Abstract

We study the dynamics of oscillating discrete solitons in an array of coupled Kerr-nonlinear cavities. They emanate from stationary discrete cavity solitons due to Hopf instability and are very robust. We show that these oscillating solitons can spontaneously lose their spatial symmetry and start rocking around the equilibrium position. Moreover they can suddenly jump to adjacent resonators starting a chaotic motion along the array, resembling the Brownian motion of particles. We also identify the parameter domain where they move with constant velocity across the array.

© 2013 Optical Society of America

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  1. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
    [CrossRef]
  2. S. Flach and A. V. Gorbach, Phys. Rep. 467, 1 (2008).
    [CrossRef]
  3. N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2008).
  4. T. Ackemann, W. J. Firth, and G. Oppo, in Advances in Atomic, Molecular, and Optical Physics, Vol. 57 (Elsevier, 2009), pp. 323–421.
  5. U. Peschel, O. Egorov, and F. Lederer, Opt. Lett. 29, 1909 (2004).
    [CrossRef]
  6. O. Egorov, U. Peschel, and F. Lederer, Phys. Rev. E 71, 056612 (2005).
    [CrossRef]
  7. A. V. Yulin, A. R. Champneys, and D. V. Skryabin, Phys. Rev. A 78, 011804 (2008).
    [CrossRef]
  8. A. V. Yulin and A. R. Champneys, SIAM J. Appl. Dyn. Syst. 9, 391 (2010).
    [CrossRef]
  9. P. Lalanne, C. Sauvan, and J. P. Hugonin, Laser Photon. Rev. 2, 514 (2008).
    [CrossRef]
  10. R. E. Noskov, P. A. Belov, and Y. S. Kivshar, Phys. Rev. Lett. 108, 093901 (2012).
    [CrossRef]
  11. O. Egorov, U. Peschel, and F. Lederer, Phys. Rev. E 72, 066603 (2005).
    [CrossRef]
  12. O. Egorov, F. Lederer, and Y. S. Kivshar, Opt. Express 15, 4149 (2007).
    [CrossRef]
  13. O. Egorov, F. Lederer, and K. Staliunas, Opt. Lett. 32, 2106 (2007).
    [CrossRef]
  14. S. Fedorov, D. Michaelis, U. Peschel, C. Etrich, D. V. Skryabin, N. Rosanov, and F. Lederer, Phys. Rev. E 64, 036610 (2001).
    [CrossRef]
  15. Y. S. Kivshar and D. K. Campbell, Phys. Rev. E 48, 3077 (1993).
    [CrossRef]
  16. L. Hadžievski, A. Maluckov, M. Stepić, and D. Kip, Phys. Rev. Lett. 93, 033901 (2004).
    [CrossRef]
  17. A. V. Gorbach, S. Denisov, and S. Flach, Opt. Lett. 31, 1702 (2006).
    [CrossRef]
  18. J. L. Marín, F. Falo, P. J. Martínez, and L. M. Floría, Phys. Rev. E 63, 066603 (2001).
    [CrossRef]
  19. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Application to Physics, Biology, Chemistry, and Engineering (Westview, 2000).

2012

R. E. Noskov, P. A. Belov, and Y. S. Kivshar, Phys. Rev. Lett. 108, 093901 (2012).
[CrossRef]

2010

A. V. Yulin and A. R. Champneys, SIAM J. Appl. Dyn. Syst. 9, 391 (2010).
[CrossRef]

2008

P. Lalanne, C. Sauvan, and J. P. Hugonin, Laser Photon. Rev. 2, 514 (2008).
[CrossRef]

A. V. Yulin, A. R. Champneys, and D. V. Skryabin, Phys. Rev. A 78, 011804 (2008).
[CrossRef]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

