## Dissipative soliton acceleration in nonlinear optical lattices |

Optics Express, Vol. 20, Issue 16, pp. 18165-18172 (2012)

http://dx.doi.org/10.1364/OE.20.018165

Acrobat PDF (5783 KB)

### Abstract

An effective mechanism for dissipative soliton acceleration in nonlinear optical lattices under the presence of linear gain and nonlinear loss is presented. The key idea for soliton acceleration consists of the dynamical reduction of the amplitude of the effective potential experienced by the soliton so that its kinetic energy eventually increases. This is possible through the dependence of the effective potential amplitude on the soliton mass, which can be varied due to the presence of gain and loss mechanisms. In contrast to the case where either the linear or the nonlinear refractive index is spatially modulated, we show that when both indices are modulated with the same period we can have soliton acceleration and mass increasing as well as stable soliton propagation with constant non-oscillating velocity. The acceleration mechanism is shown to be very robust for a wide range of configurations.

© 2012 OSA

## 1. Introduction

1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature **424**, 817–823 (2003). [CrossRef] [PubMed]

4. I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. (in press), doi: [CrossRef] .

5. U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett. **23**, 1701–1703 (1998). [CrossRef]

6. T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. **83**, 4752–4755 (1999). [CrossRef]

7. K. G. Makris, D. N. Christodoulides, O. Peleg, M. Segev, and D. Kip, “Optical transitions and Rabi oscillations in waveguide arrays,” Opt. Express **16**, 10309–10314 (2008). [CrossRef] [PubMed]

8. K. Shandarova, C. E. Ruter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. **102**, 123905 (2009). [CrossRef] [PubMed]

9. Y. V. Kartashov and V. A. Vysloukh, “Anderson localization of solitons in optical lattices with random frequency modulation,” Phys. Rev. E **72**, 026606 (2005). [CrossRef]

10. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature **446**, 52–55 (2007). [CrossRef] [PubMed]

11. S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. **3**, 243–261 (2009). [CrossRef]

12. F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, “Josephson junction arrays with Bose-Einstein condensates,” Science **293**, 843–846 (2001). [CrossRef] [PubMed]

15. F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, “Nonlinear excitations in arrays of Bose–Einstein condensates,” Phys. Rev. A **64**, 436061–4360610 (2001). [CrossRef]

32. N. Akhmediev and A. Ankiewicz, eds., *Dissipative Solitons, Lect. Notes Phys.* (Springer, 2005), Vol. 661. [CrossRef]

34. F. Kh. Abdullaev, V. V. Konotop, M. Salerno, and A. V. Yulin, “Dissipative periodic waves, solitons and breathers of the nonlinear Schrödinger equation with complex potentials,” Phys. Rev. E **82**, 056606 (2010). [CrossRef]

## 2. Model and results

*z*and

*x*are the normalized longitudinal and transverse coordinates, respectively;

*n*(

*x*) describes the normalized transverse modulation of the linear and nonlinear refractive index which, for simplicity, is considered to vary as

*n*(

*x*) = sin(

*Kx*);

*α*(

_{i}*i*= 1, 2) are the modulation amplitudes of the linear and the nonlinear refractive index,

*σ*(

_{i}*i*= 1, 2) are the normalized coefficients of linear gain (

*σ*

_{1}< 0), and two- photon absorption (

*σ*

_{2}> 0), respectively. In order to study soliton dynamics, we utilize a perturbative approach [23

23. D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London, Ser. A **361**, 413–446 (1978). [CrossRef]

24. Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. **61**, 763–915 (1989). [CrossRef]

*u*=

*η*sech[

*η*(

*x*−

*x*

_{0})] exp[

*i*(

*vx*/2 + 2

*ϕ*)], with

*dϕ*/

*dz*=

*η*

^{2}/2 −

*v*

^{2}/8. The soliton is treated as an effective particle moving under the effect of the perturbation

*R*(

*x*,

*u*;

