## Power dependent soliton location and stability in complex photonic structures

Optics Express, Vol. 16, Issue 16, pp. 12124-12138 (2008)

http://dx.doi.org/10.1364/OE.16.012124

Acrobat PDF (1499 KB)

### Abstract

The presence of spatial inhomogeneity in a nonlinear medium results in the breaking of the translational invariance of the underlying propagation equation. As a result traveling wave soliton solutions do not exist in general for such systems, while stationary solitons are located in fixed positions with respect to the inhomogeneous spatial structure. In simple photonic structures with monochromatic modulation of the linear refractive index, soliton position and stability do not depend on the characteristics of the soliton such as power, width and propagation constant. In this work, we show that for more complex photonic structures where either one of the refractive indices (linear or nonlinear) is modulated by more than one wavenumbers, or both of them are modulated, soliton position and stability depends strongly on its characteristics. The latter results in additional functionality related to soliton discrimination in such structures. The respective power (or width / propagation constant) dependent bifurcations are studied in terms of a Melnikov-type theory. The latter is used for the determination of the specific positions, with respect to the spatial structure, where solitons can be located. A wide variety of cases are studied, including solitons in periodic and quasiperiodic lattices where both the linear and the nonlinear refractive index are spatially modulated. The investigation of a wide variety of inhomogeneities provides physical insight for the design of a spatial structure and the control of the position and stability of a localized wave.

© 2008 Optical Society of America

## 1. Introduction

6. N.K. Efremidis, S. Sears, D.N. Christodoulides, J.W. Fleischer, and M. Segev, ”Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E **66**, 046602 (2002). [CrossRef]

7. J.W. Fleischer, T. Carmon, M. Segev, N.K. Efremidis, and D.N. Christodoulides, ”Observation of Discrete Solitons in Optically Induced Real Time Waveguide Arrays,” Phys. Rev. Lett. **90**, 023902 (2003). [CrossRef] [PubMed]

8. D. Neshev, E. Ostrovskaya, Y.S. Kivshar, and W. Krolikowski, ”Spatial solitons in optically induced gratings,” Opt. Lett. **28**, 710–712 (2003). [CrossRef] [PubMed]

9. D. Neshev, A.A. Sukhorukov, Y.S. Kivshar, and W. Krolikowski, ”Observation of transverse instabilities in optically induced lattices,” Opt. Lett. **29**, 259–261 (2004). [CrossRef] [PubMed]

10. Y. Kominis and K. Hizanidis, ”Continuous-wave-controlled steering of spatial solitons,” J. Opt. Soc. Am. B **21**, 562–567 (2004). [CrossRef]

11. Y. Kominis and K. Hizanidis, ”Optimal multidimensional solitary wave steering,” J. Opt. Soc. Am. B **22**, 1360–1365 (2005). [CrossRef]

12. Z. Chen, H. Martin, E. Eugenieva, J. Xu, and J. Yang, ”Formation of discrete solitons in light-induced photonic lattices,” Opt. Express **13**, 1816–1826 (2005). [CrossRef] [PubMed]

13. C.R. Rosberg, D.N. Neshev, A.A. Sukhorukov, Y.S. Kivshar, and W. Krolikowski, ”Tunable positive and negative refraction in optically induced photonic lattices,” Opt. Lett. **30**, 2293–2295 (2005). [CrossRef] [PubMed]

14. T. Song, S.M. Liu, R. Guo, Z.H. Liu, N. Zhu, and Y.M. Gao, ”Observation of composite gap solitons in optically induced nonlinear lattices in LiNbO3:Fe crystal,” Opt. Express , **14**, 1924–1932 (2006). [CrossRef] [PubMed]

15. I. Tsopelas, Y. Kominis, and K. Hizanidis, ”Soliton dynamics and interactions in dynamically photoinduced lattices,” Phys. Rev. E **74**, 036613 (2006). [CrossRef]

16. I. Tsopelas, Y. Kominis, and K. Hizanidis, ”Dark soliton dynamics and interactions in continuous-wave-induced lattices,” Phys. Rev. E **76**, 046609 (2007). [CrossRef]

