Abstract

We numerically show the generation of robust vortex clusters embedded in a two-dimensional beam propagating in a dissipative medium described by the generic cubic-quintic complex Ginzburg-Landau equation with an inhomogeneous effective diffusion term, which is asymmetrical in the two transverse directions and periodically modulated in the longitudinal direction. We show the generation of stable optical vortex clusters for different values of the winding number (topological charge) of the input optical beam. We have found that the number of individual vortex solitons that form the robust vortex cluster is equal to the winding number of the input beam. We have obtained the relationships between the amplitudes and oscillation periods of the inhomogeneous effective diffusion and the cubic gain and diffusion (viscosity) parameters, which depict the regions of existence and stability of vortex clusters. The obtained results offer a method to form robust vortex clusters embedded in two-dimensional optical beams, and we envisage potential applications in the area of structured light.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. S. Trillo and W. Torruellas, Spatial Solitons (Springer, 2001).
  2. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).
  3. W. J. Firth, Self-Organization in Optical Systems and Applications in Information Technology (Springer, 1995).
  4. N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, Dissipative Solitons: From Optics to Biology and Medicine (Springer, 2008).
  5. R. Kuszelewicz, S. Barbay, G. Tissoni, and G. Almuneau, “Editorial on dissipative optical solitons,” Eur. Phys. J. D 59(1), 1–2 (2010).
  6. 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).
    [PubMed]
  7. Z. Chen and K. McCarthy, “Spatial soliton pixels from partially incoherent light,” Opt. Lett. 27(22), 2019–2021 (2002).
    [PubMed]
  8. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
    [PubMed]
  9. Y. V. Kartashov, A. A. Egorov, L. Torner, and D. N. Christodoulides, “Stable soliton complexes in two-dimensional photonic lattices,” Opt. Lett. 29(16), 1918–1920 (2004).
    [PubMed]
  10. N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, 2002).
  11. B. A. Malomed, “Complex Ginzburg-Landau equation,” in Encyclopedia of Nonlinear Science, A. Scott, ed. (Routledge, 2005).
  12. P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics,” J. Opt. B 6(9), R60–R75 (2004).
  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).
    [PubMed]
  14. 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).
    [PubMed]
  15. 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(3), 319–326 (2006).
  16. D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability of dissipative optical solitons in thethree-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. A 75(75), 033811 (2007).
  17. W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77(2), 023814 (2008).
  18. 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(11), 2204–2210 (2009).
  19. I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74(11), 99–143 (2002).
  20. N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: complex Ginzburg-Landau equation approach,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(5 Pt 2), 056602 (2001).
    [PubMed]
  21. 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]
  22. Y. J. He, B. A. Malomed, F. W. Ye, and B. B. Hu, “Dynamics of dissipative spatial solitons over a sharp potential,” J. Opt. Soc. Am. B 27(5), 1139–1142 (2010).
  23. P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6(2), 84–92 (2012).
  24. C. Fernandez-Oto, G. J. De Valcárcel, M. Tlidi, K. Panajotov, and K. Staliunas, “Phase-bistable patterns and cavity solitons induced by spatially periodic injection into vertical-cavity surface-emitting lasers,” Phys. Rev. A 89(5), 055802 (2014).
  25. B. A. Malomed, “Spatial solitons supported by localized gain,” J. Opt. Soc. Am. B 31(10), 2460–2475 (2014).
  26. M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimirov, and M. G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Philos Trans A Math Phys Eng Sci 372(2027), 20140101 (2014).
    [PubMed]
  27. N. N. Rosanov, G. B. Sochilin, V. D. Vinokurova, and N. V. Vysotina, “Spatial and temporal structures in cavities with oscillating boundaries,” Philos Trans A Math Phys Eng Sci 372(2027), 20140012 (2014).
    [PubMed]
  28. D. Mihalache, D. Mazilu, V. Skarka, B. A. Malomed, H. Leblond, N. B. Aleksić, and F. Lederer, “Stable topological modes in two-dimensional Ginzburg-Landau models with trapping potentials,” Phys. Rev. A 82(2), 023813 (2010).
  29. V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105(21), 213901 (2010).
    [PubMed]
  30. V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90(2), 023845 (2014).
  31. B. Liu, X.-D. He, and S.-J. Li, “Continuous emission of fundamental solitons from vortices in dissipative media by a radial-azimuthal potential,” Opt. Express 21(5), 5561–5566 (2013).
    [PubMed]
  32. B. Liu, Y.-F. Liu, and X.-D. He, “Impact of phase on collision between vortex solitons in three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Opt. Express 22(21), 26203–26211 (2014).
    [PubMed]
  33. V. L. Kalashnikov and E. Sorokin, “Dissipative raman solitons,” Opt. Express 22(24), 30118–30126 (2014).
    [PubMed]
  34. Y. F. Song, H. Zhang, L. M. Zhao, D. Y. Shen, and D. Y. Tang, “Coexistence and interaction of vector and bound vector solitons in a dispersion-managed fiber laser mode locked by graphene,” Opt. Express 24(2), 1814–1822 (2016).
    [PubMed]
  35. D. Mihalache, “Localized optical structures: an overview of recent theoretical and experimental developments,” Proc. Romanian Acad. A 16(1), 62–69 (2015).
  36. D. Mihalache, “Multidimensional localized structures in optical and matter-wave media: a topical survey of recent literature,” Rom. Rep. Phys. 69(1), 403 (2017).
  37. V. Skarka, N. B. Aleksić, W. Krolikowski, D. N. Christodoulides, S. Rakotoarimalala, B. N. Aleksić, and M. Belić, “Self-structuring of stable dissipative breathing vortex solitons in a colloidal nanosuspension,” Opt. Express 25(9), 10090–10102 (2017).
    [PubMed]
  38. G. A. Swartzlander and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69(17), 2503–2506 (1992).
    [PubMed]
  39. B. Luther-Davies, J. Christou, V. Tikhonenko, and Y. S. Kivshar, “Optical vortex solitons: experiment versus theory,” J. Opt. Soc. Am. B 14(11), 3045–3053 (1997).
  40. A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,”, Proc. SPIE 5508, 16–31 (2005).
  41. V. Tikhonenko and N. N. Akhmediev, “Excitation of vortex solitons in a Gaussian beam configuration,” Opt. Commun. 126(1), 108–112 (1996).
  42. A. H. Carlsson, J. N. Malmberg, D. Anderson, M. Lisak, E. A. Ostrovskaya, T. J. Alexander, and Y. S. Kivshar, “Linear and nonlinear waveguides induced by optical vortex solitons,” Opt. Lett. 25(9), 660–662 (2000).
    [PubMed]
  43. Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298(2–3), 81–197 (1998).
  44. D. Andrews, Structured light and its applications: An introduction to phase-structured beams and nanoscale optical forces (Academic, 2008).
  45. S. K. Adhikari, “Stable spatial and spatiotemporal optical soliton in the core of an optical vortex,” Phys. Rev. E. 92(4–1), 042926 (2015).
  46. J. R. Salgueiro, A. H. Carlsson, E. Ostrovskaya, and Y. Kivshar, “Second-harmonic generation in vortex-induced waveguides,” Opt. Lett. 29(6), 593–595 (2004).
    [PubMed]
  47. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
  48. C. T. Law, X. Zhang, and G. A. Swartzlander, “Waveguiding properties of optical vortex solitons,” Opt. Lett. 25(1), 55–57 (2000).
    [PubMed]
  49. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986).
    [PubMed]
  50. Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, “Dynamics of optical vortex solitons,” Opt. Commun. 152(1–3), 198–206 (1998).
  51. I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130(130), 385–392 (1996).
  52. D. Rozas and G. A. Swartzlander., “Observed rotational enhancement of nonlinear optical vortices,” Opt. Lett. 25(2), 126–128 (2000).
    [PubMed]
  53. D. Rozas, C. T. Law, and G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14(11), 3054–3065 (1997).
  54. D. Neshev, A. Dreischuh, M. Assa, and S. Dinev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. 151(4–6), 413–421 (1998).
  55. C. Huang, F. Ye, B. A. Malomed, Y. V. Kartashov, and X. Chen, “Solitary vortices supported by localized parametric gain,” Opt. Lett. 38(13), 2177–2180 (2013).
    [PubMed]
  56. J. Zeng and B. A. Malomed, “Localized dark solitons and vortices in defocusing media with spatially inhomogeneous nonlinearity,” Phys. Rev. E 95(5), 052214 (2017).
  57. A. S. Reyna and C. B. de Araújo, “Guiding and confinement of light induced by optical vortex solitons in a cubic-quintic medium,” Opt. Lett. 41(1), 191–194 (2016).
    [PubMed]
  58. F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Twin-vortex solitons in nonlocal nonlinear media,” Opt. Lett. 35(5), 628–630 (2010).
    [PubMed]
  59. M. A. Porras and F. Ramos, “Quasi-ideal dynamics of vortex solitons embedded in flattop nonlinear Bessel beams,” Opt. Lett. 42(17), 3275–3278 (2017).
    [PubMed]

