Abstract

We introduce a one-dimensional model of a cavity with the Kerr nonlinearity and saturated gain designed so as to hold solitons in the state of shuttle motion. The solitons are always unstable in the cavity bounded by the usual potential barriers, due to accumulation of noise generated by the linear gain. Complete stabilization of the shuttling soliton is achieved if the linear barrier potentials are replaced by nonlinear ones, which trap the soliton, being transparent to the radiation. The removal of the noise from the cavity is additionally facilitated by an external ramp potential. The stable dynamical regimes are found numerically, and their basic properties are explained analytically.

© 2013 Optical Society of America

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  1. W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
    [CrossRef]
  2. C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. J. Phys. Special Topics 173, 233 (2009).
    [CrossRef]
  3. Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, Europhys. Lett. 91, 34003 (2010).
    [CrossRef]
  4. O. V. Borovkova, V. E. Lobanov, and B. A. Malomed, Europhys. Lett. 97, 44003 (2012).
    [CrossRef]
  5. C. H. Tsang, B. A. Malomed, C.-K. Lam, and K. W. Chow, Eur. Phys. J. D 59, 81 (2010).
    [CrossRef]
  6. D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, Opt. Lett. 36, 1200 (2011).
    [CrossRef]
  7. Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, Phys. Rev. A 83, 041806 (2011).
    [CrossRef]
  8. D. A. Zezyulin, V. V. Konotop, and G. L. Alfimov, Phys. Rev. E 82, 056213 (2010).
    [CrossRef]
  9. C. H. Tsang, B. A. Malomed, and K. W. Chow, Phys. Rev. E 84, 066609 (2011).
    [CrossRef]
  10. F. K. Abdullaev, V. V. Konotop, M. Salerno, and A. V. Yulin, Phys. Rev. E 82, 056606 (2010).
    [CrossRef]
  11. J. Hukriede, D. Runde, and D. Kip, J. Phys. D 36, R1 (2003).
  12. V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. Lett. 105, 213901 (2010).
    [CrossRef]
  13. V. E. Lobanov, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 36, 85 (2011).
    [CrossRef]
  14. O. V. Borovkova, Y. V. Kartashov, V. E. Lobanov, V. A. Vysloukh, and L. Torner, Opt. Lett. 36, 3783 (2011).
    [CrossRef]
  15. C. Huang, F. Ye, B. A. Malomed, Y. V. Kartashov, and X. Chen, Opt. Lett. 38, 2177 (2013).
    [CrossRef]
  16. B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, Phys. Rev. E 86, 036608 (2012).
    [CrossRef]
  17. W. J. Firth, G. K. Harkness, A. Lord, J. M. McSloy, D. Gomila, and P. Colet, J. Opt. Soc. Am. B 19, 747 (2002).
    [CrossRef]
  18. P. Mandel and M. Tlidi, J. Opt. B Quant. Semiclass. Opt. 6, R60 (2004).
    [CrossRef]
  19. A. G. Vladimirov, D. V. Skryabin, G. Kozyreff, P. Mandel, and M. Tlidi, Opt. Express 14, 1 (2006).
    [CrossRef]
  20. Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jaeger, Phys. Rev. Lett. 100, 013907 (2008).
    [CrossRef]
  21. M. Tlidi, A. G. Vladimirov, D. Pieroux, and D. Turaev, Phys. Rev. Lett. 103, 103904 (2009).
    [CrossRef]
  22. C. Yin, D. Mihalache, and Y. He, J. Opt. Soc. Am. B 28, 342 (2011).
    [CrossRef]
  23. P. Y. P. Chen, B. A. Malomed, and P. L. Chu, Phys. Rev. E 71, 066601 (2005).
    [CrossRef]
  24. N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, 2002).
  25. J. N. Kutz, SIAM Rev. 48, 629 (2006).
    [CrossRef]
  26. Y. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
    [CrossRef]
  27. M. van Hecke, E. de Wit, and W. van Saarloos, Phys. Rev. Lett. 75, 3830 (1995).
    [CrossRef]
  28. Y. V. Kartashov, B. A. Malomed, and L. Torner, Rev. Mod. Phys. 83, 247 (2011).
    [CrossRef]

2013

2012

B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, Phys. Rev. E 86, 036608 (2012).
[CrossRef]

O. V. Borovkova, V. E. Lobanov, and B. A. Malomed, Europhys. Lett. 97, 44003 (2012).
[CrossRef]

2011

2010

Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, Europhys. Lett. 91, 34003 (2010).
[CrossRef]

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. Lett. 105, 213901 (2010).
[CrossRef]