18. D.N. Christodoulides and R.I. Joseph, ”Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. **13**, 794–796 (1988). [CrossRef] [PubMed]

19. D.C. Hutchings, ”Theory of Ultrafast Nonlinear Refraction in Semiconductor Superlattices,” IEEE J. Sel. Top. Quantum Electron. **10**, 1124–1132 (2004). [CrossRef]

20. L. Berge, V.K. Mezentsev, J.J. Rasmussen, P.L. Christiansen, and Y.B. Gaididei, ”Self-guiding light in layered nonlinear media,” Opt. Lett. **25**, 1037–1039 (2000). [CrossRef]

21. D.E. Pelinovsky, P.G. Kevrekidis, and D.J. Frantzeskakis, ”Averaging for Solitons with Nonlinearity Management,” Phys. Rev. Lett. **91**, 240201 (2003). [CrossRef] [PubMed]

22. H. Sakaguchi and B.A. Malomed, ”Resonant nonlinearity management for nonlinear Schrodinger solitons,” Phys. Rev. E **70**, 066613 (2004). [CrossRef]

23. Y.V. Kartashov and V.A. Vysloukh, ”Resonant phenomena in nonlinearly managed lattice solitons,” Phys. Rev. E **70**, 026606 (2004). [CrossRef]

24. G. Fibich, Y. Sivan, and M.I. Weinstein, ”Bound states of nonlinear Schrodinger equations with a periodic nonlinear microstructure,” Physica D **217**, 31–57 (2006). [CrossRef]

25. F. Abdullaev, A. Abdumalikov, and R. Galimzyanov, ”Gap solitons in Bose-Einstein condensates in linear and nonlinear optical lattices,” Phys. Lett. A **367**, 149–155 (2007). [CrossRef]

26. Z. Rapti, P.G. Kevrekidis, V.V. Konotop, and C.K.R.T. Jones, ”Solitary waves under the competition of linear and nonlinear periodic potentials,” J. Phys. A: Math. Theor. **40**, 14151–14163 (2007). [CrossRef]

27. R. Hao, R. Yang, L. Li, and G. Zhou, ”Solutions for the propagation of light in nonlinear optical media with spatially inhomogeneous nonlinearities,” Opt. Commun. **281**, 1256–1262 (2008). [CrossRef]

28. J. Belmonte-Beitia, V.M. Perez-Garcia, V. Vekslerchik, and P.J. Torres, ”Lie Symmetries and Solitons in Nonlinear Systems with Spatially Inhomogeneous Nonlinearities,” Phys. Rev. Lett. **98**, 064102 (2007). [CrossRef] [PubMed]

29. Y. Kominis, ”Analytical solitary wave solutions of the nonlinear Kronig-Penney model in photonic structures,” Phys. Rev. E **73**, 066619 (2006). [CrossRef]

30. Y. Kominis and K. Hizanidis, ”Lattice solitons in self-defocusing optical media: Analytical solutions of the nonlinear Kronig-Penney model,” Opt. Lett. **31**, 2888–2890 (2006). [CrossRef] [PubMed]

31. Y. Kominis, A. Papadopoulos, and K. Hizanidis, ”Surface solitons in waveguide arrays: Analytical solutions,” Opt. Express **15**, 10041–10051 (2007). [CrossRef] [PubMed]

26. Z. Rapti, P.G. Kevrekidis, V.V. Konotop, and C.K.R.T. Jones, ”Solitary waves under the competition of linear and nonlinear periodic potentials,” J. Phys. A: Math. Theor. **40**, 14151–14163 (2007). [CrossRef]

32. R. Morandotti, U. Peschel, J.S. Aitchison, H.S. Eisenberg, and Y. Silberberg, ”Dynamics of Discrete Solitons in Optical Waveguide Arrays,” Phys. Rev. Lett. **83**, 2726 – 2729 (1999). [CrossRef]

