Abstract

We study the rich dynamics of dissipative spatial solitons in optical media described by the complex Ginzburg–Landau equation in the presence of periodic, sinusoidal-type spatially inhomogeneous losses. It is revealed that in the case when the soliton is launched at the point where the periodic spatial modulation loss profile has its zero value, the gradient force of the inhomogeneous loss easily induces three generic propagation scenarios: (a) soliton transverse drift, (b) persistent swing around the soliton input launching position, and (c) damped oscillations near or even far from the input position. The soliton exhibiting damped oscillations eventually evolves into a stable one, whose output position can be controlled by the amplitude of the inhomogeneous loss profile. Conversely, when the launching point coincides with an extremum (a maximum or a minimum) of the sinusoidal-type loss landscape, both soliton transverse drift and soliton damped oscillations occur due to transverse modulation instability. Moreover, in this case, depending on the balance between the amplitude of the inhomogeneous loss modulation profile and the homogeneous linear loss coefficient, either the launched soliton can maintain its stable propagation at the input position or a stable plump dissipative soliton can be formed while preserving the launching point.

© 2012 Optical Society of America

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  7. O. V. Borovkova, V. E. Lobanov, Y. V. Kartashov, and L. Torner, “Rotating vortex solitons supported by localized gain,” Opt. Lett. 36, 1936–1938 (2011).
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    [CrossRef]
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    [CrossRef]
  32. I. S. Aranson and L. Kramer, “The world of the complex Ginzburg–Landau equation,” Rev. Mod. Phys. 74, 99–143(2002).
    [CrossRef]
  33. 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]
  34. V. V. Afanasjev, N. Akhmediev, and J. M. Soto-Crespo, “Three forms of localized solutions of the quintic complex Ginzburg–Landau equation,” Phys. Rev. E 53, 1931–1939 (1996).
    [CrossRef]
  35. 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, 1139–1142 (2010).
    [CrossRef]
  36. B. Kneer, T. Wong, K. Vogel, W. P. Schleich, and D. F. Walls, “Generic model of an atom laser,” Phys. Rev. A 58, 4841–4853 (1998).
    [CrossRef]
  37. F. T. Arecchi, J. Bragard, and L. M. Castellano, in Bose-Einstein Condensates and Atom Lasers (Kluwer, 2002).
  38. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B Quantum Semiclass. Opt. 7, R53–R72 (2005).
    [CrossRef]
  39. D. Mihalache, “Three-dimensional Ginzburg-Landau dissipative solitons supported by a two-dimensional transverse grating,” Proc. Rom. Acad. A 11, 142–147 (2010).
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  43. O. V. Borovkova, V. E. Lobanov, Y. V. Kartashov, and L. Torner, “Stable vortex-soliton tori with multiple nested phase singularities in dissipative media,” Phys. Rev. A 85, 023814 (2012).
    [CrossRef]
  44. V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Topological light bullets supported by spatiotemporal gain,” Phys. Rev. A 85, 023804 (2012).
    [CrossRef]

2012 (5)

O. V. Borovkova, Y. V. Kartashov, V. A. Vysloukh, V. E. Lobanov, B. A. Malomed, and L. Torner, “Solitons supported by spatially inhomogeneous nonlinear losses,” Opt. Express 20, 2657–2667 (2012).
[CrossRef]

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

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

O. V. Borovkova, V. E. Lobanov, Y. V. Kartashov, and L. Torner, “Stable vortex-soliton tori with multiple nested phase singularities in dissipative media,” Phys. Rev. A 85, 023814 (2012).
[CrossRef]

V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Topological light bullets supported by spatiotemporal gain,” Phys. Rev. A 85, 023804 (2012).
[CrossRef]

2011 (6)

2010 (5)

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

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

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

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

D. Mihalache, “Three-dimensional Ginzburg-Landau dissipative solitons supported by a two-dimensional transverse grating,” Proc. Rom. Acad. A 11, 142–147 (2010).

