## Stabilization of multipole-mode solitons in mixed linear-nonlinear lattices with a ���� symmetry |

Optics Express, Vol. 21, Issue 3, pp. 3917-3925 (2013)

http://dx.doi.org/10.1364/OE.21.003917

Acrobat PDF (1045 KB)

### Abstract

We report the evolution of higher-order nonlinear states in a focusing cubic medium, where both the linear refractive index and the nonlinearity are spatially modulated by a complex optical lattice exhibiting a parity-time (𝒫𝒯) symmetry. We reveal that introduction of out-of-phase nonlinearity modulation makes possible the stabilization of higher-order solitons with number of poles up to 7, which are highly unstable in linear 𝒫𝒯 lattices. Under appropriate conditions, multipole-mode solitons with out-of-phase components in the neighboring lattice sites are completely stable provided that their power or propagation constant exceeds a critical value. Thus, our findings suggest an effective way for the realization of stable multipole-mode solitons in periodic potentials with gain-loss components.

© 2013 OSA

## 1. Introduction

1. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature (London) **422**, 147–150 (2003). [CrossRef]

2. 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]

3. Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. **83**, 247–306 (2011). [CrossRef]

4. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity,” Opt. Lett. **33**, 1747–1749 (2008). [CrossRef] [PubMed]

6. L. Dong, H. Li, C. Huang, S. Zhong, and C. Li, “Higher-charged vortices in mixed linear-nonlinear circular arrays,” Phys. Rev. A **84**, 043830 (2011). [CrossRef]

7. Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Two-dimensional solitons in nonlinear lattices,” Opt. Lett. **34**, 770–772 (2009). [CrossRef] [PubMed]

8. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic
potentials,” Phys. Rev. Lett. **100**, 030402 (2008). [CrossRef] [PubMed]

16. J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, and T. Kottos, “*𝒫𝒯*-symmetric electronics,” J. Phys. A **45**, 444029 (2012). [CrossRef]

8. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic
potentials,” Phys. Rev. Lett. **100**, 030402 (2008). [CrossRef] [PubMed]

9. B. Bagchi, C. Quesne, and M. Znojil, “Generalized continuity equation and modified normalization
in PT-dymmetric quantum mechanics,” Mod. Phys. Lett. **16**, 2047–2057 (2001). [CrossRef]

17. F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A **83**, 041805 (2011). [CrossRef]

18. S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A **85**, 023822 (2012). [CrossRef]

20. K. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett. **35**, 2928–2930 (2010). [CrossRef] [PubMed]

23. S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A **85**, 043826 (2012). [CrossRef]

18. S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A **85**, 023822 (2012). [CrossRef]

24. Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials,” Phys. Rev. A **84**, 053855 (2011). [CrossRef]

17. F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A **83**, 041805 (2011). [CrossRef]

25. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in PT-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. **285**, 3320–3324 (2012). [CrossRef]

*in-phase*mixed linear-nonlinear 𝒫𝒯 lattices were studied in [26

26. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A **85**, 013831 (2012). [CrossRef]

17. F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A **83**, 041805 (2011). [CrossRef]

18. S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A **85**, 023822 (2012). [CrossRef]

23. S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A **85**, 043826 (2012). [CrossRef]

27. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express **12**, 2831–2837 (2004). [CrossRef] [PubMed]

28. Z. Xu, Y. V. Kartashov, and L. Torner, “Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media,” Opt. Lett. **30**, 3171–3173 (2005). [CrossRef] [PubMed]

3. Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. **83**, 247–306 (2011). [CrossRef]

## 2. Theoretical model

3. Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. **83**, 247–306 (2011). [CrossRef]

*q*is the complex field amplitude;

*x*and

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

*p*and

*σ*denote the depths of modulation of refractive index and nonlinearity, respectively; the refractive index profile obeys:

*R*(

*x*) =

*V*(

*x*) +

*iW*(

*x*) = cos

^{2}(Ω

*x*/2)+

*iχ*sin(Ω

*x*), where Ω is the frequency of lattice and

*χ*is the relative magnitude of gain-loss component.

