Abstract

We study dissipative ring solitons with vorticity in the frame of the (2+1)-dimensional cubic-quintic complex Ginzburg-Landau equation. In dissipative media, radially symmetric ring structures with any vorticity m can be stable in a finite range of parameters. Beyond the region of stability, the solitons lose the radial symmetry but may remain stable, keeping the same value of the topological charge. We have found bifurcations into solitons with n-fold bending symmetry, with n independent on m. Solitons without circular symmetry can also display (m + 1)-fold modulation behaviour. A sequence of bifurcations can transform the ring soliton into a pulsating or chaotic state which keeps the same value of the topological charge as the original ring.

©2009 Optical Society of America

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  1. G. K. Batchelor,An Introduction to Fluid Dynamics, (Cambridge University Press, 1967).
  2. H. Kleinert, Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation, (World Scientific, Singapore, 2008).
  3. J. G. Cramer, R. L. Forward, M. S. Morris, M. Visser, G. Benford, and G. A. Landis, “Natural Wormholes as Gravitational Lenses,” Phys. Rev. D51, 3117–3120 (1995).
  4. V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
    [Crossref]
  5. J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov “Solitary-wave Vortices in Quadratic Nonlinear Media,” J. Opt. Soc. Am. B 15, 625–627 (1998).
    [Crossref]
  6. A. Dreischuh, G. G. Paulus, F. Zacher, F. Grasbon, and H. Walther, “Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium,” Phys. Rev. E 60, 6111–6117 (1999).
    [Crossref]
  7. N. K. Efremidis, K. Hizanidis, B. A. Malomed, and P. Di Trapani, “Three-Dimensional Vortex Solitons in Self-Defocusing Media,” Phys. Rev. Lett. 98, 113901 (2007).
    [Crossref] [PubMed]
  8. B. A. Malomed, L.-C. Crasovan, and D. Mihalache, “Stability of vortex solitons in the cubic-quintic model,” Physica D 161, 187–201 (2002).
    [Crossref]
  9. Y. J. He, B. A. Malomed, D. Mihalache, and H. Z. Wang, ”Crescent vortex solitons in strongly nonlocal nonlinear media,” Phys. Rev. A 78, 023824 (2008).
    [Crossref]
  10. D. N. Neshev, A. Dreischuh, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Observation of polychromatic vortex solitons,” Opt. Lett. 33, 1851 (2008).
    [Crossref] [PubMed]
  11. V. Tikhonenko, Y. Kivshar, V.V. Steblina, and A.A. Zozulya, “Vortex solitons in a saturable optical medium,” J. Opt. Soc. Am. B 15, 79–86 (1998).
    [Crossref]
  12. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Stable Ring-Profile Vortex Solitons in Bessel Optical Lattices,” Phys. Rev. Lett. 94, 043902 (2005).
    [Crossref] [PubMed]
  13. J. Wang and J. Yang, “Families of vortex solitons in periodic media,” Phys. Rev. A 77, 033834 (2008).
    [Crossref]
  14. J. Yang, “Stability of vortex solitons in a photorefractive optical lattice,” New J. Phys. 6, 47 (2004).
    [Crossref]
  15. T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric Vortex Solitons in Nonlinear Periodic Lattices,” Phys. Rev. Lett. 93, 063901 (2004).
    [Crossref] [PubMed]
  16. G. A. Swartzlander and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
    [Crossref] [PubMed]
  17. V. Tikhonenko and N. Akhmediev, “Excitation of vortex solitons in a Gaussian beam configuration,” Opt. Commun. 126, 108 (1996).
    [Crossref]
  18. I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L. C. Crasovan, and D. Mihalache, “Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation,” Phys. Lett. A 288, 292 (2001).
    [Crossref]
  19. H. Michinel, J. Campo-Táboas, M. L. Quiroga-Teixeiro, J. R. Salgueiro, and R. Garćia-Fernández, “Excitation of stable vortex solitons in nonlinear cubic-quintic materials,” J. Opt. B: Quantum Semiclass. Opt. 3, 314–317 (2001).
    [Crossref]
  20. H. Michinel, J. R. Salgueiro, and M. J. Paz-Alonso, “Square vortex solitons with a large angular momentum,” Phys. Rev. E 70, 066605 (2004).
    [Crossref]
  21. S. V. Fedorov, N. Rosanov, A. N. Shatsev, N. A. Veretenov, and A. G. Vladimirov, “Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with a saturable absorber,” IEEE J. Quantum Electron. 39, 197 (2003).
    [Crossref]
  22. N. N. Rosanov, “Solitons in laser systems with saturable absorption,” in: Dissipative solitons, (Eds.) N. Akhmediev and A. Ankiewicz, Lecture Notes in Physics, V. 661, Springer, Heidelberg, 2005.
  23. 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]
  24. D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed. “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77, 033817 (2008).
    [Crossref]
  25. J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, and N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
    [Crossref] [PubMed]
  26. A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo “Continuously self-focusing and continuously self-defocusing 2-D beams in dissipative media,” Phys. Rev. A 77, 033840 (2008).
    [Crossref]
  27. 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 63, 056602 (2001).
    [Crossref]