2009 (4)

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]

D. Mihalache and D. Mazilu, “Ginzburg–Landau spatiotemporal dissipative optical solitons,” Rom. Rep. Phys. 60, 749–761 (2008).

2007 (2)

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

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

2006 (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, 073904 (2006).
[CrossRef]

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

2005 (2)

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

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

2004 (1)

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

2003 (1)

2002 (2)

2001 (1)

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]

2000 (1)

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg–Landau equation,” Phys. Rev. E 63, 016605 (2000).
[CrossRef]

1999 (1)

F. T. Arecchi, S. Boccaletti, and P. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

1998 (1)

B. Kneer, T. Wong, K. Vogel, W. P. Schleich, and D. F. Walls, “Generic model of an atom laser,” Phys. Rev. A 58, 4841–4853 (1998).
[CrossRef]

1996 (2)

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

N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190–1201 (1996).
[CrossRef]

1992 (1)

L. Gil, K. Emilsson, and G.-L. Oppo, “Dynamics of spiral waves in a spatially inhomogeneous Hopf bifurcation,” Phys. Rev. A 45, R567 (1992).
[CrossRef]

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 53, 1190–1201 (1996).
[CrossRef]

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

Akhmediev, N.

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

J. M. Soto-Crespo, N. Akhmediev, C. Mejia-Cortes, and N. Devine, “Dissipative ring solitons with vorticity,” Opt. Express 17, 4236–4250 (2009).
[CrossRef]

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

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

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

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

Akhmediev, N. N.

N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190–1201 (1996).
[CrossRef]

Aleksic, N. B.

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

Ankiewicz, A.

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

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

Aranson, I. S.

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

Arecchi, F. T.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

F. T. Arecchi, J. Bragard, and L. M. Castellano, in Bose-Einstein Condensates and Atom Lasers (Kluwer, 2002).

Bakonyi, Z.

Bludov, Y. V.

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

Boccaletti, S.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

Borovkova, O. V.

Bragard, J.

F. T. Arecchi, J. Bragard, and L. M. Castellano, in Bose-Einstein Condensates and Atom Lasers (Kluwer, 2002).

Castellano, L. M.

F. T. Arecchi, J. Bragard, and L. M. Castellano, in Bose-Einstein Condensates and Atom Lasers (Kluwer, 2002).

Chang, W.

Chong, A.

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

Chow, K. W.

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

Crasovan, L.-C.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg–Landau equation,” Phys. Rev. E 63, 016605 (2000).
[CrossRef]

Devine, N.

Emilsson, K.

L. Gil, K. Emilsson, and G.-L. Oppo, “Dynamics of spiral waves in a spatially inhomogeneous Hopf bifurcation,” Phys. Rev. A 45, R567 (1992).
[CrossRef]

Fedorov, S. V.

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

Gil, L.

L. Gil, K. Emilsson, and G.-L. Oppo, “Dynamics of spiral waves in a spatially inhomogeneous Hopf bifurcation,” Phys. Rev. A 45, R567 (1992).
[CrossRef]

Grelu, P.

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

Haboucha, A.

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

He, Y. J.

Holzlöhner, R.

Hu, B. B.

Huang, H. C.

Kartashov, Y. V.

V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Topological light bullets supported by spatiotemporal gain,” Phys. Rev. A 85, 023804 (2012).
[CrossRef]

O. V. Borovkova, V. E. Lobanov, Y. V. Kartashov, and L. Torner, “Stable vortex-soliton tori with multiple nested phase singularities in dissipative media,” Phys. Rev. A 85, 023814 (2012).
[CrossRef]

O. V. Borovkova, Y. V. Kartashov, V. A. Vysloukh, V. E. Lobanov, B. A. Malomed, and L. Torner, “Solitons supported by spatially inhomogeneous nonlinear losses,” Opt. Express 20, 2657–2667 (2012).
[CrossRef]

O. V. Borovkova, V. E. Lobanov, Y. V. Kartashov, and L. Torner, “Rotating vortex solitons supported by localized gain,” Opt. Lett. 36, 1936–1938 (2011).
[CrossRef]

V. E. Lobanov, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Stable radially symmetric and azimuthally modulated vortex solitons supported by localized gain,” Opt. Lett. 36, 85–87 (2011).
[CrossRef]

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

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

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

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

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef]

Kneer, B.