*R*(

*x*) satisfies the 𝒫𝒯 symmetry, i.e.,

*V*(−

*x*) =

*V*(

*x*) and

*W*(−

*x*) = −

*W*(

*x*). The linear modulation of refractive index with a 𝒫𝒯 symmetry has been realized in experiment by Rüter

*et al.*[11

11. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. **6**, 192–195 (2010). [CrossRef]

**83**, 041805 (2011). [CrossRef]

*q*(

*x,z*) =

*u*(

*x*)exp(

*ibz*), which can be characterized by the complex soliton profile

*u*(

*x*) and the propagation constant

*b*. After substituting

*q*(

*x,z*) into Eq. (1), one obtains the following nonlinear differential equation: from which the stationary solution can be solved either by a Newton iterative method or by the squared operator iteration method [30

30. J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods
for solitary waves in general nonlinear wave equations,” Stud. Appl.
Math. **118**, 153–197 (2007). [CrossRef]

*b*, relative magnitude

*χ*, nonlinearity-modulation depth

*σ*, lattice depth

*p*, and modulation frequency Ω. Through the rest of this paper, we are going to use a specific configuration by setting

*p*≡ 4 and Ω ≡ 4. Note that the lattice period can obviously be rescaled. When the linear lattice is deep (for large

*p*), the nonlinearity modulation can be ignored and the system degenerates into the case reported in [8

8. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic
potentials,” Phys. Rev. Lett. **100**, 030402 (2008). [CrossRef] [PubMed]

**85**, 023822 (2012). [CrossRef]

*p*), the system is similar to the model proposed by Fatkhulla

*et al.*[17

**83**, 041805 (2011). [CrossRef]

**100**, 030402 (2008). [CrossRef] [PubMed]

9. B. Bagchi, C. Quesne, and M. Znojil, “Generalized continuity equation and modified normalization
in PT-dymmetric quantum mechanics,” Mod. Phys. Lett. **16**, 2047–2057 (2001). [CrossRef]

*r*and

*i*here and below stand for the real and imaginary parts.

*q*(

*x,z*) = {

*u*(

*x*) + [

*v*(

*x*) −

*w*(

*x*)]exp(

*λz*) + [

*v*(

*x*) +

*w*(

*x*)]

^{*}exp(

*λ*

^{*}

*z*)}exp(

*ibz*), where

*v*,

*w*≪ 1 are the infinitesimal perturbations, and superscript

^{*}represents the complex conjugation. Substituting the perturbed solution into Eq. (1) and linearizing it around

*u*yields an eigenvalue problem:

## 3. Multipole-mode solitons and their stabilities

*χ*= 0.5 (phase transition point), above which the eigenvalue spectra become complex, and linear waves amplify exponentially during propagation. Thus, any solitons would also be unstable to perturbations. In the following discussions, we will focus on the properties of solitons in the semi-infinite gap of 𝒫𝒯 lattice with

*χ*< 0.5. We first consider a special configuration with the relative magnitude of gain-loss component

*χ*= 0.2 and nonlinearity-modulation depth

*σ*= 0.2. The influence of

*χ*and

*σ*on the stability of solitons will be addressed later.

*χ*and

*σ*, power

*P*increases monotonically with the growth of propagation constant

*b*. There exists a lower cutoff of propagation constant(

*b*= 2.24), below which no soliton solutions can be found [Fig. 1(b)]. The amplitude ratio between the imaginary and real parts is a nonmonotonic function of propagation constant. It increases at small

_{co}*b*and reaches a maximum, afterwards, it decreases with the growth of

*b*.

*b*grows, the peak of soliton increases and soliton becomes more localized. Transverse power-flow density due to the nontrivial phase structure of complex solitons may arise in complex potentials. Following Ref. [8

**100**, 030402 (2008). [CrossRef] [PubMed]

*b*, the power-flow density is positive in the lattice sites and negative in the space between them. It becomes completely positive if

*b*exceeds a critical value. Interestingly, a local minimum appears in the central lattice channel at large

*b*[Fig. 1(e)]. This property has not been revealed in other systems.

26. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A **85**, 013831 (2012). [CrossRef]

*q*(

*x,z*= 0) =

*u*(

*x*)[1 +

*ρ*(

*x*)], where

*ρ*(

*x*) is a random function with Gaussian distribution. An example of stable propagation is illustrated in Fig. 1(f).

*b*[Fig. 2(a)]. In comparison with Fig. 1(b), the amplitude ratio between the imaginary and real parts increases at small and moderate

*b*and decreases rapidly when the nonlinearity is strong. It implies that the strong modulation of nonlinearity can suppress the imaginary part of solitons. In other words, the nonlinearity modulation effectively counteracts the role played by imaginary lattice, since the emergence of imaginary part of soliton is solely due to the existence of imaginary lattice. Two representative profiles of dipole solitons are shown in Figs. 2(b) and 2(c). The symmetric center of dipoles is shifted to

*x*

_{0}=

*π*/4 and the symmetries of the real and imaginary parts exhibit as:

*u*[−(

_{r}*x*+

*π*/4)] = −

*u*(

_{r}*x*+

*π*/4) and

*u*[−(

_{i}*x*+

*π*/4)] =

*u*(

_{i}*x*+

*π*/4). The two peaks of the modulus of dipole soliton are of the same height, due to the symmetries of the real and imaginary parts [Fig. 2(c)]. With the decrease of

*b*, dipole soliton becomes broader and extends to several lattice sites.

*λ*and find that all imaginary parts of

*λ*equal zero when

*b*is smaller than the critical value

*b*. It confirms the prediction according to the V-K criterion. Oscillatory instability corresponding to

_{in}*λ*with nonzero real and imaginary parts may arise if

*b*>

*b*[Fig. 2(d)]. By comparing with fundamental solitons, the instability of dipoles may be attributed to the repulsive interaction between the two parts beside the symmetrical center. Dipole solitons are completely stable when the propagation constant exceeds a critical value. In other words, dipoles are stable at higher power, which is again in contrast to the stability of fundamental solitons in in-phase mixed linear-nonlinear 𝒫𝒯 lattices [26

_{in}26. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A **85**, 013831 (2012). [CrossRef]

*b*= 3.44 and 5.70 are displayed in Figs. 2(e) and 2(f), respectively. After a short distance, the unstable dipole soliton undergoes an asymmetric distortion, similar to the evolution of unstable solitons in linear 𝒫𝒯 lattices [18

**85**, 023822 (2012). [CrossRef]

21. H. Wang and J. Wang, “Defect solitons in parity-time periodic potentials,” Opt. Express **19**, 4030–4035 (2011). [CrossRef] [PubMed]

27. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express **12**, 2831–2837 (2004). [CrossRef] [PubMed]

*b*, the power of triple soliton approximates to three times of the power of fundamental soliton. From Fig. 3(a), one can also infer that the amplitude of imaginary part increases with the growth of the pole number of solitons.

*b*and increase simultaneously with

*b*. The physical reason is that the out-of-phase nonlinearity modulation partly weakens the modulation of linear lattice, and thus the poles of high-power solitons experience a saturation effect, just similar to the profiles of solitons in competing or saturable nonlinear media.

*b*[Fig. 3(d)]. Similar to dipole solitons, triples can propagate stably provided that their power exceeds a critical value. We should note that the instability region expands slowly with the growth of the number of soliton poles. Direct propagation simulations on triple solitons verify the prediction of linear stability analysis, see e.g., Figs. 3(e) and 3(f).

*χ*and the depth of nonlinearity modulation

*σ*on the stability of multipole-mode solitons, we investigate the evolution of solitons in lattices with relatively large

*χ*and

*σ*, taking dipole and triple solitons as examples. At fixed

*b*and

*σ*, while the real part of dipole soliton remains unchanged, the peak of imaginary part increases with

*χ*[Fig. 4(a)]. Yet, dipole solitons are still completely stable at higher power [Fig. 4(c)]. Thus, the growth of

*χ*leads to the shrink of stability region. At fixed

*χ*and

*b*, the real part of triple soliton decreases and the imaginary part increases with the growth of

*σ*[Fig. 4(b)]. If

*σ*is large, an instability region of triple solitons appears at higher power [Fig. 4(d)], since too strong nonlinearity modulation destroys the stability of high-power solitons. The growth of the depth of nonlinearity modulation

*σ*can be seen as an effective decrease of the depth of linear lattice

*p*. Thus, one can infer from Fig. 4(d) that, for fixed

*σ*, the decrease of

*p*will lead to the shrinkage of the stability domain of higher-order solitons.

27. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express **12**, 2831–2837 (2004). [CrossRef] [PubMed]

32. M. Trippenbach, E. Infeld, J. Gocalek, M. Matuszewski, M. Oberthaler, and B. A. Malomed, “Spontaneous symmetry breaking of gap solitons and phase transitions in double-well traps,” Phys. Rev. A **78**, 013603 (2008). [CrossRef]

*u*(

*x*) =

*A*exp[−(

*x/d*)

^{2}]exp(

*iθx*), where

*A*and

*d*are the amplitude and width of the input beam, respectively;

*θ*characterizes the tilt angle or the phase. Figure 6(a) displays an example of the excitation of stable fundamental soliton by a Gaussian beam. As expected, the excited stable soliton immediately transforms into a sech-type soliton if the 𝒫𝒯 lattice is removed at a certain propagation distance. Similar phenomenon occurs for the excited dipole soliton by a broader input Gaussian beam [Fig. 6(b)]. The asymmetric splitting of the excited dipole soliton in the absence of optical lattice manifest its initial inner phase structure.

33. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Packing, unpacking, and steering of multicolor solitons in opticallattices,” Opt. Lett. **29**, 1399–1401 (2004). [CrossRef] [PubMed]

## 4. Summary

## Acknowledgments

## References and links

1. | J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature (London) |

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. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity,” Opt. Lett. |

5. | F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Light bullets in Bessel optical lattices with spatially modulated nonlinearity,” Opt. Express |

6. | L. Dong, H. Li, C. Huang, S. Zhong, and C. Li, “Higher-charged vortices in mixed linear-nonlinear circular arrays,” Phys. Rev. A |

7. | Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Two-dimensional solitons in nonlinear lattices,” Opt. Lett. |

8. | Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic
potentials,” Phys. Rev. Lett. |

9. | B. Bagchi, C. Quesne, and M. Znojil, “Generalized continuity equation and modified normalization
in PT-dymmetric quantum mechanics,” Mod. Phys. Lett. |

10. | A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical
potentials,” Phys. Rev. Lett. |

11. | C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. |

12. | L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A |

13. | Y. He and D. Mihalache, “Spatial solitons in parity-time-symmetric mixed linear-nonlinear optical lattices: Recent theoretical results,” Rom. Rep. Phys. |

14. | Y. He and D. Mihalache, “Lattice solitons in optical media described by the complex Ginzburg-Landau model with |

15. | J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active |

16. | J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, and T. Kottos, “ |

17. | F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A |

18. | S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A |

19. | C. Li, H. Liu, and L. Dong, “Multi-stable solitons in PT-symmetric optical lattices,” Opt. Express |

20. | K. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett. |

21. | H. Wang and J. Wang, “Defect solitons in parity-time periodic potentials,” Opt. Express |

22. | Z. Lu and Z.-M. Zhang, “Defect solitons in parity-time symmetric superlattices,” Opt. Express |

23. | S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A |

24. | Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials,” Phys. Rev. A |

25. | Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in PT-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. |

26. | Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A |

27. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express |

28. | Z. Xu, Y. V. Kartashov, and L. Torner, “Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media,” Opt. Lett. |

29. | N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” ArXiv e-prints (2013). |

30. | J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods
for solitary waves in general nonlinear wave equations,” Stud. Appl.
Math. |

31. | N. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. |

32. | M. Trippenbach, E. Infeld, J. Gocalek, M. Matuszewski, M. Oberthaler, and B. A. Malomed, “Spontaneous symmetry breaking of gap solitons and phase transitions in double-well traps,” Phys. Rev. A |

33. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Packing, unpacking, and steering of multicolor solitons in opticallattices,” Opt. Lett. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.6135) Nonlinear optics : Spatial solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: January 8, 2013