*α*

_{1},

*α*

_{2},

*σ*

_{1},

*σ*

_{2}) =

*n*(

*x*)(

*α*

_{1}+

*α*

_{2}|

*u*|

^{2})

*u*−

*i*(

*σ*

_{1}+

*σ*

_{2}|

*u*|

^{2})

*u*corresponding to the rhs of Eq. (1). The effective particle mass

*m*= ∫ |

*u*|

^{2}

*dx*= 2

*η*, momentum

*p*=

*mv*and position

*x*

_{0}, vary according to the equations with In the absence of dissipation (

*σ*= 0,

_{i}*i*= 1, 2) the effective particle motion, given by Eqs. (2)–(4), describes two distinct dynamical behaviors corresponding to trapped solitons oscillating around a potential minimum and traveling solitons moving transversely across the potential, depending on the initial energy of the effective particle. From Eq. (5) it is obvious that, in the same periodic medium, solitons of different mass move in potentials having different amplitude. The dependence of the potential strength on the soliton mass reflects the relation between the spatial width of soliton (∼

*η*

^{−1}) and the spatial period of the lattice (∼

*K*

^{−1}) [35, 36

36. H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A **81**, 013624 (2010). [CrossRef]

*m*launched at a local maximum of the effective potential with zero initial velocity. Firstly we investigate the case where only the linear or the nonlinear refractive index is modulated (

_{in}*α*

_{2}= 0 or

*α*

_{1}= 0). In the absence of linear gain (

*σ*

_{1}= 0,

*σ*

_{2}> 0), the soliton will evolve with continuously decreasing mass in a potential of continuously decreasing amplitude (Fig. 1), and as a result will have a continuously increasing velocity. In that case we can only achieve acceleration of solitons having an increasing width and decreasing amplitude during propagation; such a mechanism would be useful only for a restricted propagation distance so that a specific degree of localization is retained. Under the presence of linear gain (

*σ*

_{1}≠ 0), there is fixed mass value

*m*′

_{0}, resulting in stable soliton propagation. A soliton with initial mass

*m*>

_{in}*m*′

_{0}will evolve with decreasing mass until its mass reaches the value

*m*′

_{0}and then propagate with constant shape. During the transient phase, the amplitude of effective potential has been decreased and soliton has been accelerated. Subsequently, the soliton propagates with a velocity which is highly oscillating around a nonzero mean value, as shown in Fig. 2(a).

*α*

_{1}/

*α*

_{2}>

*K*

^{2}, allows for two options for soliton acceleration. As it can be seen from Fig. 1, a soliton launched with an initial mass

*m*>

_{in}*m*

_{0}can be accelerated by reducing its mass to

*m*

_{0}if the values of the gain and loss coefficients are chosen so that

*m*′

_{0}=

*m*

_{0}. In that case, after a transient stage, the soliton eventually travels with a constant mass and as if there is no spatial modulation of the medium. In contrast to the aforementioned cases where only one of the linear or the nonlinear refractive indexes were spatially modulated, in this case soliton eventually travels with a constant, non-oscillating velocity as shown in Fig. 2(b). On the other hand, soliton acceleration with simultaneous mass increasing is possible for solitons launched with initial mass

*m*<

_{in}*m*

_{0}as it can be seen from the form dependence of the effective potential amplitude on the soliton mass in Fig. 1. In contrast to all previous cases, this is the only possibility for beam acceleration to a constant velocity [Fig. 2(b)] not only without degradation but also with enhancement of the beam localization, which is an important feature of the acceleration mechanism with respect to applications.

