Abstract

We study the dynamics of two-dimensional spatial solitons in the structured optical medium modeled by the complex Ginzburg–Landau equation with cubic–quintic nonlinearity and a spatially periodic modulation of the local gain–loss coefficient [a dissipative lattice (DL)]. The analysis, following the variation of the DL’s amplitude and period, reveals several dynamical scenarios: stable or unstable propagation of a single dissipative soliton (the unstable propagation entails generation of an irregular multisoliton cluster), transformation of the input soliton into stable or unstable regular clusters patterned as the underlying DL, and decay of the input. Most results are obtained by means of systematic simulations, but the boundary of the single-soliton stability domain is explained analytically.

© 2014 Optical Society of America

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  1. C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, “Spatial solitons supported by localized gain in nonlinear optical waveguides,” Eur. Phys. J. Special Topics 173, 233–243 (2009).
    [CrossRef]
  2. C. H. Tsang, B. A. Malomed, C.-K. Lam, and K. W. Chow, “Solitons pinned to hot spots,” Eur. Phys. J. D 59, 81–89 (2010).
    [CrossRef]
  3. C. H. Tsang, B. A. Malomed, and K. W. Chow, “Multistable dissipative structures pinned to dual hot spots,” Phys. Rev. E 84, 066609 (2011).
    [CrossRef]
  4. B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, “Pinned modes in lossy lattices with local gain and nonlinearity,” Phys. Rev. E 86, 036608 (2012).
    [CrossRef]
  5. D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, “Solitons in a medium with linear dissipation and localized gain,” Opt. Lett. 36, 1200–1202 (2011).
    [CrossRef]
  6. F. Ye, C. Huang, Y. V. Kartashov, and B. A. Malomed, “Solitons supported by localized parametric gain,” Opt. Lett. 38, 480–482 (2013).
    [CrossRef]
  7. Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and L. Torner, “Dissipative defect modes in periodic structures,” Opt. Lett. 35, 1638–1640 (2010).
    [CrossRef]
  8. Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, “Symmetry breaking and multipeaked solitons in inhomogeneous gain landscapes,” Phys. Rev. A 83, 041806(R) (2011).
    [CrossRef]
  9. O. V. Borovkova, Y. V. Kartashov, V. E. Lobanov, V. A. Vysloukh, and L. Torner, “Vortex twins and anti-twins supported by multi-ring gain landscapes,” Opt. Lett. 36, 3783–3785 (2011).
    [CrossRef]
  10. C. Huang, F. Ye, B. A. Malomed, Y. V. Kartashov, and X. Chen, “Solitary vortices supported by localized parametric gain,” Opt. Lett. 38, 2177–2180 (2013).
    [CrossRef]
  11. S. C. Fernández and V. S. Shchesnovich, “Nondecaying linear and nonlinear modes in a periodic array of spatially localized dissipations,” arXiv:1401.1687v1 (2014).
  12. Y. V. Bludov and V. V. Konotop, “Nonlinear patterns in Bose-Einstein condensates in dissipative optical lattices,” Phys. Rev. A 81, 013625 (2010).
    [CrossRef]
  13. 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, 213901 (2010).
    [CrossRef]
  14. Y. He and D. Mihalache, “Soliton drift or swing induced by spatially inhomogeneous losses in media described by the complex Ginzburg–Landau,” J. Opt. Soc. Am. B 29, 2554–2558 (2012).
    [CrossRef]
  15. Y. He and D. Mihalache, “Lattice solitons in optical media described by the complex Ginzburg-Landau model with PT-symmetric periodic potentials,” Phys. Rev. A 87, 013812 (2013).
    [CrossRef]
  16. N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Universal properties of self-organized localized structures,” Appl. Phys. B 81, 937–943 (2005).
    [CrossRef]
  17. P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics,” J. Opt. B 6, R60–R75 (2004).
    [CrossRef]
  18. L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg-Landau equation,” Phys. Rev. E 63, 016605 (2001).
    [CrossRef]
  19. J. Jimenez, Y. Noblet, P. V. Paulau, D. Gomila, and T. Ackemann, “Observation of laser vortex solitons in a self-focusing semiconductor laser,” J. Opt. 15, 044011 (2013).
    [CrossRef]
  20. C. Fernandez-Oto, M. G. Clerc, D. Escaff, and M. Tlidi, “Strong nonlocal coupling stabilizes localized structures: an analysis based on front dynamics,” Phys. Rev. Lett. 110, 174101 (2013).
    [CrossRef]
  21. 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 53, 1190–1201 (1996).
    [CrossRef]
  22. W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
    [CrossRef]
  23. N. Akhmediev and A. Ankiewicz, Dissipative Solitons, Vol. 661 of Lecture Notes in Physics (Springer, 2005).
  24. 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, 2976–2978 (2009).
    [CrossRef]
  25. 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, 073904 (2006).
    [CrossRef]
  26. H. Leblond, A. Komarov, M. Salhi, A. Haboucha, and F. Sanchez, “Cis bound states of three localized states of the cubic-quintic CGL equation,” J. Opt. A 8, 319–326 (2006).
    [CrossRef]
  27. 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, 033811 (2007).
    [CrossRef]
  28. W. Chang, N. Akhmediev, S. Wabnitz, and M. Taki, “Influence of external phase and gain-loss modulation on bound solitons in laser systems,” J. Opt. Soc. Am. B 26, 2204–2210 (2009).
    [CrossRef]
  29. N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, 2002).
  30. B. A. Malomed, “Complex Ginzburg–Landau equation,” in Encyclopedia of Nonlinear Science, A. Scott, ed. (Routledge, 2005), pp. 157–160.
  31. I. S. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys. 74, 99–143 (2002).
    [CrossRef]
  32. B. A. Malomed, “Solitary pulses in linearly coupled Ginzburg–Landau equations,” Chaos 17, 037117 (2007).
    [CrossRef]
  33. V. V. Afanasjev, N. Akhmediev, and J. M. Soto-Crespo, “Three forms of localized solutions of the quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1931–1939 (1996).
    [CrossRef]
  34. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman & Hall, 1997).
  35. N. Akhmediev and A. Ankiewicz, Dissipative Solitons: From Optics to Biology and Medicine, Vol. 751 of Lecture Notes in Physics (Springer, 2008).
  36. 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, 013903 (2006).
    [CrossRef]
  37. T. Yasuhide and K. Masanori, “Guided-mode and leaky-mode analysis by imaginary distance beam propagation method based on finite element scheme,” J. Lightwave Technol. 18, 618–623 (2000).
    [CrossRef]
  38. Y. He, D. Mihalache, B. A. Malomed, Y. Qiu, Z. Chen, and Y. Li, “Generations of polygonal soliton clusters and fundamental solitons by radially-azimuthally phase-modulated necklace-ring beams in dissipative systems,” Phys. Rev. E 85, 066206 (2012).
    [CrossRef]
  39. 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 spatial solitons by material and virtual gratings,” Phys. Scripta 82, 065404 (2010).
    [CrossRef]
  40. W.-L. Zhu, L. Luo, J.-H. Yan, and Y.-J. He, “Stable spatiotemporal dissipative soliton clusters in the complex Ginzburg–Landau equation,” J. Mod. Opt. 56, 1824–1828 (2009).
    [CrossRef]
  41. B. A. Malomed, “Potential of interaction between two- and three-dimensional solitons,” Phys. Rev. E 58, 7928–7933 (1998).
    [CrossRef]
  42. D. Mihalache, “Linear and nonlinear light bullets: recent theoretical and experimental studies,” Rom. J. Phys. 57, 352–371 (2012).
  43. H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximation,” Phys. Rep. 523, 61–126 (2013).
    [CrossRef]
  44. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).
    [CrossRef]