B. Kneer, T. Wong, K. Vogel, W. P. Schleich, and D. F. Walls, “Generic model of an atom laser,” Phys. Rev. A 58, 4841–4853 (1998).
[CrossRef]

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 Pure Appl. Opt. 8, 319–326 (2006).
[CrossRef]

Konotop, V. V.

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

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

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

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

Kramer, L.

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

Lam, C.-K.

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

Leblond, H.

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

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

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

Lederer, F.

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

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef]

Z. Bakonyi, D. Michaelis, U. Peschel, G. Onishchukov, and F. Lederer, “Dissipative solitons and their critical slowing down near a supercritical bifurcation,” J. Opt. Soc. Am. B 19, 487–491 (2002).
[CrossRef]

Liu, B.

Lobanov, V. E.

Malomed, B. A.

O. V. Borovkova, Y. V. Kartashov, V. A. Vysloukh, V. E. Lobanov, B. A. Malomed, and L. Torner, “Solitons supported by spatially inhomogeneous nonlinear losses,” Opt. Express 20, 2657–2667 (2012).
[CrossRef]

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, 1139–1142 (2010).
[CrossRef]

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

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

Y. J. He, B. A. Malomed, D. Mihalache, B. Liu, H. C. Huang, H. Yang, and H. Z. Wang, “Bound states of one-, two-, and three-dimensional solitons in complex Ginzburg–Landau equations with a linear potential,” Opt. Lett. 34, 2976–2978 (2009).
[CrossRef]

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

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

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef]

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

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg–Landau equation,” Phys. Rev. E 63, 016605 (2000).
[CrossRef]

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

Mandel, P.

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

Mazilu, D.

D. Mihalache and D. Mazilu, “Ginzburg–Landau spatiotemporal dissipative optical solitons,” Rom. Rep. Phys. 60, 749–761 (2008).

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

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef]

Mejia-Cortes, C.

Menyuk, C. R.

Michaelis, D.

Mihalache, D.

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

D. Mihalache, “Topological dissipative nonlinear modes in two- and three-dimensional Ginzburg–Landau models with trapping potentials,” Rom. Rep. Phys. 63, 9–24 (2011).

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

D. Mihalache, “Three-dimensional Ginzburg-Landau dissipative solitons supported by a two-dimensional transverse grating,” Proc. Rom. Acad. A 11, 142–147 (2010).

Y. J. He, B. A. Malomed, D. Mihalache, B. Liu, H. C. Huang, H. Yang, and H. Z. Wang, “Bound states of one-, two-, and three-dimensional solitons in complex Ginzburg–Landau equations with a linear potential,” Opt. Lett. 34, 2976–2978 (2009).
[CrossRef]

D. Mihalache and D. Mazilu, “Ginzburg–Landau spatiotemporal dissipative optical solitons,” Rom. Rep. Phys. 60, 749–761 (2008).

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

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef]

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

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg–Landau equation,” Phys. Rev. E 63, 016605 (2000).
[CrossRef]

Onishchukov, G.

Oppo, G.-L.

L. Gil, K. Emilsson, and G.-L. Oppo, “Dynamics of spiral waves in a spatially inhomogeneous Hopf bifurcation,” Phys. Rev. A 45, R567 (1992).
[CrossRef]

Peschel, U.

Ramazza, P.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

Renninger, W. H.

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

Rosanov, N. N.

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

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

Salhi, M.