*v*as a function of the initial soliton mass

_{out}*m*is shown in Fig. 3(a) for spatial modulations having different periods, as obtained from the effective particle Eqs. (2)–(4). Note that the initial soliton position located at the maxima of the respective effective potential differs by 2

_{in}*π*/

*K*for the cases

*m*>

_{in}*m*

_{0}and

*m*<

_{in}*m*

_{0}, due to sign reversal of the potential. The maxima of the potential are saddle points for soliton dynamics in the absence of gain and loss. The mass dependence on

*z*, due to the presence of gain and loss, results in an additional degree of freedom of the effective particle system, which becomes nonintegrable. Complex dependence of the soliton evolution on the initial conditions occurs in the region around a saddle point corresponding to a potential maximum, so that a stationary soliton can be accelerated to either positive or negative velocity. However, this region, corresponding to a chaotic separatrix layer is very narrow. Therefore, we can always choose an initial soliton position which is close enough to a potential maximum corresponding to high initial particle energy but slightly displaced from it in order to determine the sign of the final soliton velocity. Such a displacement from the potential maximum of 1% with respect to the lattice period has been used in all results. As it can be seen in Fig. 3(a), the modulation period

*K*determines the value mass

*m*

_{0}for which the soliton ”experiences” no effective potential as well as the maximum output velocity for given modulation amplitudes (

*α*,

_{i}*i*= 1, 2). On the other hand, the output velocity scales with the square root of

*α*so that an increased modulation amplitude results in increased output velocities. In Fig. 3(a) and (b), the two cases of soliton acceleration with decreasing and increasing mass, respectively, are shown, as given by the effective particle Eqs. (2)–(4) and numerical simulations of the original NLS Eq. (1). The results are shown to be in remarkable agreement with respect to the initial stage of the acceleration process as well as to the final velocities given by the slopes of the respective lines.

_{i}*m*>

_{in}*m*

_{0}and

*m*<

_{in}*m*

_{0}. In comparison to the case of weaker lattices where the soliton mass attains a fixed value after a transient stage (as predicted by the effective particle approach), the soliton mass undergoes periodic oscillations. However, the averaged value of the soliton mass is smaller or larger than

*m*for the case where

_{in}*m*>

_{in}*m*

_{0}or

*m*<

_{in}*m*

_{0}in accordance to results of the effective particle approach.

## 3. Discussion and conclusions

## Acknowledgments

## References and links

1. | D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature |

2. | F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. |

3. | Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. |

4. | I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. (in press), doi: [CrossRef] . |

5. | U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett. |

6. | T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. |

7. | K. G. Makris, D. N. Christodoulides, O. Peleg, M. Segev, and D. Kip, “Optical transitions and Rabi oscillations in waveguide arrays,” Opt. Express |

8. | K. Shandarova, C. E. Ruter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. |

9. | Y. V. Kartashov and V. A. Vysloukh, “Anderson localization of solitons in optical lattices with random frequency modulation,” Phys. Rev. E |

10. | T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature |

11. | S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. |

12. | F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, “Josephson junction arrays with Bose-Einstein condensates,” Science |

13. | S. Burger, F. S. Cataliotti, C. Fort, F. Minardi, M. Inguscio, M. L. Chiofalo, and M. P. Tosi, “Superfluid and dissipative dynamics of a Bose–Einstein condensate in a periodic optical potential,” Phys. Rev. Lett. |

14. | A. Trombettoni and A. Smerzi, “Discrete solitons and breathers with dilute Bose-Einstein condensates,” Phys. Rev. Lett. |

15. | F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, “Nonlinear excitations in arrays of Bose–Einstein condensates,” Phys. Rev. A |

16. | J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A |

17. | J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. |

18. | G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. |

19. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. |

20. | A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics |

21. | I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. |

22. | R. El-Ganainy, K. G. Makris, M. A. Miri, D. N. Christodoulides, and Z. Chen, “Discrete beam acceleration in uniform waveguide arrays,” Phys. Rev. A |

23. | D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London, Ser. A |

24. | Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. |

25. | V. A. Brazhnyi, V. V. Konotop, and V. Kuzmiak, “Dynamics of matter solitons in weakly modulated optical lattices,” Phys. Rev. A |