2013 (6)

Y. He and D. Mihalache, “Lattice solitons in optical media described by the complex Ginzburg-Landau model with PT-symmetric periodic potentials,” Phys. Rev. A 87, 013812 (2013).
[CrossRef]

J. Jimenez, Y. Noblet, P. V. Paulau, D. Gomila, and T. Ackemann, “Observation of laser vortex solitons in a self-focusing semiconductor laser,” J. Opt. 15, 044011 (2013).
[CrossRef]

C. Fernandez-Oto, M. G. Clerc, D. Escaff, and M. Tlidi, “Strong nonlocal coupling stabilizes localized structures: an analysis based on front dynamics,” Phys. Rev. Lett. 110, 174101 (2013).
[CrossRef]

H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximation,” Phys. Rep. 523, 61–126 (2013).
[CrossRef]

F. Ye, C. Huang, Y. V. Kartashov, and B. A. Malomed, “Solitons supported by localized parametric gain,” Opt. Lett. 38, 480–482 (2013).
[CrossRef]

C. Huang, F. Ye, B. A. Malomed, Y. V. Kartashov, and X. Chen, “Solitary vortices supported by localized parametric gain,” Opt. Lett. 38, 2177–2180 (2013).
[CrossRef]

2012 (4)

Y. He and D. Mihalache, “Soliton drift or swing induced by spatially inhomogeneous losses in media described by the complex Ginzburg–Landau,” J. Opt. Soc. Am. B 29, 2554–2558 (2012).
[CrossRef]

D. Mihalache, “Linear and nonlinear light bullets: recent theoretical and experimental studies,” Rom. J. Phys. 57, 352–371 (2012).