F. K. Abdullaev, V. V. Konotop, M. Salerno, and A. V. Yulin, Phys. Rev. E 82, 056606 (2010).
[CrossRef]

D. A. Zezyulin, V. V. Konotop, and G. L. Alfimov, Phys. Rev. E 82, 056213 (2010).
[CrossRef]

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

2009

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

M. Tlidi, A. G. Vladimirov, D. Pieroux, and D. Turaev, Phys. Rev. Lett. 103, 103904 (2009).
[CrossRef]

2008

Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jaeger, Phys. Rev. Lett. 100, 013907 (2008).
[CrossRef]

2006

2005

P. Y. P. Chen, B. A. Malomed, and P. L. Chu, Phys. Rev. E 71, 066601 (2005).
[CrossRef]

2004

P. Mandel and M. Tlidi, J. Opt. B Quant. Semiclass. Opt. 6, R60 (2004).
[CrossRef]

2003

W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
[CrossRef]

J. Hukriede, D. Runde, and D. Kip, J. Phys. D 36, R1 (2003).

2002

1995

M. van Hecke, E. de Wit, and W. van Saarloos, Phys. Rev. Lett. 75, 3830 (1995).
[CrossRef]

1989

Y. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

Abdullaev, F. K.

F. K. Abdullaev, V. V. Konotop, M. Salerno, and A. V. Yulin, Phys. Rev. E 82, 056606 (2010).
[CrossRef]

Ackemann, T.

Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jaeger, Phys. Rev. Lett. 100, 013907 (2008).
[CrossRef]

Aleksic, N. B.

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. Lett. 105, 213901 (2010).
[CrossRef]

Alfimov, G. L.

D. A. Zezyulin, V. V. Konotop, and G. L. Alfimov, Phys. Rev. E 82, 056213 (2010).
[CrossRef]

Borovkova, O. V.

O. V. Borovkova, V. E. Lobanov, and B. A. Malomed, Europhys. Lett. 97, 44003 (2012).
[CrossRef]

O. V. Borovkova, Y. V. Kartashov, V. E. Lobanov, V. A. Vysloukh, and L. Torner, Opt. Lett. 36, 3783 (2011).
[CrossRef]

Chen, P. Y. P.

P. Y. P. Chen, B. A. Malomed, and P. L. Chu, Phys. Rev. E 71, 066601 (2005).
[CrossRef]

Chen, X.

Chow, K. W.

B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, Phys. Rev. E 86, 036608 (2012).
[CrossRef]

C. H. Tsang, B. A. Malomed, and K. W. Chow, Phys. Rev. E 84, 066609 (2011).
[CrossRef]

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. Special Topics 173, 233 (2009).
[CrossRef]

Chu, P. L.

P. Y. P. Chen, B. A. Malomed, and P. L. Chu, Phys. Rev. E 71, 066601 (2005).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
[CrossRef]

Colet, P.

de Wit, E.

M. van Hecke, E. de Wit, and W. van Saarloos, Phys. Rev. Lett. 75, 3830 (1995).
[CrossRef]

Ding, E.

B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, Phys. Rev. E 86, 036608 (2012).
[CrossRef]

Firth, W. J.

Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jaeger, Phys. Rev. Lett. 100, 013907 (2008).
[CrossRef]

W. J. Firth, G. K. Harkness, A. Lord, J. M. McSloy, D. Gomila, and P. Colet, J. Opt. Soc. Am. B 19, 747 (2002).
[CrossRef]

Gomila, D.

Harkness, G. K.

He, Y.

Huang, C.

Hukriede, J.

J. Hukriede, D. Runde, and D. Kip, J. Phys. D 36, R1 (2003).

Jaeger, R.

Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jaeger, Phys. Rev. Lett. 100, 013907 (2008).
[CrossRef]

Kartashov, Y. V.

Kip, D.

J. Hukriede, D. Runde, and D. Kip, J. Phys. D 36, R1 (2003).

Kivshar, Y. S.

Y. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

Konotop, V. V.

Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, Phys. Rev. A 83, 041806 (2011).
[CrossRef]

D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, Opt. Lett. 36, 1200 (2011).
[CrossRef]

D. A. Zezyulin, V. V. Konotop, and G. L. Alfimov, Phys. Rev. E 82, 056213 (2010).
[CrossRef]

F. K. Abdullaev, V. V. Konotop, M. Salerno, and A. V. Yulin, Phys. Rev. E 82, 056606 (2010).
[CrossRef]

Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, Europhys. Lett. 91, 34003 (2010).
[CrossRef]

Kozyreff, G.

Kutz, J. N.

J. N. Kutz, SIAM Rev. 48, 629 (2006).
[CrossRef]

Lai, S. K.

