Abstract

The problems of the existence, stability, and transversal motion of the discrete dark localized modes in the lattices with saturable nonlinearity are investigated analytically and numerically. The stability analysis shows existence of regions of the parametric space with eigenvalue spectrum branches with non-zeroth real part, which indicates possibility for the propagation of stable on-site and inter-site dark localized modes. The analysis based on the conserved system quantities reveals the existence of regions with a vanishing Peierls-Nabarro barrier which allows transverse motion of the dark breathers. Propagation of the stable on-site and inter-site dark breathers and their free transversal motion are observed numerically.

© 2007 Optical Society of America

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References

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  1. P. G. Kevrekidis, K. Ø. Rasmussen, and A. R. Bishop, "The Discrete Nonlinear Schrödinger Eauation: a Survey of Recent Results," Int. J. mod. Phys. B 15,2833-2900 (2001).
    [CrossRef]
  2. A. A. Sukhorukov, Yu. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, "Spatial optical solitons in waveguide arrays," IEEE J. Quantum Electron. 39,31-50 (2003).
    [CrossRef]
  3. E. Trias, J. J. Mazo, and T. P. Orlando, "Dicrete breathers in nonlinear lattices: experimental detection in Josephson array," Phys. Rev. Lett. 84,741-744 (2000).
    [CrossRef] [PubMed]
  4. U. T. Schwarz, L. Q. English, and A. J. Sievers, "Experimental generation and observation of intrisic localized spin wave modes in an antiferromagnet," Phys. Rev. Lett. 83,223-226 (1999).
    [CrossRef]
  5. P. J. Y. Louis, E. A. Ostrovskaya, and Yu. S. Kivshar, "Dispersion control for matter waves and gap solitons in optical superlattices," J. Opt. B: Quantum Semiclass. Opt. 6,S309-S317 (2004).Q1
    [CrossRef]
  6. Yu. S. Kivshar and B. Luther-Davies, "Dark optical solitons: physics and applications," Phys. Rep. 298,81-197 (1998).
    [CrossRef]
  7. B. Sanchez-Rey and M. Johansson, "Exact numerical solutions for dark waves on the discrete nonlinear Schr¨odinger equation," Phys. Rev. E 71,036627(2005).
    [CrossRef]
  8. E. Smirnov, C. E. Rüter, M, Stepić, D. Kip, and V. Shandarov, "Formation and light guiding properties of dark solions in one-dimensional waveguide arrays," Phys. Rev. E 74, 065601(R) (2006).
    [CrossRef]
  9. E. P. Fitrakis, P. G. Kevrekidis, H. Susanto, and D. J. Frantzeskakis, "Dark solitons in discrete lattices: Saturable versus cubic nonlinearitis," arXiv:nlin.PS/0608023 (2006).
  10. T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis, and J. Cuevas, "Radiationless traveling waves in saturable nonlinear Schr¨odinger lattices," Phys. Rev. Lett. 97,124101 (2006).
    [CrossRef] [PubMed]
  11. Lj. Hadzievski, A. Maluckov, M. Stepić, and D. Kip, "Power controlled soliton stability and steering in lattices with saturable nonlinearity," Phys. Rev. Lett. 93,033901 (2004).
    [CrossRef] [PubMed]
  12. S. Wiggins, Global Bifurcations and Chaos: Analytical Methods (Springer-Verlag New York Inc., 1988).

2006 (1)

T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis, and J. Cuevas, "Radiationless traveling waves in saturable nonlinear Schr¨odinger lattices," Phys. Rev. Lett. 97,124101 (2006).
[CrossRef] [PubMed]

2005 (1)

B. Sanchez-Rey and M. Johansson, "Exact numerical solutions for dark waves on the discrete nonlinear Schr¨odinger equation," Phys. Rev. E 71,036627(2005).
[CrossRef]

2004 (2)

Lj. Hadzievski, A. Maluckov, M. Stepić, and D. Kip, "Power controlled soliton stability and steering in lattices with saturable nonlinearity," Phys. Rev. Lett. 93,033901 (2004).
[CrossRef] [PubMed]

P. J. Y. Louis, E. A. Ostrovskaya, and Yu. S. Kivshar, "Dispersion control for matter waves and gap solitons in optical superlattices," J. Opt. B: Quantum Semiclass. Opt. 6,S309-S317 (2004).Q1
[CrossRef]