2008 (5)

Y. J. He, B. A. Malomed, D. Mihalache, and H. Z. Wang, ”Crescent vortex solitons in strongly nonlocal nonlinear media,” Phys. Rev. A 78, 023824 (2008).
[Crossref]

D. N. Neshev, A. Dreischuh, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Observation of polychromatic vortex solitons,” Opt. Lett. 33, 1851 (2008).
[Crossref] [PubMed]

J. Wang and J. Yang, “Families of vortex solitons in periodic media,” Phys. Rev. A 77, 033834 (2008).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed. “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77, 033817 (2008).
[Crossref]

A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo “Continuously self-focusing and continuously self-defocusing 2-D beams in dissipative media,” Phys. Rev. A 77, 033840 (2008).
[Crossref]

2007 (1)

N. K. Efremidis, K. Hizanidis, B. A. Malomed, and P. Di Trapani, “Three-Dimensional Vortex Solitons in Self-Defocusing Media,” Phys. Rev. Lett. 98, 113901 (2007).
[Crossref] [PubMed]

2005 (1)

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Stable Ring-Profile Vortex Solitons in Bessel Optical Lattices,” Phys. Rev. Lett. 94, 043902 (2005).
[Crossref] [PubMed]

2004 (3)

J. Yang, “Stability of vortex solitons in a photorefractive optical lattice,” New J. Phys. 6, 47 (2004).
[Crossref]

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric Vortex Solitons in Nonlinear Periodic Lattices,” Phys. Rev. Lett. 93, 063901 (2004).
[Crossref] [PubMed]

H. Michinel, J. R. Salgueiro, and M. J. Paz-Alonso, “Square vortex solitons with a large angular momentum,” Phys. Rev. E 70, 066605 (2004).
[Crossref]

2003 (1)

S. V. Fedorov, N. Rosanov, A. N. Shatsev, N. A. Veretenov, and A. G. Vladimirov, “Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with a saturable absorber,” IEEE J. Quantum Electron. 39, 197 (2003).
[Crossref]

2002 (1)

B. A. Malomed, L.-C. Crasovan, and D. Mihalache, “Stability of vortex solitons in the cubic-quintic model,” Physica D 161, 187–201 (2002).
[Crossref]

2001 (4)

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L. C. Crasovan, and D. Mihalache, “Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation,” Phys. Lett. A 288, 292 (2001).
[Crossref]

H. Michinel, J. Campo-Táboas, M. L. Quiroga-Teixeiro, J. R. Salgueiro, and R. Garćia-Fernández, “Excitation of stable vortex solitons in nonlinear cubic-quintic materials,” J. Opt. B: Quantum Semiclass. Opt. 3, 314–317 (2001).
[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]

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 63, 056602 (2001).
[Crossref]

1999 (1)

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grasbon, and H. Walther, “Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium,” Phys. Rev. E 60, 6111–6117 (1999).
[Crossref]

1998 (2)

1996 (1)

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

1995 (1)

J. G. Cramer, R. L. Forward, M. S. Morris, M. Visser, G. Benford, and G. A. Landis, “Natural Wormholes as Gravitational Lenses,” Phys. Rev. D51, 3117–3120 (1995).

