Abstract

Dissipative media admit the existence of two types of stationary self-organized beams: continuously self-focused and continuously self-defocused. Each beam is stable inside of a certain region of its existence. Beyond these two regions, beams loose their stability, and new dynamical behaviors appear. We present several types of instabilities related to each beam configuration and give examples of beam dynamics in the areas adjacent to the two regions. We observed that, in one case beams loose the radial symmetry while in the other one the radial symmetry is conserved during complicated beam transformations.

© 2008 Optical Society of America

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  1. (Eds.) N. Akhmediev and A. Ankiewicz, Dissipative solitons Lecture Notes in Physics, V. 661 (Springer, Heidelberg, 2005).
    [CrossRef]
  2. Dissipative Solitons: From optics to biology and medicine Lecture Notes in Physics, V 751, (Eds.) N. Akhmediev and A. Ankiewicz, (Springer, Berlin-Heidelberg, 2008).
  3. M. Tlidi, M. Taki, and T. Kolokolnikov, "Dissipative Localized Structures in Extended Systems," Chaos 17, 037101, 2007.
    [CrossRef] [PubMed]
  4. M. Tlidi, A. Vladimirov and P. Mandel, "Curvature instability in passive diffractive resonators," Phys. Rev. Lett. 98, 233901, (2002).
    [CrossRef]
  5. I. S. Aranson and L. Kramer, "The world of the complex Ginzburg-Landau equation," Rev. Mod. Phys. 74, 99 (2002).
    [CrossRef]
  6. O. Descalzi, G. During and E. Tirapegui, "On the stable hole solutions in the complex Ginzburg-Landau equation," Physica A 356, 66-71 (2005).
    [CrossRef]
  7. H. A. Haus, "Theory of mode locking with a slow saturable absorber," IEEE Journ. of Quantum Electron.,  QE-11 (9), 736 (1975).
    [CrossRef]
  8. W. H. Renninger, A. Chong, and F. W. Wise, "Dissipative solitons in normal-dispersion fiber lasers," Phys. Rev. A 77, 023814 (2008).
    [CrossRef]
  9. A. Komarov, H. Leblond, and F. Sanchez, "Quintic complex Ginzburg-Landau model for ring fiber lasers," Phys. Rev. E 72, 025604(R) (2005).
    [CrossRef]
  10. M. Taki, N. Ouarzazi, H. Ward and P. Glorieux, "Nonlinear front propagation in optical parametric oscillators," J. Opt. Soc. Am. B 17, 997 (2000).
    [CrossRef]
  11. N. N. Rosanov, Solitons in laser systems with saturable absorption, Dissipative solitons Lecture Notes in Physics, V. 661, (Eds.) N. Akhmediev and A. Ankiewicz, (Springer, Heidelberg, 2005).
  12. E. A. Ultanir and G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, "Stable dissipative solitons in semiconductor optical amplifiers," Phys. Rev. Lett. 90, 253903 (2003).
    [CrossRef] [PubMed]
  13. C. Montes, A. Mikhailov, A. Picozzi, and F. Ginovart, "Dissipative three-wave structures in stimulated backscattering. I. A subluminous solitary attractor," Phys. Rev. E 55, 1086 (1997).
    [CrossRef]
  14. R. J. Deissler and H. R. Brand, "Periodic, quasiperiodic, and chaotic localized solutions of the quintic complex Ginzburg-Landau equation," Phys. Rev. Lett. 72, 478 - 481 (1994).
    [CrossRef] [PubMed]
  15. X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, "Cavity solitons in a driven VCSEL above threshold," J. of Sel. Topics on Quant.Electr. 12, 339-351 (2006).
    [CrossRef]
  16. 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]
  17. J. M. Soto-Crespo, N. Akhmediev y A. Ankiewicz. "Pulsating, creeping and erupting solitons in dissipative systems," Phys.Rev. Lett. 85, 2937 (2000).
    [CrossRef]
  18. 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,"Phys. Rev. E. 63, 056602 (2001).
    [CrossRef]
  19. J. M. Soto-Crespo, Ph. Grelu, and N. Akhmediev"Optical bullets and "rockets" in nonlinear dissipative systems and their transformations and interactions," Opt. Express 14, 4013 (2006).
    [CrossRef] [PubMed]
  20. J. M. Soto-Crespo, N. Akhmediev and Ph. Grelu "Optical bullets and double bullet complexes in dissipative systems," Phys. Rev. E 74, 046612 (2006).
    [CrossRef]
  21. N. AkhmedievJ.M. Soto-Crespo and Ph. Grelu, "Vibrating and shaking soliton pairs in dissipative systems." Phys. Lett. A 364, 413 (2007).
    [CrossRef]
  22. E. Tsoy and N. Akhmediev, "Bifurcations from stationary to pulsating solitons in the cubic quintic complex Ginzburg Landau equation," Phys. Lett. A 343, 417-422 (2005).
    [CrossRef]