33. A.A. Sukhorukov and Y.S. Kivshar, ”Soliton control and Bloch-wave filtering in periodic photonic lattices,” Opt. Lett. **30**, 1849–1851 (2005). [CrossRef] [PubMed]

34. Z. Xu, Y.V. Kartashov, and L. Torner, ”Soliton Mobility in Nonlocal Optical Lattices,” Phys. Rev. Lett. **95**, 113901 (2005). [CrossRef] [PubMed]

35. R.A. Vicencio and M. Johansson ”Discrete soliton mobility in two-dimensional waveguide arrays with saturable nonlinearity,” Phys. Rev. E **73**046602 (2006). [CrossRef]

36. A.A. Sukhorukov, ”Enhanced soliton transport in quasiperiodic lattices with introduced aperiodicity,” Phys. Rev. Lett. **96**, 113902 (2006). [CrossRef] [PubMed]

37. T.R.O. Melvin, A.R. Champneys, P.G. Kevrekidis, and J. Cuevas, ”Radiationless Traveling Waves in Saturable Nonlinear Schrodinger Lattices,” Phys. Rev. Lett. **97**, 124101 (2006). [CrossRef] [PubMed]

38. D.E. Pelinovsky, ”Translationally invariant nonlinear Schrodinge lattices,” Nonlinearity **19**, 26952716 (2006). [CrossRef]

39. Y.V. Kartashov, V.A. Vysloukh, and L. Torner, ”Soliton percolation in random optical lattices,” Opt. Express **15**, 12409–12417 (2007). [CrossRef] [PubMed]

40. H. Sakaguchi and B.A. Malomed, ”Gap solitons in quasiperiodic optical lattices,” Phys. Rev. E **74**, 026601 (2006). [CrossRef]

41. N.K. Efremidis and D.N. Christodoulides, ”Lattice solitons in Bose-Einstein condensates,” Phys. Rev. A **67**, 063608 (2003). [CrossRef]

42. P.J.Y. Louis, E.A. Ostrovskaya, C.M. Savage, and Y.S. Kivshar, ”Bose-Einstein condensates in optical lattices: Band-gap structure and solitons,” Phys. Rev. A **67**, 013602 (2003). [CrossRef]

43. D.E. Pelinovsky, A.A. Sukhorukov, and Y.S. Kivshar, ”Bifurcations and stability of gap solitons in periodic potentials,” Phys. Rev. E **70**, 036618 (2004). [CrossRef]

## 2. Model and stationary solutions

*x*is the transverse coordinate normalized to

*x*

_{0},

*z*is the propagation distance normalized to

*z*

_{0}= 2

*kx*

^{2}

_{0}, and ψ is the electric field ampltude normalized to

*I*

^{1/2}

_{0}with

*I*

_{0}= (

*n*

_{2}

*kk*

_{0}

*x*

^{2}

_{0})

^{-1}. The functions

*n*

_{0}(

*x*) and

*n*

_{2}(

*x*) describe the transverse variation of the linear and the nonlinear refractive index (potential), respectively. The functions

*n*(

_{i}*x*),

*i*= 0,2 can be of any form describing periodic or quasiperiodic lattices. A normalized propagation distance

*z*= 100, corresponds to an actual propagation length of 10.7-24.3

*mm*, for the case of a nonlinear material of AlGaAs type (

*n*= 3.34,

*n*

_{2}= 1.5×10

^{-13}

*cm*

^{2}/

*W*), when the transverse coordinate is normalized to

*X*

_{0}= 2-3

*μm*.

*u*(

*x*) a real function describing the transverse wave profile and β the propagation constant. The corresponding stationary equation is the following:

*M*(

*x*

_{0}) can be written as

*x*).

