Abstract

We present a variety of dissipative solitons and breathing modes in a medium with localized gain and homogeneous linear dissipation. The system possesses a number of unusual properties, like exponentially localized modes in both focusing and defocusing media, existence of modes in focusing media at negative propagation constant values, simultaneous existence of stable symmetric and antisymmetric localized modes when the gain landscape possesses two local maxima, as well as the existence of stable breathing solutions.

© 2011 Optical Society of America

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References

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  1. N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, 2010).
  2. N.Akhmediev and A.Ankiewicz, eds., Dissipative Solitons: From Optics to Biology and Medicine (Springer, 2008).
  3. C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. J. Phys. Spec. Top. 173, 233 (2009).
    [CrossRef]
  4. Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and L. Torner, Opt. Lett. 35, 1638 (2010).
    [CrossRef] [PubMed]
  5. M. O. Williams, C. W. McGrath, and J. N. Kutz, Opt. Express 18, 11671 (2010).
    [CrossRef] [PubMed]
  6. Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and L. Torner, Opt. Lett. 35, 3177 (2010).
    [CrossRef] [PubMed]
  7. Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, Opt. Lett. 36, 82 (2011).
    [CrossRef] [PubMed]
  8. C. H. Tsang, B. A. Malomed, C. K. Lam, and K. W. Chow, Eur. Phys. J. D 59, 81 (2010).
    [CrossRef]
  9. A. E. Siegman, J. Opt. Soc. Am. A 20, 1617 (2003).
    [CrossRef]

2011 (1)

2010 (4)

2009 (1)

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. J. Phys. Spec. Top. 173, 233 (2009).
[CrossRef]

2003 (1)

Chow, K. W.

C. H. Tsang, B. A. Malomed, C. K. Lam, and K. W. Chow, Eur. Phys. J. D 59, 81 (2010).
[CrossRef]

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. J. Phys. Spec. Top. 173, 233 (2009).
[CrossRef]

Kartashov, Y. V.

Konotop, V. V.

Kutz, J. N.

Lam, C. K.

C. H. Tsang, B. A. Malomed, C. K. Lam, and K. W. Chow, Eur. Phys. J. D 59, 81 (2010).
[CrossRef]

Lam, C.-K.

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. J. Phys. Spec. Top. 173, 233 (2009).
[CrossRef]

Malomed, B. A.

C. H. Tsang, B. A. Malomed, C. K. Lam, and K. W. Chow, Eur. Phys. J. D 59, 81 (2010).
[CrossRef]

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. J. Phys. Spec. Top. 173, 233 (2009).
[CrossRef]

McGrath, C. W.

Rosanov, N. N.

N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, 2010).

Siegman, A. E.

Torner, L.

Tsang, C. H.

C. H. Tsang, B. A. Malomed, C. K. Lam, and K. W. Chow, Eur. Phys. J. D 59, 81 (2010).
[CrossRef]

Vysloukh, V. A.

Wai, P. K. A.

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. J. Phys. Spec. Top. 173, 233 (2009).
[CrossRef]

Williams, M. O.

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

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. J. Phys. Spec. Top. 173, 233 (2009).
[CrossRef]

Eur. Phys. J. D (1)

C. H. Tsang, B. A. Malomed, C. K. Lam, and K. W. Chow, Eur. Phys. J. D 59, 81 (2010).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Express (1)

Opt. Lett. (3)

Other (2)

N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, 2010).

N.Akhmediev and A.Ankiewicz, eds., Dissipative Solitons: From Optics to Biology and Medicine (Springer, 2008).

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Figures (5)

Fig. 1
Fig. 1

(a) Propagation constant and (b) energy flow versus a for solitons supported by single gain channel. Red and black curves correspond to focusing and defocusing media. Dashed line indicates a value corresponding to the linear limit. Profile of (c) soliton and (d) current density in defocusing medium at a = 2.5 corresponding to the circles in (a) and (b).

Fig. 2
Fig. 2

(a) Instability increment δ r versus a for solitons in a single gain channel in focusing medium. Circles indicate points beyond which solitons exist only in defocusing medium where they are stable. (b)  δ r ( a ) for symmetric “s” and antisymmetric “a” solitons in two gain channels with d = 1 in defocusing medium.

Fig. 3
Fig. 3

(a) Propagation constant and (b) energy flow versus a for solitons supported by the two gain channels at x 0 = 6 . Solid and dashed lines correspond to the symmetric and antisymmetric modes. Red and black lines correspond to the focusing and defocusing media. The circles correspond to the solitons shown in (c), (d). (c) Unstable antisymmetric soliton in the focusing medium at a = 1.8 . (d) Stable symmetric soliton in defocusing medium at a = 2.6 .

Fig. 4
Fig. 4

Excitation of stable (a) symmetric and (b) antisymmetric solitons by symmetric and antisymmetric input beams in a system with two gain channels for a = 2.391 . (c) Excitation of a breather at a = 2.61 starting with the unstable antisymmetric soliton. (d) Switching between stable symmetric and antisymmetric solitons stimulated by abrupt change of a at z = 100 .

Fig. 5
Fig. 5

The (a) trajectory and (b) dependence U ( z ) corresponding to the dynamics shown in Fig. 4c. The bold line shows the limit cycle.

Equations (3)

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i q z = q x x i [ γ 0 γ ( x ) ] q + σ | q | 2 q ,
u x x b u + σ u 3 j 2 u 3 = 0 , j x + [ 1 γ ( x ) ] u 2 = 0.
b U = | w x | 2 d x σ | w | 4 d x .

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