Y. He, D. Mihalache, B. A. Malomed, Y. Qiu, Z. Chen, and Y. Li, “Generations of polygonal soliton clusters and fundamental solitons by radially-azimuthally phase-modulated necklace-ring beams in dissipative systems,” Phys. Rev. E 85, 066206 (2012).
[CrossRef]

B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, “Pinned modes in lossy lattices with local gain and nonlinearity,” Phys. Rev. E 86, 036608 (2012).
[CrossRef]

2011 (4)

Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, “Symmetry breaking and multipeaked solitons in inhomogeneous gain landscapes,” Phys. Rev. A 83, 041806(R) (2011).
[CrossRef]

C. H. Tsang, B. A. Malomed, and K. W. Chow, “Multistable dissipative structures pinned to dual hot spots,” Phys. Rev. E 84, 066609 (2011).
[CrossRef]

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

O. V. Borovkova, Y. V. Kartashov, V. E. Lobanov, V. A. Vysloukh, and L. Torner, “Vortex twins and anti-twins supported by multi-ring gain landscapes,” Opt. Lett. 36, 3783–3785 (2011).
[CrossRef]

2010 (5)

Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and L. Torner, “Dissipative defect modes in periodic structures,” Opt. Lett. 35, 1638–1640 (2010).
[CrossRef]

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 spatial solitons by material and virtual gratings,” Phys. Scripta 82, 065404 (2010).
[CrossRef]

Y. V. Bludov and V. V. Konotop, “Nonlinear patterns in Bose-Einstein condensates in dissipative optical lattices,” Phys. Rev. A 81, 013625 (2010).
[CrossRef]

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, 213901 (2010).
[CrossRef]

C. H. Tsang, B. A. Malomed, C.-K. Lam, and K. W. Chow, “Solitons pinned to hot spots,” Eur. Phys. J. D 59, 81–89 (2010).
[CrossRef]

2009 (4)

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, “Spatial solitons supported by localized gain in nonlinear optical waveguides,” Eur. Phys. J. Special Topics 173, 233–243 (2009).
[CrossRef]

W.-L. Zhu, L. Luo, J.-H. Yan, and Y.-J. He, “Stable spatiotemporal dissipative soliton clusters in the complex Ginzburg–Landau equation,” J. Mod. Opt. 56, 1824–1828 (2009).
[CrossRef]

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, 2976–2978 (2009).
[CrossRef]

W. Chang, N. Akhmediev, S. Wabnitz, and M. Taki, “Influence of external phase and gain-loss modulation on bound solitons in laser systems,” J. Opt. Soc. Am. B 26, 2204–2210 (2009).
[CrossRef]

2008 (1)

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[CrossRef]

2007 (2)

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, 033811 (2007).
[CrossRef]

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

2006 (3)

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, 013903 (2006).
[CrossRef]

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, 073904 (2006).
[CrossRef]

H. Leblond, A. Komarov, M. Salhi, A. Haboucha, and F. Sanchez, “Cis bound states of three localized states of the cubic-quintic CGL equation,” J. Opt. A 8, 319–326 (2006).
[CrossRef]

2005 (2)

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Universal properties of self-organized localized structures,” Appl. Phys. B 81, 937–943 (2005).
[CrossRef]

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

2004 (1)

P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics,” J. Opt. B 6, R60–R75 (2004).
[CrossRef]

2002 (1)

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

2001 (1)

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg-Landau equation,” Phys. Rev. E 63, 016605 (2001).
[CrossRef]

2000 (1)

1998 (1)

B. A. Malomed, “Potential of interaction between two- and three-dimensional solitons,” Phys. Rev. E 58, 7928–7933 (1998).
[CrossRef]

1996 (2)

V. V. Afanasjev, N. Akhmediev, and J. M. Soto-Crespo, “Three forms of localized solutions of the quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1931–1939 (1996).
[CrossRef]

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 53, 1190–1201 (1996).
[CrossRef]

Ackemann, T.

J. Jimenez, Y. Noblet, P. V. Paulau, D. Gomila, and T. Ackemann, “Observation of laser vortex solitons in a self-focusing semiconductor laser,” J. Opt. 15, 044011 (2013).
[CrossRef]

Afanasjev, V. V.

V. V. Afanasjev, N. Akhmediev, and J. M. Soto-Crespo, “Three forms of localized solutions of the quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1931–1939 (1996).
[CrossRef]

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 53, 1190–1201 (1996).
[CrossRef]

Akhmediev, N.

W. Chang, N. Akhmediev, S. Wabnitz, and M. Taki, “Influence of external phase and gain-loss modulation on bound solitons in laser systems,” J. Opt. Soc. Am. B 26, 2204–2210 (2009).
[CrossRef]

V. V. Afanasjev, N. Akhmediev, and J. M. Soto-Crespo, “Three forms of localized solutions of the quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1931–1939 (1996).
[CrossRef]

N. Akhmediev and A. Ankiewicz, Dissipative Solitons, Vol. 661 of Lecture Notes in Physics (Springer, 2005).

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

N. Akhmediev and A. Ankiewicz, Dissipative Solitons: From Optics to Biology and Medicine, Vol. 751 of Lecture Notes in Physics (Springer, 2008).

Akhmediev, N. N.

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 53, 1190–1201 (1996).
[CrossRef]

Aleksic, N. B.