2017 (4)

D. Mihalache, “Multidimensional localized structures in optical and matter-wave media: a topical survey of recent literature,” Rom. Rep. Phys. 69(1), 403 (2017).

V. Skarka, N. B. Aleksić, W. Krolikowski, D. N. Christodoulides, S. Rakotoarimalala, B. N. Aleksić, and M. Belić, “Self-structuring of stable dissipative breathing vortex solitons in a colloidal nanosuspension,” Opt. Express 25(9), 10090–10102 (2017).
[PubMed]

J. Zeng and B. A. Malomed, “Localized dark solitons and vortices in defocusing media with spatially inhomogeneous nonlinearity,” Phys. Rev. E 95(5), 052214 (2017).

M. A. Porras and F. Ramos, “Quasi-ideal dynamics of vortex solitons embedded in flattop nonlinear Bessel beams,” Opt. Lett. 42(17), 3275–3278 (2017).
[PubMed]

2016 (2)

2015 (1)

D. Mihalache, “Localized optical structures: an overview of recent theoretical and experimental developments,” Proc. Romanian Acad. A 16(1), 62–69 (2015).

2014 (7)

C. Fernandez-Oto, G. J. De Valcárcel, M. Tlidi, K. Panajotov, and K. Staliunas, “Phase-bistable patterns and cavity solitons induced by spatially periodic injection into vertical-cavity surface-emitting lasers,” Phys. Rev. A 89(5), 055802 (2014).

B. A. Malomed, “Spatial solitons supported by localized gain,” J. Opt. Soc. Am. B 31(10), 2460–2475 (2014).

M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimirov, and M. G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Philos Trans A Math Phys Eng Sci 372(2027), 20140101 (2014).
[PubMed]

N. N. Rosanov, G. B. Sochilin, V. D. Vinokurova, and N. V. Vysotina, “Spatial and temporal structures in cavities with oscillating boundaries,” Philos Trans A Math Phys Eng Sci 372(2027), 20140012 (2014).
[PubMed]

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90(2), 023845 (2014).

B. Liu, Y.-F. Liu, and X.-D. He, “Impact of phase on collision between vortex solitons in three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Opt. Express 22(21), 26203–26211 (2014).
[PubMed]

V. L. Kalashnikov and E. Sorokin, “Dissipative raman solitons,” Opt. Express 22(24), 30118–30126 (2014).
[PubMed]

2013 (2)

2012 (1)

P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6(2), 84–92 (2012).

2011 (1)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).

2010 (5)

F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Twin-vortex solitons in nonlocal nonlinear media,” Opt. Lett. 35(5), 628–630 (2010).
[PubMed]

Y. J. He, B. A. Malomed, F. W. Ye, and B. B. Hu, “Dynamics of dissipative spatial solitons over a sharp potential,” J. Opt. Soc. Am. B 27(5), 1139–1142 (2010).

R. Kuszelewicz, S. Barbay, G. Tissoni, and G. Almuneau, “Editorial on dissipative optical solitons,” Eur. Phys. J. D 59(1), 1–2 (2010).