B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, Phys. Rev. E 86, 036608 (2012).
[CrossRef]

Lam, C.-K.

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. Special Topics 173, 233 (2009).
[CrossRef]

Leblond, H.

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. Lett. 105, 213901 (2010).
[CrossRef]

Lobanov, V. E.

Lord, A.

Mak, W. C. K.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
[CrossRef]

Malomed, B. A.

C. Huang, F. Ye, B. A. Malomed, Y. V. Kartashov, and X. Chen, Opt. Lett. 38, 2177 (2013).
[CrossRef]

B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, Phys. Rev. E 86, 036608 (2012).
[CrossRef]

O. V. Borovkova, V. E. Lobanov, and B. A. Malomed, Europhys. Lett. 97, 44003 (2012).
[CrossRef]

C. H. Tsang, B. A. Malomed, and K. W. Chow, Phys. Rev. E 84, 066609 (2011).
[CrossRef]

Y. V. Kartashov, B. A. Malomed, and L. Torner, Rev. Mod. Phys. 83, 247 (2011).
[CrossRef]

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

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. Lett. 105, 213901 (2010).
[CrossRef]

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

P. Y. P. Chen, B. A. Malomed, and P. L. Chu, Phys. Rev. E 71, 066601 (2005).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
[CrossRef]

Y. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

Mandel, P.

McSloy, J. M.

Mihalache, D.

C. Yin, D. Mihalache, and Y. He, J. Opt. Soc. Am. B 28, 342 (2011).
[CrossRef]

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. Lett. 105, 213901 (2010).
[CrossRef]

Pieroux, D.

M. Tlidi, A. G. Vladimirov, D. Pieroux, and D. Turaev, Phys. Rev. Lett. 103, 103904 (2009).
[CrossRef]

Rosanov, N. N.

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

Runde, D.

J. Hukriede, D. Runde, and D. Kip, J. Phys. D 36, R1 (2003).

Salerno, M.

F. K. Abdullaev, V. V. Konotop, M. Salerno, and A. V. Yulin, Phys. Rev. E 82, 056606 (2010).
[CrossRef]

Skarka, V.

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. Lett. 105, 213901 (2010).
[CrossRef]

Skryabin, D. V.

Tanguy, Y.

Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jaeger, Phys. Rev. Lett. 100, 013907 (2008).
[CrossRef]

Tlidi, M.

M. Tlidi, A. G. Vladimirov, D. Pieroux, and D. Turaev, Phys. Rev. Lett. 103, 103904 (2009).
[CrossRef]

A. G. Vladimirov, D. V. Skryabin, G. Kozyreff, P. Mandel, and M. Tlidi, Opt. Express 14, 1 (2006).
[CrossRef]

P. Mandel and M. Tlidi, J. Opt. B Quant. Semiclass. Opt. 6, R60 (2004).
[CrossRef]

Torner, L.

Tsang, C. H.

C. H. Tsang, B. A. Malomed, and K. W. Chow, Phys. Rev. E 84, 066609 (2011).
[CrossRef]

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

Turaev, D.

M. Tlidi, A. G. Vladimirov, D. Pieroux, and D. Turaev, Phys. Rev. Lett. 103, 103904 (2009).
[CrossRef]

van Hecke, M.

M. van Hecke, E. de Wit, and W. van Saarloos, Phys. Rev. Lett. 75, 3830 (1995).
[CrossRef]

van Saarloos, W.

M. van Hecke, E. de Wit, and W. van Saarloos, Phys. Rev. Lett. 75, 3830 (1995).
[CrossRef]

Vladimirov, A. G.

M. Tlidi, A. G. Vladimirov, D. Pieroux, and D. Turaev, Phys. Rev. Lett. 103, 103904 (2009).
[CrossRef]

A. G. Vladimirov, D. V. Skryabin, G. Kozyreff, P. Mandel, and M. Tlidi, Opt. Express 14, 1 (2006).
[CrossRef]

Vysloukh, V. A.

V. E. Lobanov, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 36, 85 (2011).
[CrossRef]

O. V. Borovkova, Y. V. Kartashov, V. E. Lobanov, V. A. Vysloukh, and L. Torner, Opt. Lett. 36, 3783 (2011).
[CrossRef]

Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, Phys. Rev. A 83, 041806 (2011).
[CrossRef]

Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, Europhys. Lett. 91, 34003 (2010).
[CrossRef]

Wai, P. K. A.

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

Ye, F.

Yin, C.

Yulin, A. V.

F. K. Abdullaev, V. V. Konotop, M. Salerno, and A. V. Yulin, Phys. Rev. E 82, 056606 (2010).
[CrossRef]

Zezyulin, D. A.