1992 (2)

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[Crossref]

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

1991 (1)

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, and N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[Crossref] [PubMed]

Akhmediev, N.

A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo “Continuously self-focusing and continuously self-defocusing 2-D beams in dissipative media,” Phys. Rev. A 77, 033840 (2008).
[Crossref]

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 63, 056602 (2001).
[Crossref]

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

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, and N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[Crossref] [PubMed]

Alexander, T. J.

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric Vortex Solitons in Nonlinear Periodic Lattices,” Phys. Rev. Lett. 93, 063901 (2004).
[Crossref] [PubMed]

Ankiewicz, A.

A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo “Continuously self-focusing and continuously self-defocusing 2-D beams in dissipative media,” Phys. Rev. A 77, 033840 (2008).
[Crossref]

Batchelor, G. K.

G. K. Batchelor,An Introduction to Fluid Dynamics, (Cambridge University Press, 1967).

Bazhenov, V. Yu.

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[Crossref]

Benford, G.

J. G. Cramer, R. L. Forward, M. S. Morris, M. Visser, G. Benford, and G. A. Landis, “Natural Wormholes as Gravitational Lenses,” Phys. Rev. D51, 3117–3120 (1995).

Buryak, A. V.

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L. C. Crasovan, and D. Mihalache, “Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation,” Phys. Lett. A 288, 292 (2001).
[Crossref]

Campo-Táboas, J.

H. Michinel, J. Campo-Táboas, M. L. Quiroga-Teixeiro, J. R. Salgueiro, and R. Garćia-Fernández, “Excitation of stable vortex solitons in nonlinear cubic-quintic materials,” J. Opt. B: Quantum Semiclass. Opt. 3, 314–317 (2001).
[Crossref]

Cramer, J. G.

J. G. Cramer, R. L. Forward, M. S. Morris, M. Visser, G. Benford, and G. A. Landis, “Natural Wormholes as Gravitational Lenses,” Phys. Rev. D51, 3117–3120 (1995).

Crasovan, L. C.

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L. C. Crasovan, and D. Mihalache, “Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation,” Phys. Lett. A 288, 292 (2001).
[Crossref]

Crasovan, L.-C.

B. A. Malomed, L.-C. Crasovan, and D. Mihalache, “Stability of vortex solitons in the cubic-quintic model,” Physica D 161, 187–201 (2002).
[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]

Desyatnikov, A. S.

Devine, N.

A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo “Continuously self-focusing and continuously self-defocusing 2-D beams in dissipative media,” Phys. Rev. A 77, 033840 (2008).
[Crossref]

Dreischuh, A.

D. N. Neshev, A. Dreischuh, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Observation of polychromatic vortex solitons,” Opt. Lett. 33, 1851 (2008).
[Crossref] [PubMed]

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grasbon, and H. Walther, “Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium,” Phys. Rev. E 60, 6111–6117 (1999).
[Crossref]

Efremidis, N. K.

N. K. Efremidis, K. Hizanidis, B. A. Malomed, and P. Di Trapani, “Three-Dimensional Vortex Solitons in Self-Defocusing Media,” Phys. Rev. Lett. 98, 113901 (2007).
[Crossref] [PubMed]

Fedorov, S. V.

S. V. Fedorov, N. Rosanov, A. N. Shatsev, N. A. Veretenov, and A. G. Vladimirov, “Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with a saturable absorber,” IEEE J. Quantum Electron. 39, 197 (2003).
[Crossref]

Forward, R. L.

J. G. Cramer, R. L. Forward, M. S. Morris, M. Visser, G. Benford, and G. A. Landis, “Natural Wormholes as Gravitational Lenses,” Phys. Rev. D51, 3117–3120 (1995).

Garcia-Fernández, R.

H. Michinel, J. Campo-Táboas, M. L. Quiroga-Teixeiro, J. R. Salgueiro, and R. Garćia-Fernández, “Excitation of stable vortex solitons in nonlinear cubic-quintic materials,” J. Opt. B: Quantum Semiclass. Opt. 3, 314–317 (2001).
[Crossref]

Grasbon, F.