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

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105(21), 213901 (2010).
[PubMed]

2009 (1)

2008 (1)

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77(2), 023814 (2008).

2007 (1)

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability of dissipative optical solitons in thethree-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. A 75(75), 033811 (2007).

2006 (2)

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).
[PubMed]

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(3), 319–326 (2006).

2005 (1)

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,”, Proc. SPIE 5508, 16–31 (2005).

2004 (3)

2003 (1)

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[PubMed]

2002 (2)

Z. Chen and K. McCarthy, “Spatial soliton pixels from partially incoherent light,” Opt. Lett. 27(22), 2019–2021 (2002).
[PubMed]

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74(11), 99–143 (2002).

2001 (2)

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: complex Ginzburg-Landau equation approach,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(5 Pt 2), 056602 (2001).
[PubMed]

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]

2000 (3)

1998 (3)

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, “Dynamics of optical vortex solitons,” Opt. Commun. 152(1–3), 198–206 (1998).

D. Neshev, A. Dreischuh, M. Assa, and S. Dinev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. 151(4–6), 413–421 (1998).

Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298(2–3), 81–197 (1998).

1997 (2)

1996 (4)

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130(130), 385–392 (1996).

V. Tikhonenko and N. N. Akhmediev, “Excitation of vortex solitons in a Gaussian beam configuration,” Opt. Commun. 126(1), 108–112 (1996).

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).
[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).
[PubMed]

1992 (1)

G. A. Swartzlander and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69(17), 2503–2506 (1992).
[PubMed]

1986 (1)

Afanasjev, V. V.

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).
[PubMed]

Akhmediev, N.

P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6(2), 84–92 (2012).

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(11), 2204–2210 (2009).

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: complex Ginzburg-Landau equation approach,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(5 Pt 2), 056602 (2001).
[PubMed]

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 Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(1), 1190–1201 (1996).
[PubMed]

V. Tikhonenko and N. N. Akhmediev, “Excitation of vortex solitons in a Gaussian beam configuration,” Opt. Commun. 126(1), 108–112 (1996).

Aleksic, B. N.

V. Skarka, N. B. Aleksić, W. Krolikowski, D. N. Christodoulides, S. Rakotoarimalala, B. N. Aleksić, and M. Belić, “Self-structuring of stable dissipative breathing vortex solitons in a colloidal nanosuspension,” Opt. Express 25(9), 10090–10102 (2017).
[PubMed]

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90(2), 023845 (2014).

Aleksic, N. B.

V. Skarka, N. B. Aleksić, W. Krolikowski, D. N. Christodoulides, S. Rakotoarimalala, B. N. Aleksić, and M. Belić, “Self-structuring of stable dissipative breathing vortex solitons in a colloidal nanosuspension,” Opt. Express 25(9), 10090–10102 (2017).
[PubMed]

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90(2), 023845 (2014).

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

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105(21), 213901 (2010).
[PubMed]

Alexander, T. J.

Almuneau, G.

R. Kuszelewicz, S. Barbay, G. Tissoni, and G. Almuneau, “Editorial on dissipative optical solitons,” Eur. Phys. J. D 59(1), 1–2 (2010).

Anderson, D.

Aranson, I. S.

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74(11), 99–143 (2002).

Ashkin, A.

Assa, M.

D. Neshev, A. Dreischuh, M. Assa, and S. Dinev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. 151(4–6), 413–421 (1998).

Barbay, S.

R. Kuszelewicz, S. Barbay, G. Tissoni, and G. Almuneau, “Editorial on dissipative optical solitons,” Eur. Phys. J. D 59(1), 1–2 (2010).

Belic, M.

Bjorkholm, J. E.

Carlsson, A. H.

Chang, W.

Chen, X.

Chen, Z.

Chong, A.

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77(2), 023814 (2008).

Christodoulides, D. N.

Christou, J.

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, “Dynamics of optical vortex solitons,” Opt. Commun. 152(1–3), 198–206 (1998).

B. Luther-Davies, J. Christou, V. Tikhonenko, and Y. S. Kivshar, “Optical vortex solitons: experiment versus theory,” J. Opt. Soc. Am. B 14(11), 3045–3053 (1997).

Chu, S.

Clerc, M. G.

M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimirov, and M. G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Philos Trans A Math Phys Eng Sci 372(2027), 20140101 (2014).
[PubMed]

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(7), 073904 (2006).
[PubMed]

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]

de Araújo, C. B.

De Valcárcel, G. J.

C. Fernandez-Oto, G. J. De Valcárcel, M. Tlidi, K. Panajotov, and K. Staliunas, “Phase-bistable patterns and cavity solitons induced by spatially periodic injection into vertical-cavity surface-emitting lasers,” Phys. Rev. A 89(5), 055802 (2014).

Desyatnikov, A. S.

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,”, Proc. SPIE 5508, 16–31 (2005).

Dinev, S.

D. Neshev, A. Dreischuh, M. Assa, and S. Dinev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. 151(4–6), 413–421 (1998).

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130(130), 385–392 (1996).

Dreischuh, A.

D. Neshev, A. Dreischuh, M. Assa, and S. Dinev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. 151(4–6), 413–421 (1998).

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130(130), 385–392 (1996).

Dziedzic, J. M.

Efremidis, N. K.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[PubMed]

Egorov, A. A.

Fernandez-Oto, C.

C. Fernandez-Oto, G. J. De Valcárcel, M. Tlidi, K. Panajotov, and K. Staliunas, “Phase-bistable patterns and cavity solitons induced by spatially periodic injection into vertical-cavity surface-emitting lasers,” Phys. Rev. A 89(5), 055802 (2014).

Firth, W. J.

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).
[PubMed]

Fleischer, J. W.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[PubMed]

Grelu, P.

P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6(2), 84–92 (2012).

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(3), 319–326 (2006).

He, X.-D.

He, Y. J.

Hu, B.

Hu, B. B.

Huang, C.

Kalashnikov, V. L.

Kartashov, Y. V.