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, 213901 (2010).
[CrossRef]

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, 013903 (2006).
[CrossRef]

Ankiewicz, A.

N. Akhmediev and A. Ankiewicz, Dissipative Solitons, Vol. 661 of Lecture Notes in Physics (Springer, 2005).

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

N. Akhmediev and A. Ankiewicz, Dissipative Solitons: From Optics to Biology and Medicine, Vol. 751 of Lecture Notes in Physics (Springer, 2008).

Aranson, I. S.

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

Bludov, Y. V.

Y. V. Bludov and V. V. Konotop, “Nonlinear patterns in Bose-Einstein condensates in dissipative optical lattices,” Phys. Rev. A 81, 013625 (2010).
[CrossRef]

Borovkova, O. V.

Chang, W.

Chen, X.

Chen, Z.

Y. He, D. Mihalache, B. A. Malomed, Y. Qiu, Z. Chen, and Y. Li, “Generations of polygonal soliton clusters and fundamental solitons by radially-azimuthally phase-modulated necklace-ring beams in dissipative systems,” Phys. Rev. E 85, 066206 (2012).
[CrossRef]

Chong, A.

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[CrossRef]

Chow, K. W.

B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, “Pinned modes in lossy lattices with local gain and nonlinearity,” Phys. Rev. E 86, 036608 (2012).
[CrossRef]

C. H. Tsang, B. A. Malomed, and K. W. Chow, “Multistable dissipative structures pinned to dual hot spots,” Phys. Rev. E 84, 066609 (2011).
[CrossRef]

C. H. Tsang, B. A. Malomed, C.-K. Lam, and K. W. Chow, “Solitons pinned to hot spots,” Eur. Phys. J. D 59, 81–89 (2010).
[CrossRef]

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, “Spatial solitons supported by localized gain in nonlinear optical waveguides,” Eur. Phys. J. Special Topics 173, 233–243 (2009).
[CrossRef]

Clerc, M. G.

C. Fernandez-Oto, M. G. Clerc, D. Escaff, and M. Tlidi, “Strong nonlocal coupling stabilizes localized structures: an analysis based on front dynamics,” Phys. Rev. Lett. 110, 174101 (2013).
[CrossRef]

Crasovan, L.-C.

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, 073904 (2006).
[CrossRef]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg-Landau equation,” Phys. Rev. E 63, 016605 (2001).
[CrossRef]

Ding, E.

B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, “Pinned modes in lossy lattices with local gain and nonlinearity,” Phys. Rev. E 86, 036608 (2012).
[CrossRef]

Dong, J.

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 spatial solitons by material and virtual gratings,” Phys. Scripta 82, 065404 (2010).
[CrossRef]

Escaff, D.

C. Fernandez-Oto, M. G. Clerc, D. Escaff, and M. Tlidi, “Strong nonlocal coupling stabilizes localized structures: an analysis based on front dynamics,” Phys. Rev. Lett. 110, 174101 (2013).
[CrossRef]

Fedorov, S. V.

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Universal properties of self-organized localized structures,” Appl. Phys. B 81, 937–943 (2005).
[CrossRef]

Fernández, S. C.

S. C. Fernández and V. S. Shchesnovich, “Nondecaying linear and nonlinear modes in a periodic array of spatially localized dissipations,” arXiv:1401.1687v1 (2014).

Fernandez-Oto, C.

C. Fernandez-Oto, M. G. Clerc, D. Escaff, and M. Tlidi, “Strong nonlocal coupling stabilizes localized structures: an analysis based on front dynamics,” Phys. Rev. Lett. 110, 174101 (2013).
[CrossRef]

Gomila, D.

J. Jimenez, Y. Noblet, P. V. Paulau, D. Gomila, and T. Ackemann, “Observation of laser vortex solitons in a self-focusing semiconductor laser,” J. Opt. 15, 044011 (2013).
[CrossRef]

Haboucha, A.

H. Leblond, A. Komarov, M. Salhi, A. Haboucha, and F. Sanchez, “Cis bound states of three localized states of the cubic-quintic CGL equation,” J. Opt. A 8, 319–326 (2006).
[CrossRef]

He, Y.

Y. He and D. Mihalache, “Lattice solitons in optical media described by the complex Ginzburg-Landau model with PT-symmetric periodic potentials,” Phys. Rev. A 87, 013812 (2013).
[CrossRef]

Y. He and D. Mihalache, “Soliton drift or swing induced by spatially inhomogeneous losses in media described by the complex Ginzburg–Landau,” J. Opt. Soc. Am. B 29, 2554–2558 (2012).
[CrossRef]

Y. He, D. Mihalache, B. A. Malomed, Y. Qiu, Z. Chen, and Y. Li, “Generations of polygonal soliton clusters and fundamental solitons by radially-azimuthally phase-modulated necklace-ring beams in dissipative systems,” Phys. Rev. E 85, 066206 (2012).
[CrossRef]

He, Y. J.