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grasbon, and H. Walther, “Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium,” Phys. Rev. E 60, 6111–6117 (1999).
[Crossref]

He, Y. J.

Y. J. He, B. A. Malomed, D. Mihalache, and H. Z. Wang, ”Crescent vortex solitons in strongly nonlocal nonlinear media,” Phys. Rev. A 78, 023824 (2008).
[Crossref]

Heatley, D. R.

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, and N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[Crossref] [PubMed]

Hizanidis, K.

N. K. Efremidis, K. Hizanidis, B. A. Malomed, and P. Di Trapani, “Three-Dimensional Vortex Solitons in Self-Defocusing Media,” Phys. Rev. Lett. 98, 113901 (2007).
[Crossref] [PubMed]

Kartashov, Y. V.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Stable Ring-Profile Vortex Solitons in Bessel Optical Lattices,” Phys. Rev. Lett. 94, 043902 (2005).
[Crossref] [PubMed]

Kivshar, Y.

Kivshar, Y. S.

D. N. Neshev, A. Dreischuh, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Observation of polychromatic vortex solitons,” Opt. Lett. 33, 1851 (2008).
[Crossref] [PubMed]

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric Vortex Solitons in Nonlinear Periodic Lattices,” Phys. Rev. Lett. 93, 063901 (2004).
[Crossref] [PubMed]

Kleinert, H.

H. Kleinert, Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation, (World Scientific, Singapore, 2008).

Krolikowski, W.

Landis, G. A.

J. G. Cramer, R. L. Forward, M. S. Morris, M. Visser, G. Benford, and G. A. Landis, “Natural Wormholes as Gravitational Lenses,” Phys. Rev. D51, 3117–3120 (1995).

Law, C. T.

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

Leblond, H.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed. “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77, 033817 (2008).
[Crossref]

Lederer, F.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed. “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77, 033817 (2008).
[Crossref]

Malomed, B. A.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed. “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77, 033817 (2008).
[Crossref]

Y. J. He, B. A. Malomed, D. Mihalache, and H. Z. Wang, ”Crescent vortex solitons in strongly nonlocal nonlinear media,” Phys. Rev. A 78, 023824 (2008).
[Crossref]

N. K. Efremidis, K. Hizanidis, B. A. Malomed, and P. Di Trapani, “Three-Dimensional Vortex Solitons in Self-Defocusing Media,” Phys. Rev. Lett. 98, 113901 (2007).
[Crossref] [PubMed]

B. A. Malomed, L.-C. Crasovan, and D. Mihalache, “Stability of vortex solitons in the cubic-quintic model,” Physica D 161, 187–201 (2002).
[Crossref]

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L. C. Crasovan, and D. Mihalache, “Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation,” Phys. Lett. A 288, 292 (2001).
[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]

Mazilu, D.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed. “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77, 033817 (2008).
[Crossref]

Michinel, H.

H. Michinel, J. R. Salgueiro, and M. J. Paz-Alonso, “Square vortex solitons with a large angular momentum,” Phys. Rev. E 70, 066605 (2004).
[Crossref]

H. Michinel, J. Campo-Táboas, M. L. Quiroga-Teixeiro, J. R. Salgueiro, and R. Garćia-Fernández, “Excitation of stable vortex solitons in nonlinear cubic-quintic materials,” J. Opt. B: Quantum Semiclass. Opt. 3, 314–317 (2001).
[Crossref]

Mihalache, D.

Y. J. He, B. A. Malomed, D. Mihalache, and H. Z. Wang, ”Crescent vortex solitons in strongly nonlocal nonlinear media,” Phys. Rev. A 78, 023824 (2008).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed. “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77, 033817 (2008).
[Crossref]

B. A. Malomed, L.-C. Crasovan, and D. Mihalache, “Stability of vortex solitons in the cubic-quintic model,” Physica D 161, 187–201 (2002).
[Crossref]

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L. C. Crasovan, and D. Mihalache, “Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation,” Phys. Lett. A 288, 292 (2001).
[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]

Morris, M. S.