Revised Manuscript: February 3, 2013

Manuscript Accepted: February 3, 2013

Published: February 8, 2013

**Citation**

Changming Huang, Chunyan Li, and Liangwei Dong, "Stabilization of multipole-mode solitons in mixed linear-nonlinear lattices with a 𝒫𝒯 symmetry," Opt. Express **21**, 3917-3925 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-3917

Sort: Year | Journal | Reset

### References

- J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature (London)422, 147–150 (2003). [CrossRef]
- 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, 247–306 (2011). [CrossRef]
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity,” Opt. Lett.33, 1747–1749 (2008). [CrossRef] [PubMed]
- F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Light bullets in Bessel optical lattices with spatially modulated nonlinearity,” Opt. Express17, 11328–11334 (2009). [CrossRef] [PubMed]
- L. Dong, H. Li, C. Huang, S. Zhong, and C. Li, “Higher-charged vortices in mixed linear-nonlinear circular arrays,” Phys. Rev. A84, 043830 (2011). [CrossRef]
- Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Two-dimensional solitons in nonlinear lattices,” Opt. Lett.34, 770–772 (2009). [CrossRef] [PubMed]
- Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett.100, 030402 (2008). [CrossRef] [PubMed]
- B. Bagchi, C. Quesne, and M. Znojil, “Generalized continuity equation and modified normalization in PT-dymmetric quantum mechanics,” Mod. Phys. Lett.16, 2047–2057 (2001). [CrossRef]
- A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett.103, 093902 (2009). [CrossRef] [PubMed]
- C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys.6, 192–195 (2010). [CrossRef]
- L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A13, 46–54 (2012).
- Y. He and D. Mihalache, “Spatial solitons in parity-time-symmetric mixed linear-nonlinear optical lattices: Recent theoretical results,” Rom. Rep. Phys.64, 1243–1258 (2012).
- Y. He and D. Mihalache, “Lattice solitons in optical media described by the complex Ginzburg-Landau model with 𝒫𝒯 -symmetric periodic potentials,” Phys. Rev. A87, 013812 (2013). [CrossRef]
- J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A84, 040101 (2011). [CrossRef]
- J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, and T. Kottos, “𝒫𝒯-symmetric electronics,” J. Phys. A45, 444029 (2012). [CrossRef]
- F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A83, 041805 (2011). [CrossRef]
- S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A85, 023822 (2012). [CrossRef]
- C. Li, H. Liu, and L. Dong, “Multi-stable solitons in PT-symmetric optical lattices,” Opt. Express20, 16823–16831 (2012).
- K. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett.35, 2928–2930 (2010). [CrossRef] [PubMed]
- H. Wang and J. Wang, “Defect solitons in parity-time periodic potentials,” Opt. Express19, 4030–4035 (2011). [CrossRef] [PubMed]
- Z. Lu and Z.-M. Zhang, “Defect solitons in parity-time symmetric superlattices,” Opt. Express19, 11457–11462 (2011). [CrossRef] [PubMed]
- S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A85, 043826 (2012). [CrossRef]
- Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials,” Phys. Rev. A84, 053855 (2011). [CrossRef]
- Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in PT-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun.285, 3320–3324 (2012). [CrossRef]
- Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A85, 013831 (2012). [CrossRef]
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express12, 2831–2837 (2004). [CrossRef] [PubMed]
- Z. Xu, Y. V. Kartashov, and L. Torner, “Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media,” Opt. Lett.30, 3171–3173 (2005). [CrossRef] [PubMed]
- N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” ArXiv e-prints (2013).
- J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math.118, 153–197 (2007). [CrossRef]
- N. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Izv. Vyssh. Uchebn. Zaved., Radiofiz.16, 783–789 (1973).
- M. Trippenbach, E. Infeld, J. Gocalek, M. Matuszewski, M. Oberthaler, and B. A. Malomed, “Spontaneous symmetry breaking of gap solitons and phase transitions in double-well traps,” Phys. Rev. A78, 013603 (2008). [CrossRef]
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Packing, unpacking, and steering of multicolor solitons in opticallattices,” Opt. Lett.29, 1399–1401 (2004). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.