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

Sanchez, F.

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

Schleich, W. P.

B. Kneer, T. Wong, K. Vogel, W. P. Schleich, and D. F. Walls, “Generic model of an atom laser,” Phys. Rev. A 58, 4841–4853 (1998).
[CrossRef]

Shatsev, A. N.

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

Sinkin, O. V.

Skarka, V.

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

Soto-Crespo, J. M.

J. M. Soto-Crespo, N. Akhmediev, C. Mejia-Cortes, and N. Devine, “Dissipative ring solitons with vorticity,” Opt. Express 17, 4236–4250 (2009).
[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. V. Afanasjev, N. Akhmediev, and J. M. Soto-Crespo, “Three forms of localized solutions of the quintic complex Ginzburg–Landau equation,” Phys. Rev. E 53, 1931–1939 (1996).
[CrossRef]

N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190–1201 (1996).
[CrossRef]

Taki, M.

Tlidi, M.

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

Torner, L.

V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Topological light bullets supported by spatiotemporal gain,” Phys. Rev. A 85, 023804 (2012).
[CrossRef]

O. V. Borovkova, V. E. Lobanov, Y. V. Kartashov, and L. Torner, “Stable vortex-soliton tori with multiple nested phase singularities in dissipative media,” Phys. Rev. A 85, 023814 (2012).
[CrossRef]

O. V. Borovkova, Y. V. Kartashov, V. A. Vysloukh, V. E. Lobanov, B. A. Malomed, and L. Torner, “Solitons supported by spatially inhomogeneous nonlinear losses,” Opt. Express 20, 2657–2667 (2012).
[CrossRef]

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

O. V. Borovkova, V. E. Lobanov, Y. V. Kartashov, and L. Torner, “Rotating vortex solitons supported by localized gain,” Opt. Lett. 36, 1936–1938 (2011).
[CrossRef]

V. E. Lobanov, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Stable radially symmetric and azimuthally modulated vortex solitons supported by localized gain,” Opt. Lett. 36, 85–87 (2011).
[CrossRef]

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

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef]

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

Vogel, K.

B. Kneer, T. Wong, K. Vogel, W. P. Schleich, and D. F. Walls, “Generic model of an atom laser,” Phys. Rev. A 58, 4841–4853 (1998).
[CrossRef]

Vysloukh, V. A.

Wabnitz, S.

Wai, P. K. A.

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

Walls, D. F.

B. Kneer, T. Wong, K. Vogel, W. P. Schleich, and D. F. Walls, “Generic model of an atom laser,” Phys. Rev. A 58, 4841–4853 (1998).
[CrossRef]

Wang, H. Z.

Wise, F.

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

Wise, F. W.

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

Wong, T.

B. Kneer, T. Wong, K. Vogel, W. P. Schleich, and D. F. Walls, “Generic model of an atom laser,” Phys. Rev. A 58, 4841–4853 (1998).
[CrossRef]

Yang, H.

Ye, F. W.

Zezyulin, D. A.

Zweck, J.

Appl. Phys. B (1)

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

Chaos (1)

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

Eur. Phys. J. Spec. Topics (1)

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

J. Lightwave Technol. (1)

J. Opt. A Pure Appl. Opt. (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 Pure Appl. Opt. 8, 319–326 (2006).
[CrossRef]

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

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

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

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

J. Opt. Soc. Am. B (3)

Nat. Photonics (1)

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

Opt. Express (2)

Opt. Lett. (6)

Phys. Rep. (1)

F. T. Arecchi, S. Boccaletti, and P. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

Phys. Rev. A (8)

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

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

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

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

L. Gil, K. Emilsson, and G.-L. Oppo, “Dynamics of spiral waves in a spatially inhomogeneous Hopf bifurcation,” Phys. Rev. A 45, R567 (1992).
[CrossRef]