2003 (1)

A. A. Sukhorukov, Yu. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, "Spatial optical solitons in waveguide arrays," IEEE J. Quantum Electron. 39,31-50 (2003).
[CrossRef]

2001 (1)

P. G. Kevrekidis, K. Ø. Rasmussen, and A. R. Bishop, "The Discrete Nonlinear Schrödinger Eauation: a Survey of Recent Results," Int. J. mod. Phys. B 15,2833-2900 (2001).
[CrossRef]

2000 (1)

E. Trias, J. J. Mazo, and T. P. Orlando, "Dicrete breathers in nonlinear lattices: experimental detection in Josephson array," Phys. Rev. Lett. 84,741-744 (2000).
[CrossRef] [PubMed]

1999 (1)

U. T. Schwarz, L. Q. English, and A. J. Sievers, "Experimental generation and observation of intrisic localized spin wave modes in an antiferromagnet," Phys. Rev. Lett. 83,223-226 (1999).
[CrossRef]

1998 (1)

Yu. S. Kivshar and B. Luther-Davies, "Dark optical solitons: physics and applications," Phys. Rep. 298,81-197 (1998).
[CrossRef]

Bishop, A. R.

P. G. Kevrekidis, K. Ø. Rasmussen, and A. R. Bishop, "The Discrete Nonlinear Schrödinger Eauation: a Survey of Recent Results," Int. J. mod. Phys. B 15,2833-2900 (2001).
[CrossRef]

Champneys, A. R.

T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis, and J. Cuevas, "Radiationless traveling waves in saturable nonlinear Schr¨odinger lattices," Phys. Rev. Lett. 97,124101 (2006).
[CrossRef] [PubMed]

Cuevas, J.

T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis, and J. Cuevas, "Radiationless traveling waves in saturable nonlinear Schr¨odinger lattices," Phys. Rev. Lett. 97,124101 (2006).
[CrossRef] [PubMed]

Eisenberg, H. S.

A. A. Sukhorukov, Yu. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, "Spatial optical solitons in waveguide arrays," IEEE J. Quantum Electron. 39,31-50 (2003).
[CrossRef]

English, L. Q.

U. T. Schwarz, L. Q. English, and A. J. Sievers, "Experimental generation and observation of intrisic localized spin wave modes in an antiferromagnet," Phys. Rev. Lett. 83,223-226 (1999).
[CrossRef]

Hadzievski, Lj.

Lj. Hadzievski, A. Maluckov, M. Stepić, and D. Kip, "Power controlled soliton stability and steering in lattices with saturable nonlinearity," Phys. Rev. Lett. 93,033901 (2004).
[CrossRef] [PubMed]

Johansson, M.

B. Sanchez-Rey and M. Johansson, "Exact numerical solutions for dark waves on the discrete nonlinear Schr¨odinger equation," Phys. Rev. E 71,036627(2005).
[CrossRef]

Kevrekidis, P. G.

T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis, and J. Cuevas, "Radiationless traveling waves in saturable nonlinear Schr¨odinger lattices," Phys. Rev. Lett. 97,124101 (2006).
[CrossRef] [PubMed]

P. G. Kevrekidis, K. Ø. Rasmussen, and A. R. Bishop, "The Discrete Nonlinear Schrödinger Eauation: a Survey of Recent Results," Int. J. mod. Phys. B 15,2833-2900 (2001).
[CrossRef]

Kip, D.

Lj. Hadzievski, A. Maluckov, M. Stepić, and D. Kip, "Power controlled soliton stability and steering in lattices with saturable nonlinearity," Phys. Rev. Lett. 93,033901 (2004).
[CrossRef] [PubMed]

Kivshar, Yu. S.

P. J. Y. Louis, E. A. Ostrovskaya, and Yu. S. Kivshar, "Dispersion control for matter waves and gap solitons in optical superlattices," J. Opt. B: Quantum Semiclass. Opt. 6,S309-S317 (2004).Q1
[CrossRef]

A. A. Sukhorukov, Yu. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, "Spatial optical solitons in waveguide arrays," IEEE J. Quantum Electron. 39,31-50 (2003).
[CrossRef]

Yu. S. Kivshar and B. Luther-Davies, "Dark optical solitons: physics and applications," Phys. Rep. 298,81-197 (1998).
[CrossRef]

Louis, P. J. Y.