26. | R. Scharf and A. R. Bishop, “Length-scale competition for the one-dimensional nonlinear Schrödinger equation with spatially periodic potentials,” Phys. Rev. E |

27. | D. Polleti, T. J. Alexander, E. A. Ostrovskaya, B. Li, and Y. S. Kivshar, “Dynamics of matter-wave solitons in a ratchet potential,” Phys. Rev. Lett. |

28. | G. Assanto, L. A. Cisneros, A. A. Minzoni, B. D. Skuse, N. F. Smyth, and A. L Worthy, “Soliton steering by longitudinal modulation of the nonlinearity in waveguide arrays,” Phys. Rev. Lett. |

29. | M. Rietmann, R. Carretero-Gonzalez, and R. Chacon, “Controlling directed transport of matter wave solitons using the ratchet effect,” Phys. Rev. A |

30. | P. Papagiannis, Y. Kominis, and K. Hizanidis, “Power- and momentum-dependent soliton dynamics in lattices with longitudinal modulation,” Phys. Rev. A |

31. | S. Flach, O. Yevtushenko, and Y. Zolotaryuk, “Directed current due to broken time-space symmetry,” Phys. Rev. Lett. |

32. | N. Akhmediev and A. Ankiewicz, eds., |

33. | F. Kh. Abdullaev, A. Gammal, H. L. F. da Luz, and L. Tomio, “Dissipative dynamics of matter-wave solitons in a nonlinear optical lattice,” Phys. Rev. A |

34. | F. Kh. Abdullaev, V. V. Konotop, M. Salerno, and A. V. Yulin, “Dissipative periodic waves, solitons and breathers of the nonlinear Schrödinger equation with complex potentials,” Phys. Rev. E |

35. | H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. A |

36. | H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A |

37. | B. A. Malomed, “Evolution of nonsoliton and ’quasi-classical’ wavetrains in nonlinear Schrödinger and Korteweg–De Vries equations with dissipative perturbations,” Physica D |

**OCIS Codes**

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

(190.4720) Nonlinear optics : Optical nonlinearities of condensed matter

(190.6135) Nonlinear optics : Spatial solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: June 13, 2012

Revised Manuscript: July 16, 2012

Manuscript Accepted: July 16, 2012

Published: July 23, 2012

**Citation**

Yannis Kominis, Panagiotis Papagiannis, and Sotiris Droulias, "Dissipative soliton acceleration in nonlinear optical lattices," Opt. Express **20**, 18165-18172 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-18165