J. G. Cramer, R. L. Forward, M. S. Morris, M. Visser, G. Benford, and G. A. Landis, “Natural Wormholes as Gravitational Lenses,” Phys. Rev. D51, 3117–3120 (1995).

Neshev, D. N.

Paulus, G. G.

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grasbon, and H. Walther, “Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium,” Phys. Rev. E 60, 6111–6117 (1999).
[Crossref]

Paz-Alonso, M. J.

H. Michinel, J. R. Salgueiro, and M. J. Paz-Alonso, “Square vortex solitons with a large angular momentum,” Phys. Rev. E 70, 066605 (2004).
[Crossref]

Petrov, D. V.

Quiroga-Teixeiro, M. L.

H. Michinel, J. Campo-Táboas, M. L. Quiroga-Teixeiro, J. R. Salgueiro, and R. Garćia-Fernández, “Excitation of stable vortex solitons in nonlinear cubic-quintic materials,” J. Opt. B: Quantum Semiclass. Opt. 3, 314–317 (2001).
[Crossref]

Rosanov, N.

S. V. Fedorov, N. Rosanov, A. N. Shatsev, N. A. Veretenov, and A. G. Vladimirov, “Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with a saturable absorber,” IEEE J. Quantum Electron. 39, 197 (2003).
[Crossref]

Rosanov, N. N.

N. N. Rosanov, “Solitons in laser systems with saturable absorption,” in: Dissipative solitons, (Eds.) N. Akhmediev and A. Ankiewicz, Lecture Notes in Physics, V. 661, Springer, Heidelberg, 2005.

Salgueiro, J. R.

H. Michinel, J. R. Salgueiro, and M. J. Paz-Alonso, “Square vortex solitons with a large angular momentum,” Phys. Rev. E 70, 066605 (2004).
[Crossref]

H. Michinel, J. Campo-Táboas, M. L. Quiroga-Teixeiro, J. R. Salgueiro, and R. Garćia-Fernández, “Excitation of stable vortex solitons in nonlinear cubic-quintic materials,” J. Opt. B: Quantum Semiclass. Opt. 3, 314–317 (2001).
[Crossref]

Sammut, R. A.

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L. C. Crasovan, and D. Mihalache, “Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation,” Phys. Lett. A 288, 292 (2001).
[Crossref]

Shatsev, A. N.

S. V. Fedorov, N. Rosanov, A. N. Shatsev, N. A. Veretenov, and A. G. Vladimirov, “Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with a saturable absorber,” IEEE J. Quantum Electron. 39, 197 (2003).
[Crossref]

Shvedov, V.

Soskin, M. S.

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[Crossref]

Soto-Crespo, J. M.

A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo “Continuously self-focusing and continuously self-defocusing 2-D beams in dissipative media,” Phys. Rev. A 77, 033840 (2008).
[Crossref]

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 63, 056602 (2001).
[Crossref]

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov “Solitary-wave Vortices in Quadratic Nonlinear Media,” J. Opt. Soc. Am. B 15, 625–627 (1998).
[Crossref]

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, and N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[Crossref] [PubMed]

Steblina, V.V.

Sukhorukov, A. A.

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric Vortex Solitons in Nonlinear Periodic Lattices,” Phys. Rev. Lett. 93, 063901 (2004).
[Crossref] [PubMed]

Swartzlander, G. A.

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

Tikhonenko, V.

V. Tikhonenko, Y. Kivshar, V.V. Steblina, and A.A. Zozulya, “Vortex solitons in a saturable optical medium,” J. Opt. Soc. Am. B 15, 79–86 (1998).
[Crossref]

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

Torner, L.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Stable Ring-Profile Vortex Solitons in Bessel Optical Lattices,” Phys. Rev. Lett. 94, 043902 (2005).
[Crossref] [PubMed]

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov “Solitary-wave Vortices in Quadratic Nonlinear Media,” J. Opt. Soc. Am. B 15, 625–627 (1998).
[Crossref]

Torres, J. P.