Kivshar, Y.

Kivshar, Y. S.

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,”, Proc. SPIE 5508, 16–31 (2005).

A. H. Carlsson, J. N. Malmberg, D. Anderson, M. Lisak, E. A. Ostrovskaya, T. J. Alexander, and Y. S. Kivshar, “Linear and nonlinear waveguides induced by optical vortex solitons,” Opt. Lett. 25(9), 660–662 (2000).
[PubMed]

Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298(2–3), 81–197 (1998).

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, “Dynamics of optical vortex solitons,” Opt. Commun. 152(1–3), 198–206 (1998).

B. Luther-Davies, J. Christou, V. Tikhonenko, and Y. S. Kivshar, “Optical vortex solitons: experiment versus theory,” J. Opt. Soc. Am. B 14(11), 3045–3053 (1997).

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(3), 319–326 (2006).

Kramer, L.

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74(11), 99–143 (2002).

Krolikowski, W.

Kuszelewicz, R.

R. Kuszelewicz, S. Barbay, G. Tissoni, and G. Almuneau, “Editorial on dissipative optical solitons,” Eur. Phys. J. D 59(1), 1–2 (2010).

Law, C. T.

Leblond, H.

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90(2), 023845 (2014).

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

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105(21), 213901 (2010).
[PubMed]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability of dissipative optical solitons in thethree-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. A 75(75), 033811 (2007).

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(3), 319–326 (2006).

Lederer, F.

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

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability of dissipative optical solitons in thethree-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. A 75(75), 033811 (2007).

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).
[PubMed]

Lekic, M.

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90(2), 023845 (2014).

Li, S.-J.

Lisak, M.

Liu, B.

Liu, Y.-F.

Luther-Davies, B.

Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298(2–3), 81–197 (1998).

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, “Dynamics of optical vortex solitons,” Opt. Commun. 152(1–3), 198–206 (1998).

B. Luther-Davies, J. Christou, V. Tikhonenko, and Y. S. Kivshar, “Optical vortex solitons: experiment versus theory,” J. Opt. Soc. Am. B 14(11), 3045–3053 (1997).

Malmberg, J. N.

Malomed, B. A.

J. Zeng and B. A. Malomed, “Localized dark solitons and vortices in defocusing media with spatially inhomogeneous nonlinearity,” Phys. Rev. E 95(5), 052214 (2017).

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90(2), 023845 (2014).

B. A. Malomed, “Spatial solitons supported by localized gain,” J. Opt. Soc. Am. B 31(10), 2460–2475 (2014).

C. Huang, F. Ye, B. A. Malomed, Y. V. Kartashov, and X. Chen, “Solitary vortices supported by localized parametric gain,” Opt. Lett. 38(13), 2177–2180 (2013).
[PubMed]

Y. J. He, B. A. Malomed, F. W. Ye, and B. B. Hu, “Dynamics of dissipative spatial solitons over a sharp potential,” J. Opt. Soc. Am. B 27(5), 1139–1142 (2010).

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105(21), 213901 (2010).
[PubMed]

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

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability of dissipative optical solitons in thethree-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. A 75(75), 033811 (2007).

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).
[PubMed]

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]

Mandel, P.

P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics,” J. Opt. B 6(9), R60–R75 (2004).

Mazilu, D.

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

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability of dissipative optical solitons in thethree-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. A 75(75), 033811 (2007).

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).
[PubMed]

McCarthy, K.

Mihalache, D.

D. Mihalache, “Multidimensional localized structures in optical and matter-wave media: a topical survey of recent literature,” Rom. Rep. Phys. 69(1), 403 (2017).

D. Mihalache, “Localized optical structures: an overview of recent theoretical and experimental developments,” Proc. Romanian Acad. A 16(1), 62–69 (2015).

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90(2), 023845 (2014).

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

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105(21), 213901 (2010).
[PubMed]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability of dissipative optical solitons in thethree-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. A 75(75), 033811 (2007).

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).
[PubMed]

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]

Neshev, D.

D. Neshev, A. Dreischuh, M. Assa, and S. Dinev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. 151(4–6), 413–421 (1998).

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130(130), 385–392 (1996).

Ostrovskaya, E.

Ostrovskaya, E. A.

Padgett, M. J.

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).

Panajotov, K.

C. Fernandez-Oto, G. J. De Valcárcel, M. Tlidi, K. Panajotov, and K. Staliunas, “Phase-bistable patterns and cavity solitons induced by spatially periodic injection into vertical-cavity surface-emitting lasers,” Phys. Rev. A 89(5), 055802 (2014).

M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimirov, and M. G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Philos Trans A Math Phys Eng Sci 372(2027), 20140101 (2014).
[PubMed]

Pismen, L. M.

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, “Dynamics of optical vortex solitons,” Opt. Commun. 152(1–3), 198–206 (1998).

Porras, M. A.

Rakotoarimalala, S.

Ramos, F.

Renninger, W. H.

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77(2), 023814 (2008).

Reyna, A. S.

Rosanov, N. N.

N. N. Rosanov, G. B. Sochilin, V. D. Vinokurova, and N. V. Vysotina, “Spatial and temporal structures in cavities with oscillating boundaries,” Philos Trans A Math Phys Eng Sci 372(2027), 20140012 (2014).
[PubMed]

Rozas, D.

Salgueiro, J. R.

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(3), 319–326 (2006).

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(3), 319–326 (2006).

Scroggie, A. J.

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).
[PubMed]

Segev, M.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[PubMed]

Shen, D. Y.

Skarka, V.

V. Skarka, N. B. Aleksić, W. Krolikowski, D. N. Christodoulides, S. Rakotoarimalala, B. N. Aleksić, and M. Belić, “Self-structuring of stable dissipative breathing vortex solitons in a colloidal nanosuspension,” Opt. Express 25(9), 10090–10102 (2017).
[PubMed]

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90(2), 023845 (2014).

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

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105(21), 213901 (2010).
[PubMed]

Sochilin, G. B.