S. Flach and A. V. Gorbach, Phys. Rep. 467, 1 (2008).
[CrossRef]

2007

2006

2005

O. Egorov, U. Peschel, and F. Lederer, Phys. Rev. E 71, 056612 (2005).
[CrossRef]

O. Egorov, U. Peschel, and F. Lederer, Phys. Rev. E 72, 066603 (2005).
[CrossRef]

2004

L. Hadžievski, A. Maluckov, M. Stepić, and D. Kip, Phys. Rev. Lett. 93, 033901 (2004).
[CrossRef]

U. Peschel, O. Egorov, and F. Lederer, Opt. Lett. 29, 1909 (2004).
[CrossRef]

2001

J. L. Marín, F. Falo, P. J. Martínez, and L. M. Floría, Phys. Rev. E 63, 066603 (2001).
[CrossRef]

S. Fedorov, D. Michaelis, U. Peschel, C. Etrich, D. V. Skryabin, N. Rosanov, and F. Lederer, Phys. Rev. E 64, 036610 (2001).
[CrossRef]

1993

Y. S. Kivshar and D. K. Campbell, Phys. Rev. E 48, 3077 (1993).
[CrossRef]

Ackemann, T.

T. Ackemann, W. J. Firth, and G. Oppo, in Advances in Atomic, Molecular, and Optical Physics, Vol. 57 (Elsevier, 2009), pp. 323–421.

Akhmediev, N.

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2008).

Ankiewicz, A.

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2008).

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Belov, P. A.

R. E. Noskov, P. A. Belov, and Y. S. Kivshar, Phys. Rev. Lett. 108, 093901 (2012).
[CrossRef]

Campbell, D. K.

Y. S. Kivshar and D. K. Campbell, Phys. Rev. E 48, 3077 (1993).
[CrossRef]

Champneys, A. R.

A. V. Yulin and A. R. Champneys, SIAM J. Appl. Dyn. Syst. 9, 391 (2010).
[CrossRef]

A. V. Yulin, A. R. Champneys, and D. V. Skryabin, Phys. Rev. A 78, 011804 (2008).
[CrossRef]

Christodoulides, D. N.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Denisov, S.

Egorov, O.

Etrich, C.

S. Fedorov, D. Michaelis, U. Peschel, C. Etrich, D. V. Skryabin, N. Rosanov, and F. Lederer, Phys. Rev. E 64, 036610 (2001).
[CrossRef]

Falo, F.

J. L. Marín, F. Falo, P. J. Martínez, and L. M. Floría, Phys. Rev. E 63, 066603 (2001).
[CrossRef]

Fedorov, S.

S. Fedorov, D. Michaelis, U. Peschel, C. Etrich, D. V. Skryabin, N. Rosanov, and F. Lederer, Phys. Rev. E 64, 036610 (2001).
[CrossRef]

Firth, W. J.

T. Ackemann, W. J. Firth, and G. Oppo, in Advances in Atomic, Molecular, and Optical Physics, Vol. 57 (Elsevier, 2009), pp. 323–421.

Flach, S.

Floría, L. M.

J. L. Marín, F. Falo, P. J. Martínez, and L. M. Floría, Phys. Rev. E 63, 066603 (2001).
[CrossRef]

Gorbach, A. V.

Hadžievski, L.

L. Hadžievski, A. Maluckov, M. Stepić, and D. Kip, Phys. Rev. Lett. 93, 033901 (2004).
[CrossRef]

Hugonin, J. P.

P. Lalanne, C. Sauvan, and J. P. Hugonin, Laser Photon. Rev. 2, 514 (2008).
[CrossRef]

Kip, D.

L. Hadžievski, A. Maluckov, M. Stepić, and D. Kip, Phys. Rev. Lett. 93, 033901 (2004).
[CrossRef]

Kivshar, Y. S.

R. E. Noskov, P. A. Belov, and Y. S. Kivshar, Phys. Rev. Lett. 108, 093901 (2012).
[CrossRef]

O. Egorov, F. Lederer, and Y. S. Kivshar, Opt. Express 15, 4149 (2007).
[CrossRef]

Y. S. Kivshar and D. K. Campbell, Phys. Rev. E 48, 3077 (1993).
[CrossRef]

Lalanne, P.