P. J. Y. Louis, E. A. Ostrovskaya, and Yu. S. Kivshar, "Dispersion control for matter waves and gap solitons in optical superlattices," J. Opt. B: Quantum Semiclass. Opt. 6,S309-S317 (2004).Q1
[CrossRef]

Luther-Davies, B.

Yu. S. Kivshar and B. Luther-Davies, "Dark optical solitons: physics and applications," Phys. Rep. 298,81-197 (1998).
[CrossRef]

Maluckov, A.

Lj. Hadzievski, A. Maluckov, M. Stepić, and D. Kip, "Power controlled soliton stability and steering in lattices with saturable nonlinearity," Phys. Rev. Lett. 93,033901 (2004).
[CrossRef] [PubMed]

Mazo, J. J.

E. Trias, J. J. Mazo, and T. P. Orlando, "Dicrete breathers in nonlinear lattices: experimental detection in Josephson array," Phys. Rev. Lett. 84,741-744 (2000).
[CrossRef] [PubMed]

Melvin, T. R. O.

T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis, and J. Cuevas, "Radiationless traveling waves in saturable nonlinear Schr¨odinger lattices," Phys. Rev. Lett. 97,124101 (2006).
[CrossRef] [PubMed]

Orlando, T. P.

E. Trias, J. J. Mazo, and T. P. Orlando, "Dicrete breathers in nonlinear lattices: experimental detection in Josephson array," Phys. Rev. Lett. 84,741-744 (2000).
[CrossRef] [PubMed]

Ostrovskaya, E. A.

P. J. Y. Louis, E. A. Ostrovskaya, and Yu. S. Kivshar, "Dispersion control for matter waves and gap solitons in optical superlattices," J. Opt. B: Quantum Semiclass. Opt. 6,S309-S317 (2004).Q1
[CrossRef]

Rasmussen, K. Ø.

P. G. Kevrekidis, K. Ø. Rasmussen, and A. R. Bishop, "The Discrete Nonlinear Schrödinger Eauation: a Survey of Recent Results," Int. J. mod. Phys. B 15,2833-2900 (2001).
[CrossRef]

Sanchez-Rey, B.

B. Sanchez-Rey and M. Johansson, "Exact numerical solutions for dark waves on the discrete nonlinear Schr¨odinger equation," Phys. Rev. E 71,036627(2005).
[CrossRef]

Schwarz, U. T.

U. T. Schwarz, L. Q. English, and A. J. Sievers, "Experimental generation and observation of intrisic localized spin wave modes in an antiferromagnet," Phys. Rev. Lett. 83,223-226 (1999).
[CrossRef]

Sievers, A. J.

U. T. Schwarz, L. Q. English, and A. J. Sievers, "Experimental generation and observation of intrisic localized spin wave modes in an antiferromagnet," Phys. Rev. Lett. 83,223-226 (1999).
[CrossRef]

Silberberg, Y.

A. A. Sukhorukov, Yu. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, "Spatial optical solitons in waveguide arrays," IEEE J. Quantum Electron. 39,31-50 (2003).
[CrossRef]

Stepic, M.

Lj. Hadzievski, A. Maluckov, M. Stepić, and D. Kip, "Power controlled soliton stability and steering in lattices with saturable nonlinearity," Phys. Rev. Lett. 93,033901 (2004).
[CrossRef] [PubMed]

Sukhorukov, A. A.

A. A. Sukhorukov, Yu. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, "Spatial optical solitons in waveguide arrays," IEEE J. Quantum Electron. 39,31-50 (2003).
[CrossRef]

Trias, E.

E. Trias, J. J. Mazo, and T. P. Orlando, "Dicrete breathers in nonlinear lattices: experimental detection in Josephson array," Phys. Rev. Lett. 84,741-744 (2000).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

A. A. Sukhorukov, Yu. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, "Spatial optical solitons in waveguide arrays," IEEE J. Quantum Electron. 39,31-50 (2003).
[CrossRef]

Int. J. mod. Phys. B (1)

P. G. Kevrekidis, K. Ø. Rasmussen, and A. R. Bishop, "The Discrete Nonlinear Schrödinger Eauation: a Survey of Recent Results," Int. J. mod. Phys. B 15,2833-2900 (2001).
[CrossRef]