2008 (2)

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

N. AkhmedievJ.M. Soto-Crespo and Ph. Grelu, "Vibrating and shaking soliton pairs in dissipative systems." Phys. Lett. A 364, 413 (2007).
[CrossRef]

M. Tlidi, M. Taki, and T. Kolokolnikov, "Dissipative Localized Structures in Extended Systems," Chaos 17, 037101, 2007.
[CrossRef] [PubMed]

2006 (3)

J. M. Soto-Crespo, Ph. Grelu, and N. Akhmediev"Optical bullets and "rockets" in nonlinear dissipative systems and their transformations and interactions," Opt. Express 14, 4013 (2006).
[CrossRef] [PubMed]

J. M. Soto-Crespo, N. Akhmediev and Ph. Grelu "Optical bullets and double bullet complexes in dissipative systems," Phys. Rev. E 74, 046612 (2006).
[CrossRef]

X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, "Cavity solitons in a driven VCSEL above threshold," J. of Sel. Topics on Quant.Electr. 12, 339-351 (2006).
[CrossRef]

2005 (3)

E. Tsoy and N. Akhmediev, "Bifurcations from stationary to pulsating solitons in the cubic quintic complex Ginzburg Landau equation," Phys. Lett. A 343, 417-422 (2005).
[CrossRef]

O. Descalzi, G. During and E. Tirapegui, "On the stable hole solutions in the complex Ginzburg-Landau equation," Physica A 356, 66-71 (2005).
[CrossRef]

A. Komarov, H. Leblond, and F. Sanchez, "Quintic complex Ginzburg-Landau model for ring fiber lasers," Phys. Rev. E 72, 025604(R) (2005).
[CrossRef]

2003 (1)

E. A. Ultanir and G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, "Stable dissipative solitons in semiconductor optical amplifiers," Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

2002 (2)

M. Tlidi, A. Vladimirov and P. Mandel, "Curvature instability in passive diffractive resonators," Phys. Rev. Lett. 98, 233901, (2002).
[CrossRef]

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

2001 (1)

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,"Phys. Rev. E. 63, 056602 (2001).
[CrossRef]

2000 (2)

J. M. Soto-Crespo, N. Akhmediev y A. Ankiewicz. "Pulsating, creeping and erupting solitons in dissipative systems," Phys.Rev. Lett. 85, 2937 (2000).
[CrossRef]

M. Taki, N. Ouarzazi, H. Ward and P. Glorieux, "Nonlinear front propagation in optical parametric oscillators," J. Opt. Soc. Am. B 17, 997 (2000).
[CrossRef]

1997 (1)

C. Montes, A. Mikhailov, A. Picozzi, and F. Ginovart, "Dissipative three-wave structures in stimulated backscattering. I. A subluminous solitary attractor," Phys. Rev. E 55, 1086 (1997).
[CrossRef]

1994 (1)