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 spatial solitons by material and virtual gratings,” Phys. Scripta 82, 065404 (2010).
[CrossRef]

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, 2976–2978 (2009).
[CrossRef]

He, Y.-J.

W.-L. Zhu, L. Luo, J.-H. Yan, and Y.-J. He, “Stable spatiotemporal dissipative soliton clusters in the complex Ginzburg–Landau equation,” J. Mod. Opt. 56, 1824–1828 (2009).
[CrossRef]

Hu, B.

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 spatial solitons by material and virtual gratings,” Phys. Scripta 82, 065404 (2010).
[CrossRef]

Huang, C.

Huang, H. C.

Jimenez, J.

J. Jimenez, Y. Noblet, P. V. Paulau, D. Gomila, and T. Ackemann, “Observation of laser vortex solitons in a self-focusing semiconductor laser,” J. Opt. 15, 044011 (2013).
[CrossRef]

Kartashov, Y. V.

Komarov, A.

H. Leblond, A. Komarov, M. Salhi, A. Haboucha, and F. Sanchez, “Cis bound states of three localized states of the cubic-quintic CGL equation,” J. Opt. A 8, 319–326 (2006).
[CrossRef]

Konotop, V. V.

Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, “Symmetry breaking and multipeaked solitons in inhomogeneous gain landscapes,” Phys. Rev. A 83, 041806(R) (2011).
[CrossRef]

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

Y. V. Bludov and V. V. Konotop, “Nonlinear patterns in Bose-Einstein condensates in dissipative optical lattices,” Phys. Rev. A 81, 013625 (2010).
[CrossRef]

Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and L. Torner, “Dissipative defect modes in periodic structures,” Opt. Lett. 35, 1638–1640 (2010).
[CrossRef]

Kramer, L.

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

Lai, S. K.

B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, “Pinned modes in lossy lattices with local gain and nonlinearity,” Phys. Rev. E 86, 036608 (2012).
[CrossRef]

Lam, C.-K.

C. H. Tsang, B. A. Malomed, C.-K. Lam, and K. W. Chow, “Solitons pinned to hot spots,” Eur. Phys. J. D 59, 81–89 (2010).
[CrossRef]

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, “Spatial solitons supported by localized gain in nonlinear optical waveguides,” Eur. Phys. J. Special Topics 173, 233–243 (2009).
[CrossRef]

Leblond, H.

H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximation,” Phys. Rep. 523, 61–126 (2013).
[CrossRef]

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, 213901 (2010).
[CrossRef]

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, 033811 (2007).
[CrossRef]

H. Leblond, A. Komarov, M. Salhi, A. Haboucha, and F. Sanchez, “Cis bound states of three localized states of the cubic-quintic CGL equation,” J. Opt. A 8, 319–326 (2006).
[CrossRef]

Lederer, F.

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, 033811 (2007).
[CrossRef]

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, 073904 (2006).
[CrossRef]

Li, Y.

Y. He, D. Mihalache, B. A. Malomed, Y. Qiu, Z. Chen, and Y. Li, “Generations of polygonal soliton clusters and fundamental solitons by radially-azimuthally phase-modulated necklace-ring beams in dissipative systems,” Phys. Rev. E 85, 066206 (2012).
[CrossRef]

Liu, B.

Lobanov, V. E.

Luo, L.

W.-L. Zhu, L. Luo, J.-H. Yan, and Y.-J. He, “Stable spatiotemporal dissipative soliton clusters in the complex Ginzburg–Landau equation,” J. Mod. Opt. 56, 1824–1828 (2009).
[CrossRef]

Malomed, B. A.

C. Huang, F. Ye, B. A. Malomed, Y. V. Kartashov, and X. Chen, “Solitary vortices supported by localized parametric gain,” Opt. Lett. 38, 2177–2180 (2013).
[CrossRef]

F. Ye, C. Huang, Y. V. Kartashov, and B. A. Malomed, “Solitons supported by localized parametric gain,” Opt. Lett. 38, 480–482 (2013).
[CrossRef]

Y. He, D. Mihalache, B. A. Malomed, Y. Qiu, Z. Chen, and Y. Li, “Generations of polygonal soliton clusters and fundamental solitons by radially-azimuthally phase-modulated necklace-ring beams in dissipative systems,” Phys. Rev. E 85, 066206 (2012).
[CrossRef]

B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, “Pinned modes in lossy lattices with local gain and nonlinearity,” Phys. Rev. E 86, 036608 (2012).
[CrossRef]

C. H. Tsang, B. A. Malomed, and K. W. Chow, “Multistable dissipative structures pinned to dual hot spots,” Phys. Rev. E 84, 066609 (2011).
[CrossRef]

C. H. Tsang, B. A. Malomed, C.-K. Lam, and K. W. Chow, “Solitons pinned to hot spots,” Eur. Phys. J. D 59, 81–89 (2010).
[CrossRef]

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 spatial solitons by material and virtual gratings,” Phys. Scripta 82, 065404 (2010).
[CrossRef]

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, 213901 (2010).
[CrossRef]

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, 2976–2978 (2009).
[CrossRef]

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, “Spatial solitons supported by localized gain in nonlinear optical waveguides,” Eur. Phys. J. Special Topics 173, 233–243 (2009).
[CrossRef]

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, 033811 (2007).
[CrossRef]

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

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, 073904 (2006).
[CrossRef]

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

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg-Landau equation,” Phys. Rev. E 63, 016605 (2001).
[CrossRef]

B. A. Malomed, “Potential of interaction between two- and three-dimensional solitons,” Phys. Rev. E 58, 7928–7933 (1998).
[CrossRef]

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

Mandel, P.