*M*(

*x*

_{0}) for the existence of solitary stationary waves in a large variety of inhomogeneous media described by different functions

*n*

_{0}(

*x*) and

*n*

_{2}(

*x*). The most generic form for the modulation of the linear and the nonlinear refractive index is given by

## 3. Lattice solitons in media with periodically and quasiperiodically modulated linear refractive index

*A*

^{(2)}

_{m}= 0. More specifically, we start from the commonly studied case of a simple harmonic lattice where only one Fourier component (

*A*

^{(0)}

_{1}) of

*n*

_{0}(

*x*) is nonzero. The respective Melnikov function (9)

*x*

_{0},

*M*́(

*x*

_{0}) [46

46. T. Kapitula, ”Stability of waves in perturbed Hamiltonian systems,” Physica D **156**, 186–200 (2001). [CrossRef]

*x*

_{0}) as well as the stability type of the solutions does not depend on the propagation constant β.

*K*

^{(0)}

_{1}= 1,

*K*

^{(0)}

_{2}= 2, with

*A*

^{(0)}

_{1}= 1 and

*ϕ*

^{(0)}

_{1}= 0. In the following we investigate the effect of the second Fourier component on soliton formation and propagation for solitons having different β. For an analogous case of gap solitons, it has been shown in a more general setting that only two solutions bifurcate in periodic potentials on a single period if β is small [43

43. D.E. Pelinovsky, A.A. Sukhorukov, and Y.S. Kivshar, ”Bifurcations and stability of gap solitons in periodic potentials,” Phys. Rev. E **70**, 036618 (2004). [CrossRef]

*A*

^{(0)}

_{2}= 1,

*ϕ*

^{(0)}

_{2}= 0 and for propagation constants β = 0.1 (circles) and β = 1 (asterisks). It is shown that for β = 0.1 the Melnikov function has only two zeros (circles) located at

*x*= 0,

*π*, while for β = 1 two additional zeros (asterisks) appear at

*x*≃ 2.25,4.05 (in symmetric positions with respect to

*x*=

*π*). The latter correspond to asymmetric solution profiles. The effect of the second Fourier component on the solutions corresponding to different propagation constants, differs for the two cases due to the strong dependence of the Melnikov function (9) on β through the function

*F*as shown in Fig. 1. Therefore, there is no significant effect of the second Fourier component for solutions with propagation constant β = 0.1. This is a case of a power (or β) dependent bifurcation: for small β only two stationary solutions located at

*x*

_{0}= 0,

*π*exist with the former being stable and the latter being unstable, while for higher β two additional stationary solutions appear, resulting also to a change of the stability of the solution located at

*x*

_{0}=

*π*. The corresponding bifurcation (transition) value for β is given from the equation

*M*́(

*x*

_{0}=

*π*) = 0. It is worth mentioning that, in contrast to the case of a lattice modulated by a single wavenumber, the appearance of more than one wavenumbers results in nonuniform existence and stability properties with respect to β. This interesting feature suggests a power selectivity property of a polychromatic lattice which is promising for applications.

*π*] are shown). Their evolution under propagation is depicted in Fig. 4, where it is confirmed that stable and unstable solutions are alternate. (In all cases a random noise 1% of the maximum of solution amplitude has been superimposed on the amplitude and the phase of the stationary profiles for the numerical study of their stability.) It is also confirmed that the stationary solution located at the local maximum at

*x*=

*π*is unstable for β = 0.1, while it is stable for β = 1. Qualitatively similar conclusions can be made for different selections of the amplitude

*A*

^{(0)}

_{2}, the wavenumber

*K*

^{(0)}

_{2}and the phase

*ϕ*

^{(0)}

_{2}of the second Fourier component, depending on the number and location of the zeros of the equation (13). Also, cases with multiple Fourier components can be investigated on the same basis.