R. J. Deissler and H. R. Brand, "Periodic, quasiperiodic, and chaotic localized solutions of the quintic complex Ginzburg-Landau equation," Phys. Rev. Lett. 72, 478 - 481 (1994).
[CrossRef] [PubMed]

1975 (1)

H. A. Haus, "Theory of mode locking with a slow saturable absorber," IEEE Journ. of Quantum Electron.,  QE-11 (9), 736 (1975).
[CrossRef]

Akhmediev,

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,"Phys. Rev. E. 63, 056602 (2001).
[CrossRef]

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. AkhmedievJ.M. Soto-Crespo and Ph. Grelu, "Vibrating and shaking soliton pairs in dissipative systems." Phys. Lett. A 364, 413 (2007).
[CrossRef]

J. M. Soto-Crespo, N. Akhmediev and Ph. Grelu "Optical bullets and double bullet complexes in dissipative systems," Phys. Rev. E 74, 046612 (2006).
[CrossRef]

J. M. Soto-Crespo, Ph. Grelu, and N. Akhmediev"Optical bullets and "rockets" in nonlinear dissipative systems and their transformations and interactions," Opt. Express 14, 4013 (2006).
[CrossRef] [PubMed]

E. Tsoy and N. Akhmediev, "Bifurcations from stationary to pulsating solitons in the cubic quintic complex Ginzburg Landau equation," Phys. Lett. A 343, 417-422 (2005).
[CrossRef]

J. M. Soto-Crespo, N. Akhmediev y A. Ankiewicz. "Pulsating, creeping and erupting solitons in dissipative systems," Phys.Rev. Lett. 85, 2937 (2000).
[CrossRef]

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]

J. M. Soto-Crespo, N. Akhmediev y A. Ankiewicz. "Pulsating, creeping and erupting solitons in dissipative systems," Phys.Rev. Lett. 85, 2937 (2000).
[CrossRef]

Aranson, I. S.

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

Barland, S.

X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, "Cavity solitons in a driven VCSEL above threshold," J. of Sel. Topics on Quant.Electr. 12, 339-351 (2006).
[CrossRef]

Brambilla, M.

X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, "Cavity solitons in a driven VCSEL above threshold," J. of Sel. Topics on Quant.Electr. 12, 339-351 (2006).
[CrossRef]

Brand, H. R.

R. J. Deissler and H. R. Brand, "Periodic, quasiperiodic, and chaotic localized solutions of the quintic complex Ginzburg-Landau equation," Phys. Rev. Lett. 72, 478 - 481 (1994).
[CrossRef] [PubMed]

Caboche, E.

X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, "Cavity solitons in a driven VCSEL above threshold," J. of Sel. Topics on Quant.Electr. 12, 339-351 (2006).
[CrossRef]

Chong, A.

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

Deissler, R. J.

R. J. Deissler and H. R. Brand, "Periodic, quasiperiodic, and chaotic localized solutions of the quintic complex Ginzburg-Landau equation," Phys. Rev. Lett. 72, 478 - 481 (1994).
[CrossRef] [PubMed]

Descalzi, O.

O. Descalzi, G. During and E. Tirapegui, "On the stable hole solutions in the complex Ginzburg-Landau equation," Physica A 356, 66-71 (2005).
[CrossRef]

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]

During, G.

O. Descalzi, G. During and E. Tirapegui, "On the stable hole solutions in the complex Ginzburg-Landau equation," Physica A 356, 66-71 (2005).
[CrossRef]

Ginovart, F.

C. Montes, A. Mikhailov, A. Picozzi, and F. Ginovart, "Dissipative three-wave structures in stimulated backscattering. I. A subluminous solitary attractor," Phys. Rev. E 55, 1086 (1997).
[CrossRef]

Giudici, M.

X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, "Cavity solitons in a driven VCSEL above threshold," J. of Sel. Topics on Quant.Electr. 12, 339-351 (2006).
[CrossRef]

Glorieux, P.

Grelu, Ph.