Towers, I.

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L. C. Crasovan, and D. Mihalache, “Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation,” Phys. Lett. A 288, 292 (2001).
[Crossref]

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 63, 056602 (2001).
[Crossref]

Trapani, P. Di

N. K. Efremidis, K. Hizanidis, B. A. Malomed, and P. Di Trapani, “Three-Dimensional Vortex Solitons in Self-Defocusing Media,” Phys. Rev. Lett. 98, 113901 (2007).
[Crossref] [PubMed]

Vasnetsov, M. V.

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[Crossref]

Veretenov, N. A.

S. V. Fedorov, N. Rosanov, A. N. Shatsev, N. A. Veretenov, and A. G. Vladimirov, “Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with a saturable absorber,” IEEE J. Quantum Electron. 39, 197 (2003).
[Crossref]

Visser, M.

J. G. Cramer, R. L. Forward, M. S. Morris, M. Visser, G. Benford, and G. A. Landis, “Natural Wormholes as Gravitational Lenses,” Phys. Rev. D51, 3117–3120 (1995).

Vladimirov, A. G.

S. V. Fedorov, N. Rosanov, A. N. Shatsev, N. A. Veretenov, and A. G. Vladimirov, “Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with a saturable absorber,” IEEE J. Quantum Electron. 39, 197 (2003).
[Crossref]

Vysloukh, V. A.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Stable Ring-Profile Vortex Solitons in Bessel Optical Lattices,” Phys. Rev. Lett. 94, 043902 (2005).
[Crossref] [PubMed]

Walther, H.

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grasbon, and H. Walther, “Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium,” Phys. Rev. E 60, 6111–6117 (1999).
[Crossref]

Wang, H. Z.

Y. J. He, B. A. Malomed, D. Mihalache, and H. Z. Wang, ”Crescent vortex solitons in strongly nonlocal nonlinear media,” Phys. Rev. A 78, 023824 (2008).
[Crossref]

Wang, J.

J. Wang and J. Yang, “Families of vortex solitons in periodic media,” Phys. Rev. A 77, 033834 (2008).
[Crossref]

Wright, E. M.

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, and N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[Crossref] [PubMed]

Yang, J.

J. Wang and J. Yang, “Families of vortex solitons in periodic media,” Phys. Rev. A 77, 033834 (2008).
[Crossref]

J. Yang, “Stability of vortex solitons in a photorefractive optical lattice,” New J. Phys. 6, 47 (2004).
[Crossref]

Zacher, F.

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grasbon, and H. Walther, “Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium,” Phys. Rev. E 60, 6111–6117 (1999).
[Crossref]

Zozulya, A.A.

IEEE J. Quantum Electron. (1)

S. V. Fedorov, N. Rosanov, A. N. Shatsev, N. A. Veretenov, and A. G. Vladimirov, “Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with a saturable absorber,” IEEE J. Quantum Electron. 39, 197 (2003).
[Crossref]

J. Mod. Opt. (1)

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[Crossref]

J. Opt. B: Quantum Semiclass. Opt. (1)

H. Michinel, J. Campo-Táboas, M. L. Quiroga-Teixeiro, J. R. Salgueiro, and R. Garćia-Fernández, “Excitation of stable vortex solitons in nonlinear cubic-quintic materials,” J. Opt. B: Quantum Semiclass. Opt. 3, 314–317 (2001).
[Crossref]

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

New J. Phys. (1)

J. Yang, “Stability of vortex solitons in a photorefractive optical lattice,” New J. Phys. 6, 47 (2004).
[Crossref]

Opt. Commun. (1)

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

Opt. Lett. (1)

Phys. Lett. A (1)

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L. C. Crasovan, and D. Mihalache, “Stability of spinning ring solitons of the cubic-quintic nonlinear Schrödinger equation,” Phys. Lett. A 288, 292 (2001).
[Crossref]

Phys. Rev. (1)

J. G. Cramer, R. L. Forward, M. S. Morris, M. Visser, G. Benford, and G. A. Landis, “Natural Wormholes as Gravitational Lenses,” Phys. Rev. D51, 3117–3120 (1995).