N. N. Rosanov, G. B. Sochilin, V. D. Vinokurova, and N. V. Vysotina, “Spatial and temporal structures in cavities with oscillating boundaries,” Philos Trans A Math Phys Eng Sci 372(2027), 20140012 (2014).
[PubMed]

Song, Y. F.

Sorokin, E.

Soto-Crespo, J. M.

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: complex Ginzburg-Landau equation approach,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(5 Pt 2), 056602 (2001).
[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).
[PubMed]

Staliunas, K.

M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimirov, and M. G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Philos Trans A Math Phys Eng Sci 372(2027), 20140101 (2014).
[PubMed]

C. Fernandez-Oto, G. J. De Valcárcel, M. Tlidi, K. Panajotov, and K. Staliunas, “Phase-bistable patterns and cavity solitons induced by spatially periodic injection into vertical-cavity surface-emitting lasers,” Phys. Rev. A 89(5), 055802 (2014).

Swartzlander, G. A.

Taki, M.

Tang, D. Y.

Tikhonenko, V.

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, “Dynamics of optical vortex solitons,” Opt. Commun. 152(1–3), 198–206 (1998).

B. Luther-Davies, J. Christou, V. Tikhonenko, and Y. S. Kivshar, “Optical vortex solitons: experiment versus theory,” J. Opt. Soc. Am. B 14(11), 3045–3053 (1997).

V. Tikhonenko and N. N. Akhmediev, “Excitation of vortex solitons in a Gaussian beam configuration,” Opt. Commun. 126(1), 108–112 (1996).

Tissoni, G.

R. Kuszelewicz, S. Barbay, G. Tissoni, and G. Almuneau, “Editorial on dissipative optical solitons,” Eur. Phys. J. D 59(1), 1–2 (2010).

Tlidi, M.

M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimirov, and M. G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Philos Trans A Math Phys Eng Sci 372(2027), 20140101 (2014).
[PubMed]

C. Fernandez-Oto, G. J. De Valcárcel, M. Tlidi, K. Panajotov, and K. Staliunas, “Phase-bistable patterns and cavity solitons induced by spatially periodic injection into vertical-cavity surface-emitting lasers,” Phys. Rev. A 89(5), 055802 (2014).

P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics,” J. Opt. B 6(9), R60–R75 (2004).

Torner, L.

F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Twin-vortex solitons in nonlocal nonlinear media,” Opt. Lett. 35(5), 628–630 (2010).
[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).
[PubMed]

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,”, Proc. SPIE 5508, 16–31 (2005).

Y. V. Kartashov, A. A. Egorov, L. Torner, and D. N. Christodoulides, “Stable soliton complexes in two-dimensional photonic lattices,” Opt. Lett. 29(16), 1918–1920 (2004).
[PubMed]

Town, G.

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: complex Ginzburg-Landau equation approach,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(5 Pt 2), 056602 (2001).
[PubMed]

Velchev, I.

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130(130), 385–392 (1996).

Vinokurova, V. D.

N. N. Rosanov, G. B. Sochilin, V. D. Vinokurova, and N. V. Vysotina, “Spatial and temporal structures in cavities with oscillating boundaries,” Philos Trans A Math Phys Eng Sci 372(2027), 20140012 (2014).
[PubMed]

Vladimirov, A. G.

M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimirov, and M. G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Philos Trans A Math Phys Eng Sci 372(2027), 20140101 (2014).
[PubMed]

Vysotina, N. V.

N. N. Rosanov, G. B. Sochilin, V. D. Vinokurova, and N. V. Vysotina, “Spatial and temporal structures in cavities with oscillating boundaries,” Philos Trans A Math Phys Eng Sci 372(2027), 20140012 (2014).
[PubMed]

Wabnitz, S.

Wise, F. W.

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77(2), 023814 (2008).

Yao, A. M.

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).

Ye, F.

Ye, F. W.

Zeng, J.

J. Zeng and B. A. Malomed, “Localized dark solitons and vortices in defocusing media with spatially inhomogeneous nonlinearity,” Phys. Rev. E 95(5), 052214 (2017).

Zhang, H.

Zhang, X.

Zhao, L. M.

Adv. Opt. Photonics (1)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).

Eur. Phys. J. D (1)

R. Kuszelewicz, S. Barbay, G. Tissoni, and G. Almuneau, “Editorial on dissipative optical solitons,” Eur. Phys. J. D 59(1), 1–2 (2010).

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(3), 319–326 (2006).

J. Opt. B (1)

P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics,” J. Opt. B 6(9), R60–R75 (2004).

J. Opt. Soc. Am. B (5)

Nat. Photonics (1)

P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6(2), 84–92 (2012).

Nature (1)

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[PubMed]

Opt. Commun. (4)

D. Neshev, A. Dreischuh, M. Assa, and S. Dinev, “Motion control of ensembles of ordered optical vortices generated on finite extent background,” Opt. Commun. 151(4–6), 413–421 (1998).

Y. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies, and L. M. Pismen, “Dynamics of optical vortex solitons,” Opt. Commun. 152(1–3), 198–206 (1998).

I. Velchev, A. Dreischuh, D. Neshev, and S. Dinev, “Interactions of optical vortex solitons superimposed on different background beams,” Opt. Commun. 130(130), 385–392 (1996).

V. Tikhonenko and N. N. Akhmediev, “Excitation of vortex solitons in a Gaussian beam configuration,” Opt. Commun. 126(1), 108–112 (1996).