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

P. J. Y. Louis, E. A. Ostrovskaya, and Yu. S. Kivshar, "Dispersion control for matter waves and gap solitons in optical superlattices," J. Opt. B: Quantum Semiclass. Opt. 6,S309-S317 (2004).Q1
[CrossRef]

Phys. Rep. (1)

Yu. S. Kivshar and B. Luther-Davies, "Dark optical solitons: physics and applications," Phys. Rep. 298,81-197 (1998).
[CrossRef]

Phys. Rev. E (1)

B. Sanchez-Rey and M. Johansson, "Exact numerical solutions for dark waves on the discrete nonlinear Schr¨odinger equation," Phys. Rev. E 71,036627(2005).
[CrossRef]

Phys. Rev. Lett. (4)

E. Trias, J. J. Mazo, and T. P. Orlando, "Dicrete breathers in nonlinear lattices: experimental detection in Josephson array," Phys. Rev. Lett. 84,741-744 (2000).
[CrossRef] [PubMed]

U. T. Schwarz, L. Q. English, and A. J. Sievers, "Experimental generation and observation of intrisic localized spin wave modes in an antiferromagnet," Phys. Rev. Lett. 83,223-226 (1999).
[CrossRef]

T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis, and J. Cuevas, "Radiationless traveling waves in saturable nonlinear Schr¨odinger lattices," Phys. Rev. Lett. 97,124101 (2006).
[CrossRef] [PubMed]

Lj. Hadzievski, A. Maluckov, M. Stepić, and D. Kip, "Power controlled soliton stability and steering in lattices with saturable nonlinearity," Phys. Rev. Lett. 93,033901 (2004).
[CrossRef] [PubMed]

Other (3)

S. Wiggins, Global Bifurcations and Chaos: Analytical Methods (Springer-Verlag New York Inc., 1988).

E. Smirnov, C. E. Rüter, M, Stepić, D. Kip, and V. Shandarov, "Formation and light guiding properties of dark solions in one-dimensional waveguide arrays," Phys. Rev. E 74, 065601(R) (2006).
[CrossRef]

E. P. Fitrakis, P. G. Kevrekidis, H. Susanto, and D. J. Frantzeskakis, "Dark solitons in discrete lattices: Saturable versus cubic nonlinearitis," arXiv:nlin.PS/0608023 (2006).

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Figures (3)

Fig. 1.
Fig. 1.

Eigenvalues spectrum for the on-site (a) and the inter-site (b) dark localized mode. Numerical results are given with symbols, analytical with lines: squares (solid lines) for extremal imaginary EV, solid circles (dashed lines) for the real and circles for the imaginary part of EV discrete spectrum. The continuous EV spectrum is displayed as a shaded region.

Fig. 2.
Fig. 2.

Illustration of the stable propagation of the unstaggered dark breathers: a) the on-site near the left boundary of the existence region, ω= 0.08 and b) inter-site near the right boundary of the existence region, ω= 8.9.

Fig. 3.
Fig. 3.

The grand canonical free energy of the on-site and inter-site dark soliton (a). Free transverse motion of the discrete dark breathers for b) ω= 0.08 and c) ω= 8.9.

Equations (9)

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

i U n t + U n + 1 + U n 1 2 U n + γ U n 1 + U n 2 = 0 ,
ωϕ n + ϕ n + 1 + ϕ n 1 2 ϕ n + γϕ n ( 1 + ϕ n 2 ) = 0 .
U c 2 = ( γ ω ) ω .
i ε n t ( ω + 2 ) ε n + ε n + 1 + ε n 1 + γ ( 1 + ϕ n 2 ) 2 ( ε ϕ n 2 ε n * ) = 0 .
d dt [ f n g n ] = [ 0 H + H 0 ] [ f n g n ] M [ f n g n ] ,
H ij + = ( ω + 2 ) δ ij δ i , j + 1 δ i , j 1 γ 1 + ϕ n 2 δ ij , H ij = H ij + + 2 γ ϕ n 2 ( 1 + ϕ n 2 ) 2 δ ij ,
Ω 2 = 4 [ γ ( 1 + cos K P cos q + ω ) ω 2 ] γ .
Ω uo = ± i ( γ ω 2 ) 2 2 , Ω ui = ± i 2 ( 1 + ω ω 2 γ ) ,
P c = n ( U cw 2 U n 2 ) , H c = H + γ [ P c ( 1 + U cw 2 ) N ln ( 1 + U cw ) ] .

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