P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics,” J. Opt. B 6, R60–R75 (2004).
[CrossRef]

Masanori, K.

Mazilu, D.

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, 033811 (2007).
[CrossRef]

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, 073904 (2006).
[CrossRef]

Mihalache, D.

Y. He and D. Mihalache, “Lattice solitons in optical media described by the complex Ginzburg-Landau model with PT-symmetric periodic potentials,” Phys. Rev. A 87, 013812 (2013).
[CrossRef]

H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximation,” Phys. Rep. 523, 61–126 (2013).
[CrossRef]

D. Mihalache, “Linear and nonlinear light bullets: recent theoretical and experimental studies,” Rom. J. Phys. 57, 352–371 (2012).

Y. He, D. Mihalache, B. A. Malomed, Y. Qiu, Z. Chen, and Y. Li, “Generations of polygonal soliton clusters and fundamental solitons by radially-azimuthally phase-modulated necklace-ring beams in dissipative systems,” Phys. Rev. E 85, 066206 (2012).
[CrossRef]

Y. He and D. Mihalache, “Soliton drift or swing induced by spatially inhomogeneous losses in media described by the complex Ginzburg–Landau,” J. Opt. Soc. Am. B 29, 2554–2558 (2012).
[CrossRef]

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, 213901 (2010).
[CrossRef]

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, 2976–2978 (2009).
[CrossRef]

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, 033811 (2007).
[CrossRef]

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, 073904 (2006).
[CrossRef]

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

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg-Landau equation,” Phys. Rev. E 63, 016605 (2001).
[CrossRef]

Noblet, Y.

J. Jimenez, Y. Noblet, P. V. Paulau, D. Gomila, and T. Ackemann, “Observation of laser vortex solitons in a self-focusing semiconductor laser,” J. Opt. 15, 044011 (2013).
[CrossRef]

Paulau, P. V.

J. Jimenez, Y. Noblet, P. V. Paulau, D. Gomila, and T. Ackemann, “Observation of laser vortex solitons in a self-focusing semiconductor laser,” J. Opt. 15, 044011 (2013).
[CrossRef]

Qiu, Y.

Y. He, D. Mihalache, B. A. Malomed, Y. Qiu, Z. Chen, and Y. Li, “Generations of polygonal soliton clusters and fundamental solitons by radially-azimuthally phase-modulated necklace-ring beams in dissipative systems,” Phys. Rev. E 85, 066206 (2012).
[CrossRef]

Qiu, Z.

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 spatial solitons by material and virtual gratings,” Phys. Scripta 82, 065404 (2010).
[CrossRef]

Renninger, W. H.

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[CrossRef]

Rosanov, N. N.

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Universal properties of self-organized localized structures,” Appl. Phys. B 81, 937–943 (2005).
[CrossRef]

N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, 2002).

Salhi, M.

H. Leblond, A. Komarov, M. Salhi, A. Haboucha, and F. Sanchez, “Cis bound states of three localized states of the cubic-quintic CGL equation,” J. Opt. A 8, 319–326 (2006).
[CrossRef]

Sanchez, F.

H. Leblond, A. Komarov, M. Salhi, A. Haboucha, and F. Sanchez, “Cis bound states of three localized states of the cubic-quintic CGL equation,” J. Opt. A 8, 319–326 (2006).
[CrossRef]

Shatsev, A. N.

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Universal properties of self-organized localized structures,” Appl. Phys. B 81, 937–943 (2005).
[CrossRef]

Shchesnovich, V. S.

S. C. Fernández and V. S. Shchesnovich, “Nondecaying linear and nonlinear modes in a periodic array of spatially localized dissipations,” arXiv:1401.1687v1 (2014).

Skarka, V.

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, 213901 (2010).
[CrossRef]

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, 013903 (2006).
[CrossRef]

Soto-Crespo, J. M.

V. V. Afanasjev, N. Akhmediev, and J. M. Soto-Crespo, “Three forms of localized solutions of the quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1931–1939 (1996).
[CrossRef]

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 53, 1190–1201 (1996).
[CrossRef]

Taki, M.

Tlidi, M.

C. Fernandez-Oto, M. G. Clerc, D. Escaff, and M. Tlidi, “Strong nonlocal coupling stabilizes localized structures: an analysis based on front dynamics,” Phys. Rev. Lett. 110, 174101 (2013).
[CrossRef]

P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics,” J. Opt. B 6, R60–R75 (2004).
[CrossRef]

Torner, L.