D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, Opt. Lett. 36, 1200 (2011).
[CrossRef]

D. A. Zezyulin, V. V. Konotop, and G. L. Alfimov, Phys. Rev. E 82, 056213 (2010).
[CrossRef]

Eur. J. Phys. Special Topics

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

Eur. Phys. J. D

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

Europhys. Lett.

Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, Europhys. Lett. 91, 34003 (2010).
[CrossRef]

O. V. Borovkova, V. E. Lobanov, and B. A. Malomed, Europhys. Lett. 97, 44003 (2012).
[CrossRef]

J. Opt. B Quant. Semiclass. Opt.

P. Mandel and M. Tlidi, J. Opt. B Quant. Semiclass. Opt. 6, R60 (2004).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. D

J. Hukriede, D. Runde, and D. Kip, J. Phys. D 36, R1 (2003).

Opt. Express

Opt. Lett.

Phys. Rev. A

Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, Phys. Rev. A 83, 041806 (2011).
[CrossRef]

Phys. Rev. E

D. A. Zezyulin, V. V. Konotop, and G. L. Alfimov, Phys. Rev. E 82, 056213 (2010).
[CrossRef]

C. H. Tsang, B. A. Malomed, and K. W. Chow, Phys. Rev. E 84, 066609 (2011).
[CrossRef]

F. K. Abdullaev, V. V. Konotop, M. Salerno, and A. V. Yulin, Phys. Rev. E 82, 056606 (2010).
[CrossRef]

B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, Phys. Rev. E 86, 036608 (2012).
[CrossRef]

P. Y. P. Chen, B. A. Malomed, and P. L. Chu, Phys. Rev. E 71, 066601 (2005).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
[CrossRef]

Phys. Rev. Lett.

M. van Hecke, E. de Wit, and W. van Saarloos, Phys. Rev. Lett. 75, 3830 (1995).
[CrossRef]

Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jaeger, Phys. Rev. Lett. 100, 013907 (2008).
[CrossRef]

M. Tlidi, A. G. Vladimirov, D. Pieroux, and D. Turaev, Phys. Rev. Lett. 103, 103904 (2009).
[CrossRef]

V. Skarka, N. B. Aleksić, H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. Lett. 105, 213901 (2010).
[CrossRef]

Rev. Mod. Phys.

Y. V. Kartashov, B. A. Malomed, and L. Torner, Rev. Mod. Phys. 83, 247 (2011).
[CrossRef]

Y. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

SIAM Rev.

J. N. Kutz, SIAM Rev. 48, 629 (2006).
[CrossRef]

Other

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

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Figures (5)

Fig. 1.
Fig. 1.

Typical example of the unstable motion of the soliton in the cavity fenced by the δ-functional linear potential barriers with ε0=10; see Eq. (1). The initial tilt (velocity) is C0=4, and the gain is Γ0=0.05. Shown is a stage of the evolution at which the instability is evident.

Fig. 2.
Fig. 2.

Typical example of the stable shuttle motion of the soliton in the cavity fenced by the nonlinear δ-functional barriers with ε2=0.1; see Eq. (3), C0 and Γ0 being the same as in Fig. 1.

Fig. 3.
Fig. 3.

Completely overlapping blue and dashed red curves show amplitudes of the stably shuttling soliton, as found from simulations of Eq. (3), and as predicted by Eq. (2). In agreement with the other prediction of the analysis, the amplitude does not depend on the soliton’s tilt (velocity).

Fig. 4.
Fig. 4.

Evolution of the soliton’s amplitudes toward equilibrium value (2) in simulations of Eq. (3) with Γ0=1/15. Periodic spikes correspond to transient compression of the soliton hitting the nonlinear potential barriers.

Fig. 5.
Fig. 5.

(a) Evolution of the soliton’s velocity in the simulations of Eq. (3), with Γ0=0.05, from an initial value Cin=4 toward C0=1.15. Periodic spikes correspond to the soliton rebounding from the barriers. (b) Upper bound C0(Γ0) for velocities of stably shuttling solitons with ε2=0.1. The dashed curve is the best fit to the analytically estimated dependence, C0Γ03/4.

Equations (4)

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

iuz+(1/2)uxx+|u|2u=ε0[δ(x+L/2)+δ(xL/2)]u+igL(x)(Γ0Γ2|u|2)u.
η0=3Γ0/(2Γ2).
iuz+(1/2)uxx+|u|2u=ε2[δ(x+L/2)+δ(xL/2)]|u|2u+igL(x)(Γ0Γ2|u|2)u+Uramp(x)u,
U(ξ)=(ε2/2)η4sech4(ηξ),

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