N. AkhmedievJ.M. Soto-Crespo and Ph. Grelu, "Vibrating and shaking soliton pairs in dissipative systems." Phys. Lett. A 364, 413 (2007).
[CrossRef]

J. M. Soto-Crespo, N. Akhmediev and Ph. Grelu "Optical bullets and double bullet complexes in dissipative systems," Phys. Rev. E 74, 046612 (2006).
[CrossRef]

J. M. Soto-Crespo, Ph. Grelu, and N. Akhmediev"Optical bullets and "rockets" in nonlinear dissipative systems and their transformations and interactions," Opt. Express 14, 4013 (2006).
[CrossRef] [PubMed]

Hachair, X.

X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, "Cavity solitons in a driven VCSEL above threshold," J. of Sel. Topics on Quant.Electr. 12, 339-351 (2006).
[CrossRef]

Haus, H. A.

H. A. Haus, "Theory of mode locking with a slow saturable absorber," IEEE Journ. of Quantum Electron.,  QE-11 (9), 736 (1975).
[CrossRef]

Kheradmand, R.

X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, "Cavity solitons in a driven VCSEL above threshold," J. of Sel. Topics on Quant.Electr. 12, 339-351 (2006).
[CrossRef]

Kolokolnikov, T.

M. Tlidi, M. Taki, and T. Kolokolnikov, "Dissipative Localized Structures in Extended Systems," Chaos 17, 037101, 2007.
[CrossRef] [PubMed]

Komarov, A.

A. Komarov, H. Leblond, and F. Sanchez, "Quintic complex Ginzburg-Landau model for ring fiber lasers," Phys. Rev. E 72, 025604(R) (2005).
[CrossRef]

Kramer, L.

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

Lange, C. H.

E. A. Ultanir and G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, "Stable dissipative solitons in semiconductor optical amplifiers," Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

Leblond, H.

A. Komarov, H. Leblond, and F. Sanchez, "Quintic complex Ginzburg-Landau model for ring fiber lasers," Phys. Rev. E 72, 025604(R) (2005).
[CrossRef]

Lederer, F.

E. A. Ultanir and G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, "Stable dissipative solitons in semiconductor optical amplifiers," Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

Lugiato, L. A.

X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, "Cavity solitons in a driven VCSEL above threshold," J. of Sel. Topics on Quant.Electr. 12, 339-351 (2006).
[CrossRef]

Mandel, P.

M. Tlidi, A. Vladimirov and P. Mandel, "Curvature instability in passive diffractive resonators," Phys. Rev. Lett. 98, 233901, (2002).
[CrossRef]

Michaelis, D.

E. A. Ultanir and G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, "Stable dissipative solitons in semiconductor optical amplifiers," Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

Mikhailov, A.

C. Montes, A. Mikhailov, A. Picozzi, and F. Ginovart, "Dissipative three-wave structures in stimulated backscattering. I. A subluminous solitary attractor," Phys. Rev. E 55, 1086 (1997).
[CrossRef]

Montes, C.

C. Montes, A. Mikhailov, A. Picozzi, and F. Ginovart, "Dissipative three-wave structures in stimulated backscattering. I. A subluminous solitary attractor," Phys. Rev. E 55, 1086 (1997).
[CrossRef]

Ouarzazi, N.

Pedaci, F.

X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, "Cavity solitons in a driven VCSEL above threshold," J. of Sel. Topics on Quant.Electr. 12, 339-351 (2006).
[CrossRef]

Picozzi, A.

C. Montes, A. Mikhailov, A. Picozzi, and F. Ginovart, "Dissipative three-wave structures in stimulated backscattering. I. A subluminous solitary attractor," Phys. Rev. E 55, 1086 (1997).
[CrossRef]

Prati, F.

X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, "Cavity solitons in a driven VCSEL above threshold," J. of Sel. Topics on Quant.Electr. 12, 339-351 (2006).
[CrossRef]

Protsenko, I.

X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, "Cavity solitons in a driven VCSEL above threshold," J. of Sel. Topics on Quant.Electr. 12, 339-351 (2006).
[CrossRef]

Renninger, W. H.