Opt. Express (5)

Opt. Lett. (11)

Z. Chen and K. McCarthy, “Spatial soliton pixels from partially incoherent light,” Opt. Lett. 27(22), 2019–2021 (2002).
[PubMed]

Y. V. Kartashov, A. A. Egorov, L. Torner, and D. N. Christodoulides, “Stable soliton complexes in two-dimensional photonic lattices,” Opt. Lett. 29(16), 1918–1920 (2004).
[PubMed]

A. H. Carlsson, J. N. Malmberg, D. Anderson, M. Lisak, E. A. Ostrovskaya, T. J. Alexander, and Y. S. Kivshar, “Linear and nonlinear waveguides induced by optical vortex solitons,” Opt. Lett. 25(9), 660–662 (2000).
[PubMed]

C. T. Law, X. Zhang, and G. A. Swartzlander, “Waveguiding properties of optical vortex solitons,” Opt. Lett. 25(1), 55–57 (2000).
[PubMed]

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986).
[PubMed]

D. Rozas and G. A. Swartzlander., “Observed rotational enhancement of nonlinear optical vortices,” Opt. Lett. 25(2), 126–128 (2000).
[PubMed]

A. S. Reyna and C. B. de Araújo, “Guiding and confinement of light induced by optical vortex solitons in a cubic-quintic medium,” Opt. Lett. 41(1), 191–194 (2016).
[PubMed]

F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Twin-vortex solitons in nonlocal nonlinear media,” Opt. Lett. 35(5), 628–630 (2010).
[PubMed]

M. A. Porras and F. Ramos, “Quasi-ideal dynamics of vortex solitons embedded in flattop nonlinear Bessel beams,” Opt. Lett. 42(17), 3275–3278 (2017).
[PubMed]

C. Huang, F. Ye, B. A. Malomed, Y. V. Kartashov, and X. Chen, “Solitary vortices supported by localized parametric gain,” Opt. Lett. 38(13), 2177–2180 (2013).
[PubMed]

J. R. Salgueiro, A. H. Carlsson, E. Ostrovskaya, and Y. Kivshar, “Second-harmonic generation in vortex-induced waveguides,” Opt. Lett. 29(6), 593–595 (2004).
[PubMed]

Philos Trans A Math Phys Eng Sci (2)

M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimirov, and M. G. Clerc, “Localized structures in dissipative media: from optics to plant ecology,” Philos Trans A Math Phys Eng Sci 372(2027), 20140101 (2014).
[PubMed]

N. N. Rosanov, G. B. Sochilin, V. D. Vinokurova, and N. V. Vysotina, “Spatial and temporal structures in cavities with oscillating boundaries,” Philos Trans A Math Phys Eng Sci 372(2027), 20140012 (2014).
[PubMed]

Phys. Rep. (1)

Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298(2–3), 81–197 (1998).

Phys. Rev. A (5)

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

C. Fernandez-Oto, G. J. De Valcárcel, M. Tlidi, K. Panajotov, and K. Staliunas, “Phase-bistable patterns and cavity solitons induced by spatially periodic injection into vertical-cavity surface-emitting lasers,” Phys. Rev. A 89(5), 055802 (2014).

V. Skarka, N. B. Aleksić, M. Lekić, B. N. Aleksić, B. A. Malomed, D. Mihalache, and H. Leblond, “Formation of complex two-dimensional dissipative solitons via spontaneous symmetry breaking,” Phys. Rev. A 90(2), 023845 (2014).

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Stability of dissipative optical solitons in thethree-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. A 75(75), 033811 (2007).

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77(2), 023814 (2008).

Phys. Rev. E (1)

J. Zeng and B. A. Malomed, “Localized dark solitons and vortices in defocusing media with spatially inhomogeneous nonlinearity,” Phys. Rev. E 95(5), 052214 (2017).

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: complex Ginzburg-Landau equation approach,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(5 Pt 2), 056602 (2001).
[PubMed]

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]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

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).
[PubMed]

Phys. Rev. Lett. (4)

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).
[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).
[PubMed]

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105(21), 213901 (2010).
[PubMed]

G. A. Swartzlander and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69(17), 2503–2506 (1992).
[PubMed]

Proc. Romanian Acad. A (1)

D. Mihalache, “Localized optical structures: an overview of recent theoretical and experimental developments,” Proc. Romanian Acad. A 16(1), 62–69 (2015).

Proc. SPIE (1)

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,”, Proc. SPIE 5508, 16–31 (2005).

Rev. Mod. Phys. (1)

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74(11), 99–143 (2002).

Rom. Rep. Phys. (1)

D. Mihalache, “Multidimensional localized structures in optical and matter-wave media: a topical survey of recent literature,” Rom. Rep. Phys. 69(1), 403 (2017).

Other (8)

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).

S. Trillo and W. Torruellas, Spatial Solitons (Springer, 2001).

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

W. J. Firth, Self-Organization in Optical Systems and Applications in Information Technology (Springer, 1995).

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, Dissipative Solitons: From Optics to Biology and Medicine (Springer, 2008).

D. Andrews, Structured light and its applications: An introduction to phase-structured beams and nanoscale optical forces (Academic, 2008).

S. K. Adhikari, “Stable spatial and spatiotemporal optical soliton in the core of an optical vortex,” Phys. Rev. E. 92(4–1), 042926 (2015).

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Figures (12)

