## Dynamics of dark breathers in lattices with saturable nonlinearity

Optics Express, Vol. 15, Issue 9, pp. 5687-5692 (2007)

http://dx.doi.org/10.1364/OE.15.005687

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### 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

## 1. Introduction

## 2. Stability analysis

10. T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis, and J. Cuevas, “Radiationless traveling waves in saturable nonlinear Schrödinger lattices,” Phys. Rev. Lett. **97**, 124101 (2006). [CrossRef] [PubMed]

11. Lj. Hadžievski, 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]

*U*is the normalized wave function in the

_{n}*n*-th lattice element (

*n*= 1,2…) and

*γ*is the nonlinearity parameter. For

*γ*> 0 the nonlinearity is defocusing (DF). Under the transformation

*U*(

_{n}*t*) = exp(

*iπn*)exp(-4

*it*)

*V*

_{n}^{*}(

*t*) the equation (1) is mapped into the same DNLS equation for

*V*where

_{n}*γ*< 0 corresponds to the self-focusing (SF) nonlinearity. The Eq. (1) represents a system of linearly coupled nonlinear difference-differential equations which are not integrable in general case but posses two conserved quantities, Hamiltonian

*H*= ∑

*[-*

_{n}*γ*ln(1 +|

*U*|

_{n}^{2}) + |

*U*

_{n-1}-

*U*|

_{n}^{2}] and norm (power)

*P*= ∑

*|*

_{n}*U*|

_{n}^{2}.

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]

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]

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]

*U*(

_{n}*t*) =

*ϕ*, where

_{n}e^{iωt}*ω*is the propagation parameter, leads to the steady state version of Eq. (1)

*ϕ*is independent on

_{n}*n*and we can assume

*ϕ*=

_{n}*U*. With this assumption we obtain the following solution

_{c}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]

7. B. Sanchez-Rey and M. Johansson, “Exact numerical solutions for dark waves on the discrete nonlinear Schrödinger equation,” Phys. Rev. E **71**, 036627 (2005). [CrossRef]

*ϵ*to the wave envelopes in a form

_{n}*U*(

_{n}*t*) = (

*ϕ*+

_{n}*ϵ*(

_{n}*t*))

*eiωt*. After short and simple algebraic procedure the linearized equation for the small perturbations (

*ϵ*>>

_{n}*ϕ*) is obtained

_{n}*ϵ*into the real and imaginary part

_{n}*ϵ*=

_{n}*f*+

_{n}*ig*, the evolution of the perturbation is described by the the system of two equations for the real functions. The system written in a matrix form reads

_{n}**M**, which is generally non-hermitian [7

7. B. Sanchez-Rey and M. Johansson, “Exact numerical solutions for dark waves on the discrete nonlinear Schrödinger equation,” Phys. Rev. E **71**, 036627 (2005). [CrossRef]

*N*elements has dimension 2

*N*× 2

*N*. The submatrices

*H*

^{±}(

*N*×

*N*) can be written in the explicit form

*δ*is the Kroneker symbol.

_{ij}*f*,

_{n}*g*) = (

_{n}*f*,

*g*)

*e*

^{Ωt}*e*cos(

^{iK}_{p}^{n}*qn*), where

*K*=

_{p}*π*for staggered,

*K*= 0 for unstaggered perturbations and

_{p}*q*is the wave number, the following general dispersion relation is obtained

^{2}> 0) for different CW solutions of the DNLS equation (1) [one of them is written in (3)] we can conclude: a) For DF nonlinearity the CW staggered solutions are unstable which gives a possibility for creation of the staggered bright solitons, while the CW unstaggered solutions are stable which provides a stable background for creation of the unstaggered dark solitons; b) For SF nonlinearity the CW unstaggered solutions are unstable which gives a possibility for creation of the unstaggered bright solitons, while the CW staggered solutions are stable which provides a stable background for creation of the staggered dark solitons.

## 3. Dark modes

*M*. The eigenvalue (EV) spectrum has contributions from two sources. One is a continuous part of EV spectrum which arises from the background. The corresponding EV functions are plane waves. For the CW the dispersion relation (7) gives pure imaginary eigenvalues (Ω

^{2}< 0) and consequently does not indicate instability of dark solitons. The second source is associated with the central part of the dark soliton configuration and represents the discrete part of the EV spectrum. The discrete EV spectrum by itself and through the interaction with the continuous part of the EV spectrum can be associated with the eventual instability of the dark solitons. For this reason we will focus our study to the discrete spectrum.

*n*element. These solutions can be obtained by multiplying the unstaggered patterns with the amplitudes of the corresponding CW solution (3). The approximation holds in the limit of large amplitudes which corresponds to the values of

_{c}*ω*near the left boundary of the existence domain. Substitution of the approximate dark soliton solutions into the Eq. (5) allows calculation of the discrete eigenvalues

_{uo}andΩ

*- are discrete eigenvalues of the on-site and inter-site unstaggered configurations, respectively. These expressions show that the discrete eigenvalues are pure imaginary in the region of the dark soliton existence (0 <*

_{ui}*ω*<

*γ*) and does not give indication of the soliton instability. However, it is not a proof of the soliton stability because the eigenvalues with finite real part may appear with the introduction of the exact soliton solutions in calculations.

*N*complex eigenvalues of

**M**is found numerically for a different number of lattice elements

*N*and different values of the parameters

*γ*and

*ω*. Generally, for both on-site and inter-site configurations we can observe a subset with pure imaginary eigenvalues embedded in the continuous part of the EV spectrum (Fig. 1 ; the shaded regions), which is well described with the dispersion curve (7). The density of the eigenvalues inside the shaded region increases with the increase of

*N*, approaching the continuum when

*N*→ ∞. However, a discrete part of the EV spectrum where eigenvalues with a positive real part exist for some intervals in

*ω*indicates instability of the system.