O. V. Borovkova, Y. V. Kartashov, V. E. Lobanov, V. A. Vysloukh, and L. Torner, “Vortex twins and anti-twins supported by multi-ring gain landscapes,” Opt. Lett. 36, 3783–3785 (2011).
[CrossRef]

Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and L. Torner, “Dissipative defect modes in periodic structures,” Opt. Lett. 35, 1638–1640 (2010).
[CrossRef]

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, 073904 (2006).
[CrossRef]

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

Tsang, C. H.

C. H. Tsang, B. A. Malomed, and K. W. Chow, “Multistable dissipative structures pinned to dual hot spots,” Phys. Rev. E 84, 066609 (2011).
[CrossRef]

C. H. Tsang, B. A. Malomed, C.-K. Lam, and K. W. Chow, “Solitons pinned to hot spots,” Eur. Phys. J. D 59, 81–89 (2010).
[CrossRef]

Vysloukh, V. A.

Wabnitz, S.

Wai, P. K. A.

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, “Spatial solitons supported by localized gain in nonlinear optical waveguides,” Eur. Phys. J. Special Topics 173, 233–243 (2009).
[CrossRef]

Wang, H. Z.

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 spatial solitons by material and virtual gratings,” Phys. Scripta 82, 065404 (2010).
[CrossRef]

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, 2976–2978 (2009).
[CrossRef]

Wise, F.

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

Wise, F. W.

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[CrossRef]

Yan, J.-H.

W.-L. Zhu, L. Luo, J.-H. Yan, and Y.-J. He, “Stable spatiotemporal dissipative soliton clusters in the complex Ginzburg–Landau equation,” J. Mod. Opt. 56, 1824–1828 (2009).
[CrossRef]

Yang, H.

Yasuhide, T.

Ye, F.

Zezyulin, D. A.

Zhu, W.-L.

W.-L. Zhu, L. Luo, J.-H. Yan, and Y.-J. He, “Stable spatiotemporal dissipative soliton clusters in the complex Ginzburg–Landau equation,” J. Mod. Opt. 56, 1824–1828 (2009).
[CrossRef]

Appl. Phys. B (1)

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Universal properties of self-organized localized structures,” Appl. Phys. B 81, 937–943 (2005).
[CrossRef]

Chaos (1)

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

Eur. Phys. J. D (1)

C. H. Tsang, B. A. Malomed, C.-K. Lam, and K. W. Chow, “Solitons pinned to hot spots,” Eur. Phys. J. D 59, 81–89 (2010).
[CrossRef]

Eur. Phys. J. Special Topics (1)

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, “Spatial solitons supported by localized gain in nonlinear optical waveguides,” Eur. Phys. J. Special Topics 173, 233–243 (2009).
[CrossRef]

J. Lightwave Technol. (1)

J. Mod. Opt. (1)

W.-L. Zhu, L. Luo, J.-H. Yan, and Y.-J. He, “Stable spatiotemporal dissipative soliton clusters in the complex Ginzburg–Landau equation,” J. Mod. Opt. 56, 1824–1828 (2009).
[CrossRef]

J. Opt. (1)

J. Jimenez, Y. Noblet, P. V. Paulau, D. Gomila, and T. Ackemann, “Observation of laser vortex solitons in a self-focusing semiconductor laser,” J. Opt. 15, 044011 (2013).
[CrossRef]

J. Opt. A (1)

H. Leblond, A. Komarov, M. Salhi, A. Haboucha, and F. Sanchez, “Cis bound states of three localized states of the cubic-quintic CGL equation,” J. Opt. A 8, 319–326 (2006).
[CrossRef]

J. Opt. B (2)

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

P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics,” J. Opt. B 6, R60–R75 (2004).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Lett. (6)

Phys. Rep. (1)

H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximation,” Phys. Rep. 523, 61–126 (2013).
[CrossRef]

Phys. Rev. A (5)

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[CrossRef]

Y. He and D. Mihalache, “Lattice solitons in optical media described by the complex Ginzburg-Landau model with PT-symmetric periodic potentials,” Phys. Rev. A 87, 013812 (2013).
[CrossRef]

Y. V. Bludov and V. V. Konotop, “Nonlinear patterns in Bose-Einstein condensates in dissipative optical lattices,” Phys. Rev. A 81, 013625 (2010).
[CrossRef]

Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, “Symmetry breaking and multipeaked solitons in inhomogeneous gain landscapes,” Phys. Rev. A 83, 041806(R) (2011).
[CrossRef]

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, 033811 (2007).
[CrossRef]

Phys. Rev. E (7)

Y. He, D. Mihalache, B. A. Malomed, Y. Qiu, Z. Chen, and Y. Li, “Generations of polygonal soliton clusters and fundamental solitons by radially-azimuthally phase-modulated necklace-ring beams in dissipative systems,” Phys. Rev. E 85, 066206 (2012).
[CrossRef]