Phys. Rev. A (5)

J. Wang and J. Yang, “Families of vortex solitons in periodic media,” Phys. Rev. A 77, 033834 (2008).
[Crossref]

Y. J. He, B. A. Malomed, D. Mihalache, and H. Z. Wang, ”Crescent vortex solitons in strongly nonlocal nonlinear media,” Phys. Rev. A 78, 023824 (2008).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed. “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A 77, 033817 (2008).
[Crossref]

J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, and N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636–644 (1991).
[Crossref] [PubMed]

A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo “Continuously self-focusing and continuously self-defocusing 2-D beams in dissipative media,” Phys. Rev. A 77, 033840 (2008).
[Crossref]

Phys. Rev. E (4)

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 63, 056602 (2001).
[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]

A. Dreischuh, G. G. Paulus, F. Zacher, F. Grasbon, and H. Walther, “Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium,” Phys. Rev. E 60, 6111–6117 (1999).
[Crossref]

H. Michinel, J. R. Salgueiro, and M. J. Paz-Alonso, “Square vortex solitons with a large angular momentum,” Phys. Rev. E 70, 066605 (2004).
[Crossref]

Phys. Rev. Lett. (4)

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric Vortex Solitons in Nonlinear Periodic Lattices,” Phys. Rev. Lett. 93, 063901 (2004).
[Crossref] [PubMed]

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

N. K. Efremidis, K. Hizanidis, B. A. Malomed, and P. Di Trapani, “Three-Dimensional Vortex Solitons in Self-Defocusing Media,” Phys. Rev. Lett. 98, 113901 (2007).
[Crossref] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Stable Ring-Profile Vortex Solitons in Bessel Optical Lattices,” Phys. Rev. Lett. 94, 043902 (2005).
[Crossref] [PubMed]

Physica D (1)

B. A. Malomed, L.-C. Crasovan, and D. Mihalache, “Stability of vortex solitons in the cubic-quintic model,” Physica D 161, 187–201 (2002).
[Crossref]

Other (3)

G. K. Batchelor,An Introduction to Fluid Dynamics, (Cambridge University Press, 1967).

H. Kleinert, Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation, (World Scientific, Singapore, 2008).

N. N. Rosanov, “Solitons in laser systems with saturable absorption,” in: Dissipative solitons, (Eds.) N. Akhmediev and A. Ankiewicz, Lecture Notes in Physics, V. 661, Springer, Heidelberg, 2005.

Supplementary Material (7)

» Media 1: MPG (1406 KB)     
» Media 2: MPG (1406 KB)     
» Media 3: MPG (1406 KB)     
» Media 4: MPG (1125 KB)     
» Media 5: MPG (1125 KB)     
» Media 6: MPG (1124 KB)     
» Media 7: MPG (1125 KB)     

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Figures (15)

Fig. 1.
Fig. 1. The power Q vs ε for the radially symmetric vortex dissipative solitons with m = 1,2,3,4 and 5. Note the logarithmic scale along the vertical axis.

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Fig. 2.
Fig. 2. Power Q versus ε diagram for the ring solitons with vorticity m = 1. The red and blue curves and the solid part of the green curve correspond to stable solutions, while the dashed green one is for unstable ones. Solutions on the red curve have a radially symmetric amplitude profile while other solutions are radially asymmetric. The yellow inset in the lower right corner is a magnification of the curves enclosed into the small yellow rectangle in the upper left part of the main diagram. This magnification shows more clearly the bifurcation of the pulsating vortex solutions (blue stripe) from the stationary radially asymmetric ones (green curve).

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Fig. 3.
Fig. 3. 2D vortex soliton profiles for m = 1 for the equation parameters, written in the upper left part of the figure. In the cases (a) and (b) the power, Q is constant, while for (c) the power is a periodic function of z, as it is shown on top of the panel (c).

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Fig. 4.
Fig. 4. (Upper row) Contour color plots of the soliton profiles shown in Fig. 3. (Lower row) Contour color plots of the phase profiles for the same solutions. The periodic evolution of the soliton profile shown in (c1) can be seen in the linked movie (Media 1).