*ω*as a bifurcation parameter the bifurcations of the Hopf type [12] for on-site dark modes are indicated. The bifurcation points coincide with the intersection of the discrete and continuous part of the EV spectrum (Fig. 1). The analytically calculated discrete branch for the on-site configuration (8) is in good agreement with the numerical results in the region near the left boundary of the existence region (small

*ω*) where approximation used for the analytical calculations is valid.

*ω*as a bifurcation parameter the merge of two purely real EVs after which only pure imaginary EVs remain in discrete spectrum is formally noted as a tangential (saddle-center) bifurcation [12]. However, the analytically calculated discrete branch for the inter-site configuration (8) near the left boundary of the existence region (small

*ω*) is embedded in the continuous part of the spectrum which is not consistent with the numerical results. In this case the bifurcation appears only near the upper boundary of the existence region (high

*ω*-small amplitudes) where approximate solution (8) is not valid. The correct approach in this region is a calculation of the discrete spectrum in the limit

*U*>> 1, which gives the expression consistent with the numerical results:

_{c}*ω*and indicates existence of the pure imaginary branch (

_{b}*ω*>

*ω*) associated with the existence of the neutrally stable dark inter-site breathers (Fig.1).

_{b}*Re*(Ω) ≈ 0 and

*Im*(Ω) ≠ 0 exist which indicates neutral stability and possibility for existence of the dark localized structures of the breather type. These regions for the on-site dark breather configurations are near the lower (

*ω*<

*ω*

_{b1}, large amplitudes) and upper (

*ω*>

*ω*

_{b2}, small amplitudes) boundaries of the existence region (Fig.1). The inter-site dark breather configurations can exist only in the region near the upper boundary of the existence region (

*ω*>

*ω*, small amplitudes). These results are confirmed numerically directly solving the model equation (1) as a Cauchy problem with initial conditions in a form of slightly perturbed dark solitons. The time-space evolution shows existence of the stable dark breathers in the predicted regions (Fig. 2) and instability in all other cases. This is consistent with the recently published experimental results [8

_{b}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]

## 4. Transversal motion

*P*and

*H*, which diverge for the localized dark configuration we use the complementary quantities

*P*and

_{c}*H*, where the Bloch-wave background (

_{c}*U*) is removed [1

_{cw}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]

10. T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis, and J. Cuevas, “Radiationless traveling waves in saturable nonlinear Schrödinger lattices,” Phys. Rev. Lett. **97**, 124101 (2006). [CrossRef] [PubMed]

*G*=

_{c}*H*-

_{c}*ωP*.

_{c}*G*between on-site and inter-site dark soliton configurations with the same norm

_{c}*P*has a sense of the potential barrier which arises from the discreteness of the system and can be taken as a measure of the well-known Peierls-Nabarro (PN) barrier. The

_{c}*G*(

_{c}*P*) for the unstaggered on-site and inter-site dark soliton configurations is displayed in Fig. 3. The remarkable feature of stability alternation between on-site and inter-site configurations observed for the bright solitons in lattices with SF nolinearity is absent [11

_{c}11. Lj. Hadžievski, 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]

*G*. Instead, as can be clearly seen, the energy difference Δ

_{c}*G*(

_{c}*P*) for

_{c}*P*<< 1 and Pc >> 1 vanishes. These regions coincide with the regions where our previous stability analysis predicted the existence of the dark breather configurations. These facts lead to the significant conclusion that the dark breathers in these regions are not affected by the PN barrier and can freely move across the lattice elements. As a consequence, we expect and confirm numerically that a small phase perturbation of any dark soliton configuration (on-site or inter-site) will cause creation of the moving dark breather (Fig. 3). This behavior indicates possibility for an easy experimental observation of the moving dark breathers with a similar experimental setup as in [8

_{c}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]

## 5. Conclusion

**74**, 065601(R) (2006). [CrossRef]

## Acknowledgments

## References and links

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 |

2. | A. A. Sukhorukov, Yu. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. |

3. | E. Trias, J. J. Mazo, and T. P. Orlando, “Dicrete breathers in nonlinear lattices: experimental detection in Joseph-son array,” Phys. Rev. Lett. |

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. |

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. | Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. |

7. | B. Sanchez-Rey and M. Johansson, “Exact numerical solutions for dark waves on the discrete nonlinear Schrödinger equation,” Phys. Rev. E |

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 |

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 Schrödinger lattices,” Phys. Rev. Lett. |

11. | Lj. Hadžievski, A. Maluckov, M. Stepić, and D. Kip, “Power controlled soliton stability and steering in lattices with saturable nonlinearity,” Phys. Rev. Lett. |

12. | Wiggins S. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.5330) Nonlinear optics : Photorefractive optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: March 20, 2007

Revised Manuscript: April 12, 2007

Manuscript Accepted: April 13, 2007

Published: April 25, 2007

**Citation**

Ljupco Hadzievski, Aleksandra Maluckov, and Milutin Stepic, "Dynamics of dark breathers in lattices with saturable nonlinearity," Opt. Express **15**, 5687-5692 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-9-5687

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### References

- 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]
- 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]
- 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]
- 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]
- Yu. S. Kivshar and B. Luther-Davies, "Dark optical solitons: physics and applications," Phys. Rep. 298,81-197 (1998). [CrossRef]
- 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]
- 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).
- 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]
- S. Wiggins, Global Bifurcations and Chaos: Analytical Methods (Springer-Verlag New York Inc., 1988).

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