective expanding anti-waveguiding structures with *m* jets light bullet by an initial light bullet initially placed at the center. The influence of *m* and diffusivity term (viscosity) β on the corresponding strength of potential is studied. In the case of *m* = 0 (conical geometry), these are concentric waves expanding in the radial direction.

© 2011 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]

4. 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]

5. A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo, “Continuously self-focusing and continuously self-defocusing two-dimensional beams in dissipative media,” Phys. Rev. A **77**(3), 033840 (2008). [CrossRef]

13. A. Kamagate, P. 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]

14. D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between counter-rotating solitary vortices in the three-dimensional Ginzburg-Landau equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **78**(5), 056601 (2008). [CrossRef] [PubMed]

16. 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]

17. 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]

30. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics **4**(2), 103–106 (2010). [CrossRef]

11. 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]

31. P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Light bullets and dynamic pattern formation in nonlinear dissipative systems,” Opt. Express **13**(23), 9352–9630 (2005). [CrossRef] [PubMed]

32. 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]

33. 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]

*m*(annular periods) jets fundamental STS along symmetry directions of the potential is observed. The region of strength of potential by variety of

*m*and diffusion term is obtained by performing large number of numerical simulations. Finally, the potential with

*m*= 0 gives rise to the generation of an array of concentric annular STSs.

## 2. The model

12. 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]

20. 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]

*x*,

*y*) and

*t*are the transverse coordinates and temporal coordinate.

*z*is the propagation distance. The coefficients of diffraction and cubic self-focusing nonlinearity are scaled respectively, to be 1/2 and 1.

*D*is group-velocity dispersion (GVD) coefficient. Below, we set

*D*= 1/2 is for the anomalous dispersion propagation regime. 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.

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

16. 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]

19. 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]

*a*is the strength of potential.

## 3. Results and analysis

*a*. A novel dynamical regime that the central STS continuously generates

*m*streams of secondary pulses is observed, as shown in Figs. 2(a) and (b) at (

*m*= 4,

*a*= 0.08) and (

*m*= 8,

*a*= 0.07). The gain term in the CQ CGL equation is the source of the energy necessary for continuous generation. The emitted pulses self-trap into fundamental STSs which slide along slopes of the potential. We have calculated the evolution of total amount of energy

*E*that it carries:as a function of propagation distance

*z*. The evolutions of

*E*[shown in Fig. 2(c)] reveal an obviously periodic generation of

*m*STSs.

*z*

_{1}and

*z*

_{2}stand for the propagation distance of one period at

*a*= 0.08 and 0.09, respectively. The comparison of them demonstrates that the stronger potential provides for a higher emission rate. In addition, the corresponding region of

*a*is shown in Fig. 2(d). The upper critical value reduce with the growth of

*m*, but the lower critical value (

*a*= 0.05) is almost unchanged.

*a*by varying β. The lower critical value (

*a*= 0.05) is almost unchanged, but the upper critical value significantly decreases with the growth of β. At β = 0.6, there is no appropriate strength

*a*for attaining continuous generation.

*a*= 0.04. By comparing the evolution of total energy at (

*m*= 4,

*a*= 0.02) and (

*m*= 4,

*a*= 0.03) [black and blue lines in Fig. 3(c) ], the stretching force increases with

*a*.

*m*= 4 and

*a*= 0.12, the generated pulses rapidly dissipate, failing to self-trap into secondary STSs, because the potential’s slope exceeds the critical value admitting steady motion of the dissipative solitons [19

19. 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]

*a*depending on

*m*is shown in Fig. 3(d). The periodic evolutions of

*E*are observed at (a = 0.15, m = 4) and (a = 0.2, m = 4) [red and pink lines in Fig. 3(c)].

*E*at (m = 4,

*a*= 1) [green line in Fig. 3(c)] shows that the combination of the sharp tip and steep slopes completely destroys the central STS.

*a*= 0.12. The comparison of the evolutions of

*E*in Fig. 4(c) at different

*a*demonstrates that the generation rate also increases with

*a*. The weaker potential in Eq. (4), with

*E*at

*a*= 0.02 and 0.025 [shown in Fig. 4(d)] demonstrate that the stretching force also increase with

*a*. On the other hand, a stronger potential, with

*a*= 0.45. The evolutions of

*E*at

*a*= 0.45 and 0.5 in Fig. 4(e), reveals that a stronger

*a*leads to a rapidly expanding of ring pattern. At

*E*[red line in Fig. 4(e)] reveals the central STS suffers a decay.

## 4. Conclusions

*m*and diffusivity term β. Besides, the rate of generation increases with the strength of potential. If the potential is weak, the stretch of the central STS is observed instead. But for the conical potential (

*m*= 0) readily generates arrays of concentric pulses expanding in the radial direction.