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### References

- D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature424, 817–823 (2003). [CrossRef] [PubMed]
- F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1–126 (2008). [CrossRef]
- Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys.83, 247–305 (2011). [CrossRef]
- I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. (in press), doi: . [CrossRef]
- U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett.23, 1701–1703 (1998). [CrossRef]
- T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett.83, 4752–4755 (1999). [CrossRef]
- K. G. Makris, D. N. Christodoulides, O. Peleg, M. Segev, and D. Kip, “Optical transitions and Rabi oscillations in waveguide arrays,” Opt. Express16, 10309–10314 (2008). [CrossRef] [PubMed]
- K. Shandarova, C. E. Ruter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett.102, 123905 (2009). [CrossRef] [PubMed]
- Y. V. Kartashov and V. A. Vysloukh, “Anderson localization of solitons in optical lattices with random frequency modulation,” Phys. Rev. E72, 026606 (2005). [CrossRef]
- T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446, 52–55 (2007). [CrossRef] [PubMed]
- S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev.3, 243–261 (2009). [CrossRef]
- F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, “Josephson junction arrays with Bose-Einstein condensates,” Science293, 843–846 (2001). [CrossRef] [PubMed]
- S. Burger, F. S. Cataliotti, C. Fort, F. Minardi, M. Inguscio, M. L. Chiofalo, and M. P. Tosi, “Superfluid and dissipative dynamics of a Bose–Einstein condensate in a periodic optical potential,” Phys. Rev. Lett.86, 4447–4450 (2001). [CrossRef] [PubMed]
- A. Trombettoni and A. Smerzi, “Discrete solitons and breathers with dilute Bose-Einstein condensates,” Phys. Rev. Lett.86, 2353–2356 (2001). [CrossRef] [PubMed]
- F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, “Nonlinear excitations in arrays of Bose–Einstein condensates,” Phys. Rev. A64, 436061–4360610 (2001). [CrossRef]
- J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A4, 651–654 (1987). [CrossRef]
- J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987). [CrossRef] [PubMed]
- G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32, 979–981 (2007). [CrossRef] [PubMed]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99, 213901 (2007). [CrossRef]
- A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics4, 103–106 (2010). [CrossRef]
- I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett.106, 213903 (2011). [CrossRef] [PubMed]
- R. El-Ganainy, K. G. Makris, M. A. Miri, D. N. Christodoulides, and Z. Chen, “Discrete beam acceleration in uniform waveguide arrays,” Phys. Rev. A84, 023842 (2011). [CrossRef]
- D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London, Ser. A361, 413–446 (1978). [CrossRef]
- Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys.61, 763–915 (1989). [CrossRef]
- V. A. Brazhnyi, V. V. Konotop, and V. Kuzmiak, “Dynamics of matter solitons in weakly modulated optical lattices,” Phys. Rev. A70, 043604 (2004). [CrossRef]
- R. Scharf and A. R. Bishop, “Length-scale competition for the one-dimensional nonlinear Schrödinger equation with spatially periodic potentials,” Phys. Rev. E47, 1375–1383 (1993). [CrossRef]
- D. Polleti, T. J. Alexander, E. A. Ostrovskaya, B. Li, and Y. S. Kivshar, “Dynamics of matter-wave solitons in a ratchet potential,” Phys. Rev. Lett.101, 150403 (2008). [CrossRef]
- G. Assanto, L. A. Cisneros, A. A. Minzoni, B. D. Skuse, N. F. Smyth, and A. L Worthy, “Soliton steering by longitudinal modulation of the nonlinearity in waveguide arrays,” Phys. Rev. Lett.104, 053903 (2010). [CrossRef] [PubMed]
- M. Rietmann, R. Carretero-Gonzalez, and R. Chacon, “Controlling directed transport of matter wave solitons using the ratchet effect,” Phys. Rev. A83, 053617 (2011). [CrossRef]
- P. Papagiannis, Y. Kominis, and K. Hizanidis, “Power- and momentum-dependent soliton dynamics in lattices with longitudinal modulation,” Phys. Rev. A84, 013820 (2011). [CrossRef]
- S. Flach, O. Yevtushenko, and Y. Zolotaryuk, “Directed current due to broken time-space symmetry,” Phys. Rev. Lett.84, 2358–2361 (2000). [CrossRef] [PubMed]
- N. Akhmediev and A. Ankiewicz, eds., Dissipative Solitons, Lect. Notes Phys. (Springer, 2005), Vol. 661. [CrossRef]
- F. Kh. Abdullaev, A. Gammal, H. L. F. da Luz, and L. Tomio, “Dissipative dynamics of matter-wave solitons in a nonlinear optical lattice,” Phys. Rev. A76, 043611 (2007). [CrossRef]
- F. Kh. Abdullaev, V. V. Konotop, M. Salerno, and A. V. Yulin, “Dissipative periodic waves, solitons and breathers of the nonlinear Schrödinger equation with complex potentials,” Phys. Rev. E82, 056606 (2010). [CrossRef]
- H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. A72, 046610 (2005).
- H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A81, 013624 (2010). [CrossRef]
- B. A. Malomed, “Evolution of nonsoliton and ’quasi-classical’ wavetrains in nonlinear Schrödinger and Korteweg–De Vries equations with dissipative perturbations,” Physica D29, 155–172 (1987). [CrossRef]

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