B. A. Malomed, “Potential of interaction between two- and three-dimensional solitons,” Phys. Rev. E 58, 7928–7933 (1998).
[CrossRef]

V. V. Afanasjev, N. Akhmediev, and J. M. Soto-Crespo, “Three forms of localized solutions of the quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1931–1939 (1996).
[CrossRef]

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 53, 1190–1201 (1996).
[CrossRef]

C. H. Tsang, B. A. Malomed, and K. W. Chow, “Multistable dissipative structures pinned to dual hot spots,” Phys. Rev. E 84, 066609 (2011).
[CrossRef]

B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, “Pinned modes in lossy lattices with local gain and nonlinearity,” Phys. Rev. E 86, 036608 (2012).
[CrossRef]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg-Landau equation,” Phys. Rev. E 63, 016605 (2001).
[CrossRef]

Phys. Rev. Lett. (4)

C. Fernandez-Oto, M. G. Clerc, D. Escaff, and M. Tlidi, “Strong nonlocal coupling stabilizes localized structures: an analysis based on front dynamics,” Phys. Rev. Lett. 110, 174101 (2013).
[CrossRef]

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, 213901 (2010).
[CrossRef]

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, 013903 (2006).
[CrossRef]

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, 073904 (2006).
[CrossRef]

Phys. Scripta (1)

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 spatial solitons by material and virtual gratings,” Phys. Scripta 82, 065404 (2010).
[CrossRef]

Rev. Mod. Phys. (1)

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

Rom. J. Phys. (1)

D. Mihalache, “Linear and nonlinear light bullets: recent theoretical and experimental studies,” Rom. J. Phys. 57, 352–371 (2012).

Other (6)

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

N. Akhmediev and A. Ankiewicz, Dissipative Solitons: From Optics to Biology and Medicine, Vol. 751 of Lecture Notes in Physics (Springer, 2008).

S. C. Fernández and V. S. Shchesnovich, “Nondecaying linear and nonlinear modes in a periodic array of spatially localized dissipations,” arXiv:1401.1687v1 (2014).

N. Akhmediev and A. Ankiewicz, Dissipative Solitons, Vol. 661 of Lecture Notes in Physics (Springer, 2005).

N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, 2002).

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

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Figures (4)

Fig. 1.
Fig. 1.

(a) Shape of the DL L(x,y) defined by Eq. (2) with w=5 and T=1. (b) Input dissipative soliton, which is the attractor of the CQ CGL model (1) in the absence of the DL and viscosity (β=0).

Fig. 2.
Fig. 2.

Soliton clusters produced by the periodic DL with large amplitudes, w=10 and w=5. (a), (b) Curves labeled A separate regions in the plane of (β,T) where stable soliton clusters form (above curves A) and unstable clusters emerge (below curves A) for (a) w=10 and (b) w=5. (c) Typical example of the formation of a stable cluster in the form of the soliton square lattice, for w=10, β=0.4, and T=2.5. (d) Example of the generation of an unstable cluster for w=10, β=0, and T=2.5.

Fig. 3.
Fig. 3.

Analysis of soliton dynamics in the plane of the DL’s amplitude w and diffusion coefficient β, for a fixed DL period, T=1. (a) The plotted curves divide the plane of (w,β) into five regions: the soliton-decay domain (below curve D), domains corresponding to stable (between curves C and D) and unstable (between B and C) single-soliton evolution [the shape of line C, which is the boundary of the destabilization of the soliton by background perturbations, can be explained analytically; see Eq. (4)], domains where stable soliton clusters form (between A and B), and those where an unstable soliton cluster appears (above curve A). Other panels display typical examples of the different dynamical scenarios: (b) soliton decay at w=0.1 and β=0.5, (c) formation of a stable single soliton at w=0.45 and β=0.5, (d) evolution of an unstable single soliton at w=1 and β=0.5, which leads to generation of an irregular cluster, (e) formation of a stable soliton cluster at w=3 and β=0.5, and (f) evolution of an unstable cluster at w=30 and β=0.5.

Fig. 4.
Fig. 4.

Analysis of the soliton dynamics in the plane of the DL’s amplitude w and period T, for a fixed value of the diffusion parameter, β=0.4. The plotted curves divide the plane of (w,T) into five regions: decay of the soliton (below curve D), stable and unstable single-soliton evolution (between curves C and D, and between B and C, respectively), and the generation of stable and unstable soliton clusters (between curves A and B, and above curve A, respectively). The shape of curve C can be explained analytically; see Eq. (4).

Equations (4)

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iuz+(1/2)(uxx+uyy)+|u|2uν|u|4u=iR[u]iL(x,y)u,
L(x,y)=w[sin(x/T)+sin(y/T)],
δu(x,y)[sin(x/(2T)π/4)+sin(y/(2T)π/4)],
Δα=w/2β/(4T2).

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