## Acknowledgment

## References and links

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

2. | N. Rosanov, “Solitons in laser systems with absorption,” chapter in “Dissipative solitons,” edited by N. Akhmediev and A. Ankievicz (Springer-Verlag, Berlin, Heidelberg, 2005). |

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

4. | 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 |

5. | A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo, “Continuously self-focusing and continuously self-defocusing two-dimensional beams in dissipative media,” Phys. Rev. A |

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. |

7. | A. Desyatnikov, A. Maimistov, and B. Malomed, “Three-dimensional spinning solitons in dispersive media with the cubic-quintic nonlinearity,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

8. | L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Erupting, flat-top, and composite spiral solitons in two-dimensional Ginzburg-Landau equation,” Phys. Lett. A |

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

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

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

12. | 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 |

13. | A. Kamagate, P. 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. |

14. | D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between counter-rotating solitary vortices in the three-dimensional Ginzburg-Landau equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

15. | 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 |

16. | 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. | 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 |

18. | H. Sakaguchi and B. A. Malomed, “Two-dimensional dissipative gap solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

19. | 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. |

20. | 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. |

21. | C. Yin, D. Mihalache, and Y. He, “Dynamics of two-dimensional dissipative spatial solitons interacting with an umbrella-shaped potential,” J. Opt. Soc. Am. B |

22. | D. Mihalache, “Three-dimensional Ginzburg-Landau dissipative solitons supported by a two-dimensional transverse grating,” Proc. Rom. Acad. A |

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

24. | B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B Quantum Semiclassical Opt. |

25. | S. Minardi, F. Eilenberger, Y. V. Kartashov, A. Szameit, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, L. Torner, F. Lederer, A. Tünnermann, and T. Pertsch, “Three-Dimensional Light Bullets in Arrays of Waveguides,” Phys. Rev. Lett. |

26. | F. Eilenberger, S. Minardi, A. Szameit, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, L. Torner, F. Lederer, A. Tünnermann, and T. Pertsch, “Evolution dynamics of discrete-continuous light bullets,” Phys. Rev. A |

27. | Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. |

28. | A. B. Aceves and C. De Angelis, “Spatiotemporal pulse dynamics in a periodic nonlinear waveguide,” Opt. Lett. |

29. | D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. |

30. | A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics |

31. | P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Light bullets and dynamic pattern formation in nonlinear dissipative systems,” Opt. Express |

32. | 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. |

33. | 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 |

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

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: August 17, 2011

Revised Manuscript: September 6, 2011

Manuscript Accepted: September 6, 2011

Published: September 28, 2011

**Citation**

Bin Liu and Xing-Dao He, "Continuous generation of “light bullets” in dissipative media by an annularly periodic potential," Opt. Express **19**, 20009-20014 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20009

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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]
- N. Rosanov, “Solitons in laser systems with absorption,” chapter in “Dissipative solitons,” edited by N. Akhmediev and A. Ankievicz (Springer-Verlag, Berlin, Heidelberg, 2005).
- B. A. Malomed, “Complex Ginzburg–Landau equation,” in Encyclopedia of Nonlinear Science, edited by A. Scott (Routledge, New York, 2005), p. 157.
- 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]
- A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo, “Continuously self-focusing and continuously self-defocusing two-dimensional beams in dissipative media,” Phys. Rev. A77(3), 033840 (2008). [CrossRef]
- 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. Desyatnikov, A. Maimistov, and B. Malomed, “Three-dimensional spinning solitons in dispersive media with the cubic-quintic nonlinearity,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics61(3), 3107–3113 (2000). [CrossRef]
- L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Erupting, flat-top, and composite spiral solitons in two-dimensional Ginzburg-Landau equation,” Phys. Lett. A289(1-2), 59–65 (2001). [CrossRef]
- 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]
- 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]
- 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]
- 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. A75(3), 033811 (2007). [CrossRef]
- A. Kamagate, P. 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]
- D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between counter-rotating solitary vortices in the three-dimensional Ginzburg-Landau equation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.78(5), 056601 (2008). [CrossRef] [PubMed]
- 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]
- 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]
- 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. A80(3), 033835 (2009). [CrossRef]
- 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]
- 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]
- 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]
- C. Yin, D. Mihalache, and Y. He, “Dynamics of two-dimensional dissipative spatial solitons interacting with an umbrella-shaped potential,” J. Opt. Soc. Am. B28(2), 342–346 (2011). [CrossRef]
- D. Mihalache, “Three-dimensional Ginzburg-Landau dissipative solitons supported by a two-dimensional transverse grating,” Proc. Rom. Acad. A11, 142–147 (2010).
- Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).
- 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]
- S. Minardi, F. Eilenberger, Y. V. Kartashov, A. Szameit, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, L. Torner, F. Lederer, A. Tünnermann, and T. Pertsch, “Three-Dimensional Light Bullets in Arrays of Waveguides,” Phys. Rev. Lett.105(26), 263901 (2010). [CrossRef] [PubMed]
- F. Eilenberger, S. Minardi, A. Szameit, U. Röpke, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, L. Torner, F. Lederer, A. Tünnermann, and T. Pertsch, “Evolution dynamics of discrete-continuous light bullets,” Phys. Rev. A84(1), 013836 (2011). [CrossRef]
- Y. Silberberg, “Collapse of optical pulses,” Opt. Lett.15(22), 1282–1284 (1990). [CrossRef] [PubMed]
- A. B. Aceves and C. De Angelis, “Spatiotemporal pulse dynamics in a periodic nonlinear waveguide,” Opt. Lett.18(2), 110–112 (1993). [CrossRef] [PubMed]
- D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett.105(25), 253901 (2010). [CrossRef] [PubMed]
- A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics4(2), 103–106 (2010). [CrossRef]
- P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Light bullets and dynamic pattern formation in nonlinear dissipative systems,” Opt. Express13(23), 9352–9630 (2005). [CrossRef] [PubMed]
- 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]
- 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. Express14(13), 6055–6062 (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]

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