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

Sanchez, F.

A. Komarov, H. Leblond, and F. Sanchez, "Quintic complex Ginzburg-Landau model for ring fiber lasers," Phys. Rev. E 72, 025604(R) (2005).
[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]

J. M. Soto-Crespo, N. Akhmediev and Ph. Grelu "Optical bullets and double bullet complexes in dissipative systems," Phys. Rev. E 74, 046612 (2006).
[CrossRef]

J. M. Soto-Crespo, Ph. Grelu, and N. Akhmediev"Optical bullets and "rockets" in nonlinear dissipative systems and their transformations and interactions," Opt. Express 14, 4013 (2006).
[CrossRef] [PubMed]

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,"Phys. Rev. E. 63, 056602 (2001).
[CrossRef]

J. M. Soto-Crespo, N. Akhmediev y A. Ankiewicz. "Pulsating, creeping and erupting solitons in dissipative systems," Phys.Rev. Lett. 85, 2937 (2000).
[CrossRef]

Soto-Crespo, J.M.

N. AkhmedievJ.M. Soto-Crespo and Ph. Grelu, "Vibrating and shaking soliton pairs in dissipative systems." Phys. Lett. A 364, 413 (2007).
[CrossRef]

Stegeman, G. I.

E. A. Ultanir and G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, "Stable dissipative solitons in semiconductor optical amplifiers," Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

Taki, M.

M. Tlidi, M. Taki, and T. Kolokolnikov, "Dissipative Localized Structures in Extended Systems," Chaos 17, 037101, 2007.
[CrossRef] [PubMed]

M. Taki, N. Ouarzazi, H. Ward and P. Glorieux, "Nonlinear front propagation in optical parametric oscillators," J. Opt. Soc. Am. B 17, 997 (2000).
[CrossRef]

Tirapegui, E.

O. Descalzi, G. During and E. Tirapegui, "On the stable hole solutions in the complex Ginzburg-Landau equation," Physica A 356, 66-71 (2005).
[CrossRef]

Tissoni, G.

X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, "Cavity solitons in a driven VCSEL above threshold," J. of Sel. Topics on Quant.Electr. 12, 339-351 (2006).
[CrossRef]

Tlidi, M.

M. Tlidi, M. Taki, and T. Kolokolnikov, "Dissipative Localized Structures in Extended Systems," Chaos 17, 037101, 2007.
[CrossRef] [PubMed]

M. Tlidi, A. Vladimirov and P. Mandel, "Curvature instability in passive diffractive resonators," Phys. Rev. Lett. 98, 233901, (2002).
[CrossRef]

Town, G.

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,"Phys. Rev. E. 63, 056602 (2001).
[CrossRef]

Tredicce, J. R.

X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, "Cavity solitons in a driven VCSEL above threshold," J. of Sel. Topics on Quant.Electr. 12, 339-351 (2006).
[CrossRef]

Tsoy, E.

E. Tsoy and N. Akhmediev, "Bifurcations from stationary to pulsating solitons in the cubic quintic complex Ginzburg Landau equation," Phys. Lett. A 343, 417-422 (2005).
[CrossRef]

Ultanir, E. A.

E. A. Ultanir and G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, "Stable dissipative solitons in semiconductor optical amplifiers," Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

Vladimirov, A.

M. Tlidi, A. Vladimirov and P. Mandel, "Curvature instability in passive diffractive resonators," Phys. Rev. Lett. 98, 233901, (2002).
[CrossRef]

Ward, H.

Wise, F. W.