P. Lalanne, C. Sauvan, and J. P. Hugonin, Laser Photon. Rev. 2, 514 (2008).
[CrossRef]

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

O. Egorov, F. Lederer, and Y. S. Kivshar, Opt. Express 15, 4149 (2007).
[CrossRef]

O. Egorov, F. Lederer, and K. Staliunas, Opt. Lett. 32, 2106 (2007).
[CrossRef]

O. Egorov, U. Peschel, and F. Lederer, Phys. Rev. E 71, 056612 (2005).
[CrossRef]

O. Egorov, U. Peschel, and F. Lederer, Phys. Rev. E 72, 066603 (2005).
[CrossRef]

U. Peschel, O. Egorov, and F. Lederer, Opt. Lett. 29, 1909 (2004).
[CrossRef]

S. Fedorov, D. Michaelis, U. Peschel, C. Etrich, D. V. Skryabin, N. Rosanov, and F. Lederer, Phys. Rev. E 64, 036610 (2001).
[CrossRef]

Maluckov, A.

L. Hadžievski, A. Maluckov, M. Stepić, and D. Kip, Phys. Rev. Lett. 93, 033901 (2004).
[CrossRef]

Marín, J. L.

J. L. Marín, F. Falo, P. J. Martínez, and L. M. Floría, Phys. Rev. E 63, 066603 (2001).
[CrossRef]

Martínez, P. J.

J. L. Marín, F. Falo, P. J. Martínez, and L. M. Floría, Phys. Rev. E 63, 066603 (2001).
[CrossRef]

Michaelis, D.

S. Fedorov, D. Michaelis, U. Peschel, C. Etrich, D. V. Skryabin, N. Rosanov, and F. Lederer, Phys. Rev. E 64, 036610 (2001).
[CrossRef]

Noskov, R. E.

R. E. Noskov, P. A. Belov, and Y. S. Kivshar, Phys. Rev. Lett. 108, 093901 (2012).
[CrossRef]

Oppo, G.

T. Ackemann, W. J. Firth, and G. Oppo, in Advances in Atomic, Molecular, and Optical Physics, Vol. 57 (Elsevier, 2009), pp. 323–421.

Peschel, U.

O. Egorov, U. Peschel, and F. Lederer, Phys. Rev. E 71, 056612 (2005).
[CrossRef]

O. Egorov, U. Peschel, and F. Lederer, Phys. Rev. E 72, 066603 (2005).
[CrossRef]

U. Peschel, O. Egorov, and F. Lederer, Opt. Lett. 29, 1909 (2004).
[CrossRef]

S. Fedorov, D. Michaelis, U. Peschel, C. Etrich, D. V. Skryabin, N. Rosanov, and F. Lederer, Phys. Rev. E 64, 036610 (2001).
[CrossRef]

Rosanov, N.

S. Fedorov, D. Michaelis, U. Peschel, C. Etrich, D. V. Skryabin, N. Rosanov, and F. Lederer, Phys. Rev. E 64, 036610 (2001).
[CrossRef]

Sauvan, C.

P. Lalanne, C. Sauvan, and J. P. Hugonin, Laser Photon. Rev. 2, 514 (2008).
[CrossRef]

Segev, M.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Skryabin, D. V.

A. V. Yulin, A. R. Champneys, and D. V. Skryabin, Phys. Rev. A 78, 011804 (2008).
[CrossRef]

S. Fedorov, D. Michaelis, U. Peschel, C. Etrich, D. V. Skryabin, N. Rosanov, and F. Lederer, Phys. Rev. E 64, 036610 (2001).
[CrossRef]

Staliunas, K.

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Stepic, M.

L. Hadžievski, A. Maluckov, M. Stepić, and D. Kip, Phys. Rev. Lett. 93, 033901 (2004).
[CrossRef]

Strogatz, S. H.