Fig. 1
Fig. 1 (a) Different propagation scenarios in the plane of parameters (ε, A x ) for M = 1. Below line 1: the input vortical waveform evolves into two elliptically-shaped fundamental (vorticityless) solitons; between lines 1 and 2: the generation of a single vortex soliton; between lines 2 and 3: the generation of a cluster containing several vortex solitons; the upper side of line 3: the excess gain scenario. (b) The input vortical field splits and then evolves into two elliptically-shaped fundamental solitons for A x = 1.8 and ε = 2.0 [corresponding to point M indicated in panel (a)]. (c) The generation of a single vortex soliton for A x = 1.4 and ε = 2.4 [corresponding to point N indicated in panel (a)]. (d) The generation of a cluster containing several vortex solitons for A x = 1.6 and ε = 2.8 [corresponding to point O indicated in panel (a)]. (e) The input vortical field evolves into a wavefield displaying excess gain for A x = 1.4 and ε = 3.4 [corresponding to point P indicated in panel (a)]. The other parameters are β = 0.6, A y = 1.0, and T x = T y = 4.
Fig. 2
Fig. 2 (a) Different propagation scenarios in the plane of parameters (ε, A x ) for M = 2. Below line 1: the input vortex waveform evolves into an irregular beam; between lines 1 and 2: the generation of a cluster containing two vortex solitons; between lines 2 and 3: the generation of a cluster containing several vortex solitons; the upper side of line 3: the excess gain scenario. (b) The input vortical field evolves into an irregular beam for A x = 1.2 and ε = 2.1 [corresponding to point A indicated in panel (a)]. (c) The generation of a vortex cluster composed of two individual vortex solitons for A x = 1.4 and ε = 2.5 [corresponding to point B indicated in panel (a)]. (d) The generation of a vortex cluster containing more than two individual vortex solitons for A x = 1.4 and ε = 3.0 [corresponding to point C indicated in panel (a)]. (e) The input vortical field evolves into a wavefield displaying excess gain for A x = 1.4 and ε = 3.4 [corresponding to point D indicated in panel (a)]. The other parameters are the same as in Fig. 1.
Fig. 3
Fig. 3 (a) Different propagation scenarios in the plane of parameters (ε, A x ) for M = 3. Below line 1: the input optical field evolves into an irregular beam; between lines 1 and 2: the generation of a cluster containing three vortex solitons; between lines 2 and 3: generation of a cluster containing several vortex solitons; the upper side of line 3: the excess gain scenario. (b) The input vortical field evolves into an irregular hollow beam for A x = 1.2, ε = 2.3 [corresponding to point E indicated in panel (a)]; (c) The generation of a vortex cluster composed of three individual vortex solitons for A x = 1.2 and ε = 2.5 [corresponding to point F indicated in panel (a)]; (d) The generation of a vortex cluster containing more than three individual vortex solitons for A x = 1.4 and ε = 2.9 [corresponding to point G indicated in panel (a)]. (e) The input vortical field evolves into an optical waveform displaying excess gain for A x = 1.4 and ε = 3.4 [corresponding to point H indicated in panel (a)]. The other parameters are the same as in Fig. 1.
Fig. 4
Fig. 4 (a) Different propagation scenarios in the plane of parameters (ε, A x ) for M = 4. Below line 1: the input vortical field evolves into an elliptically-shaped beam; between lines 1 and 2: the generation of a cluster containing four vortex solitons; between lines 2 and 3: the generation of a cluster containing several vortex solitons; the upper side of line 3: the excess gain scenario. (b) The input vortical waveform splits into two fundamental solitons that further coalesce and evolve into an elliptically-shaped beam for A x = 1.4, ε = 2.3 [corresponding to point I indicated in panel (a)]; (c) The generation of a vortex cluster composed of four individual vortex solitons for A x = 1.2 and ε = 2.48 [corresponding to point J indicated in panel (a)]; (d) The generation of a vortex cluster containing more than four individual vortex solitons for A x = 1.4 and ε = 2.8 [corresponding to point K indicated in panel (a)]. (e) The input vortical field evolves into an optical waveform displaying excess gain for A x = 1.2 and ε = 3.4 [corresponding to point L indicated in panel (a)]. The other parameters are the same as in Fig. 1.
Fig. 5
Fig. 5 (a) Different soliton scenarios in the plane of parameters (β, A x ) for M = 1. Below line 1: the input wavefield evolves into several vortex solitons forming a cluster nested in the beam (see in Fig. 5(b) the final plot corresponding to the propagation distance z = 36); between lines 1 and 2: the generation of a single vortex soliton; between lines 2 and 3: the input optical field splits into two fragments in the early stage of evolution and then the two splinters coalesce into a rather irregular beam containing three distinct vortices (see in Fig. 5(d) the final plot corresponding to the propagation distance z = 36); above line 3: the input optical beam is vanishing during propagation. (b) The input beam evolves into a cluster composed of several vortex solitons for A x = 1.4 and β = 0.1 [corresponding to point M indicated in panel (a)]; (c) The generation of a single vortex soliton for A x = 1.4 and β = 1.0 [corresponding to point N indicated in panel (a) ]; (d) The input beam splits and then the two splinters coalesce into a rather irregular beam that contains three distinct vortex solitons for A x = 1.8 and β = 2.5 [corresponding to point O indicated in panel (a)]. (e) The input beam is vanishing during propagation for A x = 1.6 and β = 3.0 [corresponding to point P indicated in panel (a)]. The other parameters are ε = 2.6, A y = 1.0, and T x = T y = 4.
Fig. 6
Fig. 6 (a) Different soliton scenarios in the plane of parameters (β, A x ) for M = 2. Below line 1: the input wavefield evolves into several vortex solitons forming a cluster nested in the beam, see in Fig. 6(b) the final plot corresponding to the propagation distance z = 36; between lines 1 and 2: the generation of a vortex cluster composed of two vortex solitons; between lines 2 and 3: the input optical field evolves into an elliptical beam; above line 3: the input optical beam is vanishing during propagation. (b) The input beam evolves into a cluster composed of several vortex solitons for A x = 1.2 and β = 0.25 [corresponding to point A indicated in panel (a)]; (c) The generation of a cluster formed by two vortex solitons for A x = 1.4 and β = 0.6 [corresponding to point B indicated in panel (a) ]; (d) The input beam evolves into an elliptically-shaped beam for A x = 1.6 and β = 2.25 [corresponding to point C indicated in panel (a)]. (e) The input beam is vanishing during propagation for A x = 1.6 and β = 2.6 [corresponding to point D indicated in panel (a)]. The other parameters are the same as in Fig. 5.
Fig. 7