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Fig. 5.
Fig. 5. Regions of existence of vortex solitons with m = 2. The red curve corresponds to the radially symmetric ring solitons. Solitons represented by the blue curve are radially asymmetric. The thick filled green circle indicates the location, in the parameter space, of the vortex soliton whose intensity and phase profiles are shown in Fig. 6.

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Fig. 6.
Fig. 6. Radially asymmetric ring vortex soliton with m = 2. (a) 3-D plot of the intensity profile. (b) Color contour plot of the same intensity profile in the (x, y)-plane. (c) Color contour plot of the phase profile. The system parameters are chosen at the thick filled green circle in Fig. 5.

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Fig. 7.
Fig. 7. Regions of existence of vortex solitons with m = 3 in the ε domain. The solid (dashed) red curve represents stable (unstable) radially symmetric solitons. The solid blue and green curves correspond to stable radially asymmetric ring solitons while the blue dashed curve stands for unstable ones.

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Fig. 8.
Fig. 8. 3-D intensity profiles of the radially asymmetric ring vortex solutions whose location in the parameter space is indicated in Fig. 7 by the gray filled circles. The two profiles belong to (a) the green and b) the blue branches of solitons in Fig. 7 respectively. The green profile is stable while the blue one is slightly unstable and converges, on propagation, to the green one. The profile (b) has four-fold bending symmetry while the profile in (a) is completely asymmetric. Fig. 9 shows more clearly the difference between them.

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Fig. 9.
Fig. 9. (upper row) Color contour plots of the intensity in the (x,y) plane of the ring solitons with m = 3 shown in Fig. 8. (lower row) Phase profiles for the same solitons. The right column corresponds to the upper grey filled circle in Fig. 7 while the left column to the lower circle. Evolution of each ring soliton can also be seen in animation, i.e. a1) (Media 2) b1) (Media 3).

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Fig. 10.
Fig. 10. Bifurcation diagram for the ring vortex solitons with m = 4. The upper left yellow inset is a magnified part of the diagram in the lower small yellow box. In the inset, instead or representing the allowed values of Q on propagation, only maxima (red) and minima (blue) of Q(z) are shown

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Fig. 11.
Fig. 11. Two examples of radially asymmetric ring solitons with m = 4. Each one can be seen in animation (Media 4 and Media 5).

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Fig. 12.
Fig. 12. Evolution of Q for stationary (light blue curve), quasi-periodic pulsating (magenta) and chaotic (orange) vortex solitons. The colors correspond to the thick vertical lines in Fig. 10.

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Fig. 13.
Fig. 13. (a) Three-dimensional plot of the ring soliton with m = 9 for ε = 0.745. Color contour plots of (b) the intensity and (c) the phase of the same vortex dissipative soliton.

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Fig. 14.
Fig. 14. Double periodic evolution of Q vs z for vortex dissipative soliton with m = 9 (blue curve). The yellow inset on the top of the figure shows a magnified part of the blue curve enclosed into a small yellow box to the right of z = 0.

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Fig. 15.
Fig. 15. (a) Movie (Media 6) showing the fast clockwise rotation of the nine maxima of the vortex dissipative soliton with m = 9. (b) Movie (Media 7) showing the slow rotation of the dissipative soliton with four-fold bending symmetry in the opposite direction.

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Equations (4)

Equations on this page are rendered with MathJax. Learn more.

i ψ z + D 2 ∇ ⊥ 2 ψ + ∣ ψ ∣ 2 ψ + ν ∣ ψ ∣ 4 ψ = iδψ + iε ∣ ψ ∣ 2 ψ + iβ ∇ ⊥ 2 ψ + iμ ∣ ψ ∣ 4 ψ .
∇ ⊥ 2 = ∂ ∂ x 2 + ∂ 2 ∂ y 2
Q ( z ) = ∫ − ∞ ∞ ∫ − ∞ ∞ ∣ ψ ( x , y , z ) ∣ 2 dxdy ,
ψ ( x , y , 0 ) = A 0 r m exp ( − r 2 w 2 + i m θ )

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