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

Chaos (1)

M. Tlidi, M. Taki, and T. Kolokolnikov, "Dissipative Localized Structures in Extended Systems," Chaos 17, 037101, 2007.
[CrossRef] [PubMed]

Electr. (1)

X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, "Cavity solitons in a driven VCSEL above threshold," J. of Sel. Topics on Quant.Electr. 12, 339-351 (2006).
[CrossRef]

IEEE Journ. of Quantum Electron. (1)

H. A. Haus, "Theory of mode locking with a slow saturable absorber," IEEE Journ. of Quantum Electron.,  QE-11 (9), 736 (1975).
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J. Opt. Soc. Am. B (1)

Opt. Express (1)

Phys. Lett. A (2)

N. AkhmedievJ.M. Soto-Crespo and Ph. Grelu, "Vibrating and shaking soliton pairs in dissipative systems." Phys. Lett. A 364, 413 (2007).
[CrossRef]

E. Tsoy and N. Akhmediev, "Bifurcations from stationary to pulsating solitons in the cubic quintic complex Ginzburg Landau equation," Phys. Lett. A 343, 417-422 (2005).
[CrossRef]

Phys. Rev. A (2)

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]

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

Phys. Rev. E (3)

A. Komarov, H. Leblond, and F. Sanchez, "Quintic complex Ginzburg-Landau model for ring fiber lasers," Phys. Rev. E 72, 025604(R) (2005).
[CrossRef]

J. M. Soto-Crespo, N. Akhmediev and Ph. Grelu "Optical bullets and double bullet complexes in dissipative systems," Phys. Rev. E 74, 046612 (2006).
[CrossRef]

C. Montes, A. Mikhailov, A. Picozzi, and F. Ginovart, "Dissipative three-wave structures in stimulated backscattering. I. A subluminous solitary attractor," Phys. Rev. E 55, 1086 (1997).
[CrossRef]

Phys. Rev. E. (1)

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,"Phys. Rev. E. 63, 056602 (2001).
[CrossRef]

Phys. Rev. Lett. (4)

R. J. Deissler and H. R. Brand, "Periodic, quasiperiodic, and chaotic localized solutions of the quintic complex Ginzburg-Landau equation," Phys. Rev. Lett. 72, 478 - 481 (1994).
[CrossRef] [PubMed]

E. A. Ultanir and G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, "Stable dissipative solitons in semiconductor optical amplifiers," Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

J. M. Soto-Crespo, N. Akhmediev y A. Ankiewicz. "Pulsating, creeping and erupting solitons in dissipative systems," Phys.Rev. Lett. 85, 2937 (2000).
[CrossRef]

M. Tlidi, A. Vladimirov and P. Mandel, "Curvature instability in passive diffractive resonators," Phys. Rev. Lett. 98, 233901, (2002).
[CrossRef]

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Other (3)

(Eds.) N. Akhmediev and A. Ankiewicz, Dissipative solitons Lecture Notes in Physics, V. 661 (Springer, Heidelberg, 2005).
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Dissipative Solitons: From optics to biology and medicine Lecture Notes in Physics, V 751, (Eds.) N. Akhmediev and A. Ankiewicz, (Springer, Berlin-Heidelberg, 2008).

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

Supplementary Material (3)

» Media 1: MPG (568 KB)     
» Media 2: MPG (1242 KB)     
» Media 3: MPG (636 KB)     

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Figures (12)

Fig. 1.
Fig. 1.

Regions (in blue) in the parameter space with radially symmetric stable stationary beams. The plot on the left (a) shows the region for continuously self-focusing beams (region I) and on the right (b) the one for continuously self-defocusing beams (region II). The location of these two regions in the five-dimensional parameter space is such that they cannot be represented in the same plane. Soliton solutions beyond these regions are either non-stationary or loose the radial symmetry. Below, we give examples located at the points indicated here by green, yellow and red lines. The change in color corresponds to a bifurcation. The arrows indicate the direction in which the parameters have been changed while obtaining a particular bifurcation diagram.

Fig. 2.
Fig. 2.

Bifurcation of an elliptic soliton from the self-focusing radially symmetric one.

Fig. 3.
Fig. 3.