B. Kneer, T. Wong, K. Vogel, W. P. Schleich, and D. F. Walls, “Generic model of an atom laser,” Phys. Rev. A 58, 4841–4853 (1998).
[CrossRef]

O. V. Borovkova, V. E. Lobanov, Y. V. Kartashov, and L. Torner, “Stable vortex-soliton tori with multiple nested phase singularities in dissipative media,” Phys. Rev. A 85, 023814 (2012).
[CrossRef]

V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Topological light bullets supported by spatiotemporal gain,” Phys. Rev. A 85, 023804 (2012).
[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]

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

N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190–1201 (1996).
[CrossRef]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the two-dimensional Ginzburg–Landau equation,” Phys. Rev. E 63, 016605 (2000).
[CrossRef]

Phys. Rev. Lett. (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, 073904 (2006).
[CrossRef]

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

Proc. Rom. Acad. A (1)

D. Mihalache, “Three-dimensional Ginzburg-Landau dissipative solitons supported by a two-dimensional transverse grating,” Proc. Rom. Acad. A 11, 142–147 (2010).

Rev. Mod. Phys. (1)

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

Rom. J. Phys. (1)

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

Rom. Rep. Phys. (2)

D. Mihalache, “Topological dissipative nonlinear modes in two- and three-dimensional Ginzburg–Landau models with trapping potentials,” Rom. Rep. Phys. 63, 9–24 (2011).

D. Mihalache and D. Mazilu, “Ginzburg–Landau spatiotemporal dissipative optical solitons,” Rom. Rep. Phys. 60, 749–761 (2008).

Other (5)

F. T. Arecchi, J. Bragard, and L. M. Castellano, in Bose-Einstein Condensates and Atom Lasers (Kluwer, 2002).

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

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

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

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

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Figures (6)

Fig. 1.
Fig. 1.

(a) Soliton power P versus linear loss coefficient α; (b) soliton transverse profile for α=0.4.

Fig. 2.
Fig. 2.

(a) Inhomogeneous loss modulation profile V(x)=dsin(x/5); (b) soliton dynamics described by the relation of the linear loss coefficient α to the amplitude d of the inhomogeneous loss, including excess gain (below curve 1), left drift (between curves 1 and 2), persistent swing (on the curve 2), damped oscillations (between curves 2 and 3), and decay (above curve 3). Evolutions of the solitons: (c) excess gain for α=0.1 and d=0.2, (d) left drift for α=0.36 and d=0.2, (e) persistent swing for α=0.385 and d=0.1, (f) damped oscillation for α=0.45 and d=0.2, and (g) decay for α=0.56 and d=0.2.

Fig. 3.
Fig. 3.

Different soliton scenarios in the plane (α, T) for d=0.1 in the case of loss function of the form V(x)=dsin(x/T): soliton excess gain (below curve 1), soliton drift (between curves 1 and 2), soliton persistent swing (on the curve 2), soliton damped oscillations (between curves 2 and 3), and soliton decay (above curve 3).

Fig. 4.
Fig. 4.

(a) Dependence of output center position xout on the linear loss coefficient α for the soliton exhibiting a damped oscillation with d=0.1 and d=0.2. Soliton damped oscillations for (b) α=0.42 and d=0.1, and (c) α=0.5 and d=0.2.

Fig. 5.
Fig. 5.

(a) Soliton profile (solid curve) and the loss modulation function (dashed curve), V(x)=dsin(π/2x/T), with T=5. Soliton evolutions for (b) d=0.1, (c) d=0.3, and (d) d=0.6. Here α=0.5.

Fig. 6.
Fig. 6.

(a) Soliton profile (solid curve) and the symmetric loss modulation function (dashed curve), V(x)=dsin(π/2x/T), with T=5. Soliton evolutions for (b) d=0.05, (c) d=0.12, and (d) d=0.3. Here, α=0.3.

Equations (1)

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iuz+(1/2)uxx+|u|2u+ν|u|4u=iR[u]+i[V(x)α]u,

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