## Stability conditions for moving dissipative solitons in one- and multidimensional systems with a linear potential |

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

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]

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]

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]

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]

## 2. The model

*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]

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]

*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

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]

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

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. 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*,

*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).

**96**(1), 013903 (2006). [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]

*A*=

*A*

_{0}+ ∆α,

*w*=

*w*

_{0}+ ∆

*w*, c = c

_{0}+ ∆c, L = L

_{0}+ ∆L, we derive the respective linearized equations: where we define

^{4}+

*a*

_{1}λ

^{3}+

*a*

_{2}λ

^{2}+

*a*

_{3}λ +

*a*

_{4}= 0, the stability condition amounts to the Routh-Hurwitz criterion, i.e., the coefficients

*a*

_{1,2,3}must satisfy four inequalities:where

*L*to the slope of the LRIM α, diffusivity β and parameters of the soliton,

^{2}, 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]

_{0}, w

_{0}, c

_{0}, and L

_{0}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 = α

_{cr}, up to which (at α≤α

_{cr}) the solitons are stable.

## 4. Results and analysis

*x*,

*y*,

*t*)], where ρ(

*x*,

*y*,

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

_{cr}≈0.24. On the other hand, direct simulations of Eq. (1) yield α

_{cr}≈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]

_{cr}≈0.4, and direct simulations of Eq. (1) obtain α

_{cr}≈0.41 [26

**34**(19), 2976–2978 (2009). [CrossRef] [PubMed]

_{cr}≈0.08, and direct simulations of Eq. (1) find α

_{cr}≈0.082 [26

**34**(19), 2976–2978 (2009). [CrossRef] [PubMed]

## 5. Conclusions

## Acknowledgment

## 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. |

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. |

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 |

7. | W. J. Firth and A. J. Scroggie, “Optical bullet holes: Robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. |

8. | L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, “Spatial solitons in semiconductor microcavities,” Phys. Rev. A |

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. |

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. |

11. | P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Bistable and addressable localized vortices in semiconductor lasers,” Phys. Rev. Lett. |

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. |

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 |

14. | N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. |

15. | J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, “Pulsating, creeping, and erupting solitons in dissipative systems,” Phys. Rev. Lett. |

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

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 |

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 |

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. |

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 |

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 |

23. | D. Mihalache and D. Mazilu, “Ginzburg-Landau spatiotemporal dissipative optical solitons,” Rom. Rep. Phys. |

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. |

25. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton percolation in random optical lattices,” Opt. Express |

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. |

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. |

**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

- Y. S. Kivshar, and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).
- 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]
- N. N. Akhmediev, and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, London, 1997).
- 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]
- 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.
- G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286(5444), 1518–1523 (1999). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- I. S. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys. 74(1), 99–143 (2002). [CrossRef]
- B. A. Malomed, “Complex Ginzburg-Landau equation,” in: Encyclopedia of Nonlinear Science, A. Scott, ed., (Routledge, New York, 2005) pp. 157–160.
- B. A. Malomed, “Solitary pulses in linearly coupled Ginzburg-Landau equations,” Chaos 17(3), 037117 (2007). [CrossRef] [PubMed]
- 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]
- D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. Lett. 97(7), 073904 (2006). [CrossRef] [PubMed]
- D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. A 75(3), 033811 (2007). [CrossRef]
- 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]
- D. Mihalache and D. Mazilu, “Ginzburg-Landau spatiotemporal dissipative optical solitons,” Rom. Rep. Phys. 60, 749–761 (2008).
- 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]
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton percolation in random optical lattices,” Opt. Express 15(19), 12409–12417 (2007). [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]
- 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]

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