(a) Q versus ε diagram as obtained when increasing (red dotted line) or decreasing ε (blue dashed line). (b) magnification of a portion of the upper blue curve around the hysteresis cycle between the two black arrows in (a). The red dots in this case correspond to stationary beams of elliptic shape while the blue dashed line corresponds to stationary beams without any radial symmetry. Typical examples of these two types of solutions are plotted in Figs.4a and 4b respectively for the ε values indicated here with the red and blue arrows. Black vertical lines with continuous values of Q in (b) correspond to pulsating localized solutions.

Fig. 4.
Fig. 4.

Solitons (a) of an elliptic shape and (b) without any radial symmetry (comma shaped) for the values of ε indicated in Fig.3(b) by the red and blue arrows.

Fig. 5.
Fig. 5.

A portion of the diagram in Fig.3b with a magnified scale. In this case only the maxima (QM ) and minima (Qm ) of the beam power are shown. The two branches for stationary solutions corresponding to solutions of elliptic shape (left) or highly asymmetric solutions (comma-shaped) with constant Q are shown in green. The red points represent the maxima of the curves Q(z) and the blue points the minima. The data for this plot are obtained when decreasing ε. The period-1 solutions bifurcate from the asymmetric solutions at ε=0.45. Period-doubling bifurcations appear at ε=0.446. A much more complicated type of pulsations appear at ε=0.445. These look like a beating of the two types of stationary solutions (elliptic and highly asymmetric).

Fig. 6.
Fig. 6.

Pulsating 2-D soliton profiles when the oscillating power Q takes its (a) maximal and minimal values (dashed and dotted vertical lines in Fig. 7 respectively).

Fig. 7.
Fig. 7.

(a) Periodic evolution of Q versus z for a pulsating soliton with a single period. The red and blue vertical lines show a maximum and a minimum of Q. i The corresponding profiles are shown in Fig. 6(a) and Fig. 6(b) respectively. The actual evolution during a period (marked in green in (a)) is shown in (Media 1) (b).

Fig. 8.
Fig. 8.

Dissipative soliton oscillations in two transverse dimensions. The solution converges to these oscillations at around z≈120. The power oscillations are shown in greater resolution in the upper left inset; they are almost harmonic. Color plots in the lower left insets show the intensity profiles when Q reaches two consecutive minima (left and right frames) and any maximum (central frame). The oscillations are weakly unstable and are transformed into a stable configuration at z≈260, consisting of a twin soliton profile shown at the upper right inset that rotates around its center of symmetry

Fig. 9.
Fig. 9.

(a) Periodic evolution of Q versus z for three values of epsilon. As epsilon decreases the maxima of Q increases as well as the separation between “bursts”. The red curve is horizontal for large interval of values of z during which the solution is of elliptic shape. (b) (Media 2) showing the evolution of the beam profile.

Fig. 10.
Fig. 10.

Periodic evolution of the solution for ε=0.4446 (blue curve in Fig.9). The green trajectory in (a) corresponds to the green zone in (b).

Fig. 11.
Fig. 11.

a) Trajectory of the peak intensity of the solutions in the (X,Y) plane and b) evolution of Q versus z. Six periods are plotted, each with a different color. The position of the solution behaves somehow chaotic.

Fig. 12.
Fig. 12.

(a) Bifurcation diagram at the boundary of the region II. Parameters of the system are given inside the plot. An Andronov-Hopf bifurcation of a stationary self-defocusing beam into a pulsating one occurs at β=0.55. The inset shows the power evolution with z for two cases: β=0.42 (magenta) and β=0.4 (amber). In the interval 0.41 <β < 055 the beam is pulsating. The Q(z) curve is harmonic. The evolution is chaotic at the values of β<0.41. The Q(z)-curve (amber) reveals the beam explosions. (Media 3) in (b) shows one period of evolution of the pulsating beam at β=0.42.

Equations (2)

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i ψ z + D 2 2 ψ + ψ 2 ψ + v ψ 4 ψ = i δ ψ + i ε ψ 2 ψ + i β 2 ψ + i μ ψ 4 ψ .
Q ( z ) = ψ ( x , y , z ) 2 d x d y ,

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