S. H. Strogatz, Nonlinear Dynamics and Chaos: With Application to Physics, Biology, Chemistry, and Engineering (Westview, 2000).

Yulin, A. V.

A. V. Yulin and A. R. Champneys, SIAM J. Appl. Dyn. Syst. 9, 391 (2010).
[CrossRef]

A. V. Yulin, A. R. Champneys, and D. V. Skryabin, Phys. Rev. A 78, 011804 (2008).
[CrossRef]

Laser Photon. Rev.

P. Lalanne, C. Sauvan, and J. P. Hugonin, Laser Photon. Rev. 2, 514 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rep.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

S. Flach and A. V. Gorbach, Phys. Rep. 467, 1 (2008).
[CrossRef]

Phys. Rev. A

A. V. Yulin, A. R. Champneys, and D. V. Skryabin, Phys. Rev. A 78, 011804 (2008).
[CrossRef]

Phys. Rev. E

J. L. Marín, F. Falo, P. J. Martínez, and L. M. Floría, Phys. Rev. E 63, 066603 (2001).
[CrossRef]

O. Egorov, U. Peschel, and F. Lederer, Phys. Rev. E 71, 056612 (2005).
[CrossRef]

O. Egorov, U. Peschel, and F. Lederer, Phys. Rev. E 72, 066603 (2005).
[CrossRef]

S. Fedorov, D. Michaelis, U. Peschel, C. Etrich, D. V. Skryabin, N. Rosanov, and F. Lederer, Phys. Rev. E 64, 036610 (2001).
[CrossRef]

Y. S. Kivshar and D. K. Campbell, Phys. Rev. E 48, 3077 (1993).
[CrossRef]

Phys. Rev. Lett.

L. Hadžievski, A. Maluckov, M. Stepić, and D. Kip, Phys. Rev. Lett. 93, 033901 (2004).
[CrossRef]

R. E. Noskov, P. A. Belov, and Y. S. Kivshar, Phys. Rev. Lett. 108, 093901 (2012).
[CrossRef]

SIAM J. Appl. Dyn. Syst.

A. V. Yulin and A. R. Champneys, SIAM J. Appl. Dyn. Syst. 9, 391 (2010).
[CrossRef]

Other

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2008).

T. Ackemann, W. J. Firth, and G. Oppo, in Advances in Atomic, Molecular, and Optical Physics, Vol. 57 (Elsevier, 2009), pp. 323–421.

S. H. Strogatz, Nonlinear Dynamics and Chaos: With Application to Physics, Biology, Chemistry, and Engineering (Westview, 2000).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Amplitude maximum of stationary DCS and ODCS versus the coupling constant C. Dashed lines mark unstable solutions. CB1 designates the bifurcation point where ODCSs lose their spatial symmetry. (b) Center of mass (ξ1) of DCSs and ODCSs versus the constant C. Asymmetrical ODCSs oscillate around the array sites for CB2<C<CB1. Parameters: Δ=3, E0=2.

Fig. 2.
Fig. 2.

(a) Dynamics of the ODCS profile for C=6. (b) Dynamics of asymmetrical ODCS for C=8.5. Parameters: Δ=3, E0=2.

Fig. 3.
Fig. 3.

Evolution of ODCSs in phase space represented by the center of mass ξ1(T) and the mean power ξ2(T) for different coupling: (a) C=8.5, (b) C=7.64, and (c) C=7.5. (d) Center of mass ξ1(T) dynamics of the chaotically moving ODCSs for C=7.5. Parameters: Δ=3, E0=2.

Fig. 4.
Fig. 4.

Dynamics of (a) chaotically and (b) uniformly moving ODCSs for (a) C=8.2 and (b) C=7.6. Parameters: E0=2.1, Δ=3.

Equations (2)

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

iunT+C(un+1+un12un)+(i+Δ)un+|un|2un=E0.
ξ1(T)=(j·(uj(T)b)2)ξ21(T),ξ2(T)=(uj(T)b)2,

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