Fig. 7 (a) Different propagation scenarios in the plane of parameters (β, A x ) for M = 3. Below line 1: the input wavefield evolves into several vortex solitons forming a cluster nested in the beam, see in Fig. 7(b) the final plot corresponding to the propagation distance z = 36; between lines 1 and 2: the generation of a vortex cluster composed of three vortex solitons; between lines 2 and 3: the input optical field evolves into an elliptically-shaped beam containing a nested vortex; above line 3: the input beam is vanishing during propagation. (b) The input optical field evolves into a cluster containing several vortex solitons for A x = 1.4 and β = 0.25 [corresponding to point E indicated in panel (a)]. (c) The generation of a cluster formed of three vortex solitons for A x = 1.2 and β = 0.7 [corresponding to point F indicated in panel (a)]. (d) The input optical field evolves into an elliptic beam that contains a nested vortex soliton for A x = 1.4 and β = 1.0 [corresponding to point G indicated in panel (a)]. (e) The input beam is vanishing during propagation for A x = 1.2 and β = 1.3 [corresponding to point H indicated in panel (a)]. The other parameters are the same as in Fig. 5.
Fig. 8
Fig. 8 (a) Different propagation scenarios in the plane of parameters (β, A x ) for M = 4. Below line 1: the input optical field evolves into several vortex solitons forming a cluster nested in the beam; between lines 1 and 2: the generation of a vortex cluster composed of four vortex solitons, see in Fig. 8(c) the final plot corresponding to the propagation distance z = 36; between lines 2 and 3: the input optical field evolves into an irregular beam; above line 3: the input beam is vanishing during propagation. (b) The input optical field evolves into a cluster containing several vortex solitons for A x = 1.2, β = 0.2 [corresponding to point I indicated in panel (a)]. (c) The generation of a cluster formed of four vortex solitons for A x = 1.4 and β = 0.7 [corresponding to point J indicated in panel (a)]. (d) The input optical field splits into two fragments and then the splinters coalesce into a rather irregular beam for A x = 1.6 and β = 0.8 [corresponding to point K indicated in panel (a)]. (e) The input beam is vanishing during propagation for A x = 1.6 and β = 1.1 [corresponding to point L indicated in panel (a)]. The other parameters are the same as in Fig. 5.
Fig. 9
Fig. 9 (a) Different propagation scenarios in the plane of parameters (β, T x ) for M = 1. Below line 1: the input optical field evolves into several vortex solitons nested in the beam; between lines 1 and 2: the generation of a single vortex soliton; between lines 2 and 3: the input optical field splits into two fragments in the early stage of evolution and then the two splinters coalesce into a rather irregular beam containing three distinct vortices (see in Fig. 9(d) the final plot corresponding to the propagation distance z = 36); above line 3: the input beam is vanishing during propagation. (b) The input optical field evolves into a cluster containing several vortex solitons for T x = 7 and β = 0.15 [corresponding to point M indicated in panel (a)]. (c) The generation of a single vortex soliton for T x = 5 and β = 1.5 [corresponding to point N indicated in panel (a)]. (d) The input optical field splits and then the two splinters evolve into a rather irregular beam containing three distinct vortex solitons for T x = 1 and β = 2.1 [corresponding to point O indicated in panel (a)]. (e) The input beam is vanishing during propagation for T x = 3 and β = 4.0 [corresponding to point P indicated in panel (a)]. The other parameters are ε = 2.6, A x = A y = 1.0, and T y = 4.
Fig. 10
Fig. 10 (a) Different propagation scenarios in the plane of parameters (β, T x ) for M = 2. Below line 1: the input optical field evolves into several vortex solitons nested in the beam; between lines 1 and 2: the generation of a vortex cluster composed of two vortex solitons; between lines 2 and 3: the input optical field splits into two fragments, which then coalesce during further evolution; above line 3: the input beam is vanishing during propagation. (b) The input optical field evolves into a cluster containing several vortex solitons for T x = 7 and β = 0.25 [corresponding to point A indicated in panel (a)]. (c) The generation of a cluster formed of two vortex solitons for T x = 5 and β = 0.75 [corresponding to point B indicated in panel (a)]. (d) The evolution of the input optical field for T x = 2 and β = 2.5 [corresponding to point C indicated in panel (a)]. (e) The input beam is vanishing during propagation for T x = 5 and β = 4.0 [corresponding to point D indicated in panel (a)]. The other parameters are the same as in Fig. 9.
Fig. 11
Fig. 11 (a) Different propagation scenarios in the plane of parameters (β, T x ) for M = 3. Below line 1: the input optical field evolves into several vortex solitons nested in the beam; between lines 1 and 2: the generation of a cluster formed of three vortex solitons; between lines 2 and 3: the input optical field splits into two fundamental vortex solitons that further coalesce into a rather irregular beam having a nested vortex soliton; above line 3: the input beam is vanishing during propagation. (b) The input optical field evolves into a cluster containing several vortex solitons for T x = 5 and β = 0.23 [corresponding to point E as indicated in panel (a)]. (c) The generation of a cluster formed of three vortex solitons for T x = 5 and β = 0.75 [corresponding to point F as indicated in panel (a)]. (d) The evolution of the input optical field for T x = 5 and β = 1.4 [corresponding to point G as indicated in panel (a)]. (e) The input beam is vanishing during propagation for T x = 5 and β = 1.6 [corresponding to point H as indicated in panel a)]. The other parameters are the same as in Fig. 9.
Fig. 12
Fig. 12 (a) Different propagation scenarios in the plane of parameters (β, T x ) for M = 4. Below line 1: the input optical field evolves into several vortex solitons nested in the beam; between lines 1 and 2: the generation of a cluster formed of four vortex solitons; between lines 2 and 3: the input optical field splits into two fundamental solitons that further coalesce into a rather irregular beam; above line 3: the input beam is vanishing during propagation. (b) The input optical field evolves into a cluster containing several vortex solitons for T x = 3 and β = 0.31 [corresponding to point I indicated in panel (a)]. (c) The generation of a cluster formed of four vortex solitons for T x = 5 and β = 0.7 [corresponding to point J indicated in panel (a)]. (d) The evolution of the input optical field for T x = 5 and β = 1.2 [corresponding to point K indicated in panel (a)]. (e) The input beam is vanishing during propagation for T x = 3 and β = 1.3 [corresponding to point L indicated in panel (a)]. The other parameters are the same as in Fig. 9.

Equations (3)

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i u z +(1/2)Δu+ | u | 2 u+ν | u | 4 u=iR[ u ].
Eu=β[ A x | sin( z/ T x ) | 2 u x 2 + A y | sin( z/ T y ) | 2 u y 2 ].
u=A( z )exp[ x 2 + y 2 2 w 2 ( z ) +iMθ ].

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