*M*(

*x*

_{0}) (9) is also quasiperiodic and its irregularly distributed zeros predict the location of the stationary solutions in the lattice. Additionally, the stability of a solution corresponding to

*x*

_{0}is determined by the sign of

*M*́(

*x*

_{0}). It is obvious that the information provided by the Melnikov function is even more important in this complex case. As an example we consider the case where the linear refractive index is modulated by two incommensurate wavenumbers, so that the respective parameters are

*K*

^{(0)}

_{1}= 1,

*K*

^{(0)}

_{2}=

*π*/2,

*A*

^{(0)}

_{1}=

*A*

^{(0)}

_{2}= 1 and

*ϕ*

^{(0)}

_{1}=

*ϕ*

^{(0)}

_{2}= 0. The form of the nonperiodic linear refractive index profile along with the irregularly distributed corresponding zeros of the Melnikov function, for a finite part of the lattice, are shown in Fig. 5 for propagation constants β = 0.1 (circles) and β = 1 (asterisks). In Fig. 6, the profiles of the first three solutions (on the right of the origin) are shown, while their respective evolution under propagation is depicted in Fig. 7, where the alternate character of the stability type of each solution, as predicted by the sign of

*M*́(

*x*

_{0}), is confirmed.

## 4. Lattice solitons in media with modulated linear and nonlinear refractive indices

*G*(11) appears in the corresponding Melnikov function (9) instead of the function

*F*(10). A more interesting case occurs when both the linear and the nonlinear refractive indices are spatially modulated. As an example we firstly investigate the case where both refractive indices are modulated by the same wavenumber and we consider the following parameter values

*K*

^{(0)}

_{1}=

*K*

^{(2)}

_{1}= 1,

*A*

^{(0)}

_{1}= 1 and

*ϕ*

^{(0)}

_{1}= 0. The location of the stationary solutions are given from the zeros of the corresponding Melnikov function, which, according to (9), is

*ϕ*

^{(2)}

_{1}= 0, the Melnikov function is

*x*

_{0}= 0,

*π*) are the same in both cases, (ii) Their stability type can be either the same or interchanged depending on the sign of the quantity (16). For example, let us consider the case of a nonlinear refractive index profile having

*A*

^{(2)}

_{1}= -4.8 and a set of stationary solutions corresponding to β = 1 and β = 0.1. The profiles of the solutions (corresponding to

*x*

_{0}= 0,π) are shown in Fig. 8, while their respective evolution under propagation is depicted in Fig. 9.

*x*

_{0}= 0 becomes now unstable, while the soliton at

*x*

_{0}=

*π*becomes stable. It is worth emphasizing that, since the quantity (16) depends on the propagation constant β (and therefore on the power

*P*), the stability type of the solutions is not uniform for all β, so that for different β we can have different stability type of the two solutions. As in the case where only the linear refractive index is modulated and we have more than one wavenumbers (Figs. 2, 3, 4), the stability type of the solution depends on the propagation constant (or the power); the difference here is that we have an exchange of stability type between the two solutions, while no additional solution appears.

*A*

^{(2)}

_{1}, there exist a propagation constant β, for which the Melnikov function vanishes identically. Such case corresponds to a bifurcation point in the parameter space, and the investigation based on the Melnikov function becomes inconclusive. Keeping in mind that the previously presented Melnikov method, is actually a first-order perturbation theory [44, 45], in order to conclude on the existence and the stability of stationary solutions, higher-order theory should be used. However, since higher-order calculations may become quite complicated, in the following we investigate numerically this case. As an example, we consider the case where the quantity (16) is zero for β = 0.1, corresponding to

*A*

^{(2)}

_{1}= -8.57. As shown in Fig. 10, additional stationary solutions appear in this case. Therefore, we have found 3 stationary solutions corresponding to

*x*

_{0}in the interval [0,

*π*], while a fourth solution exist in the interval (

*π*,2

*π*) symmetrically with respect to

*π*. In an analogous case for gap solitons it has been shown that four branches of solutions bifurcate in the limit of small beta [43

43. D.E. Pelinovsky, A.A. Sukhorukov, and Y.S. Kivshar, ”Bifurcations and stability of gap solitons in periodic potentials,” Phys. Rev. E **70**, 036618 (2004). [CrossRef]

*π*] for any β. The respective evolution under propagation of these solutions is depicted in Fig. 11, where it is shown that the instability of the first two solutions is significantly slower than all the previous cases corresponding to a nonvanishing Melnikov function. The latter is important from the point of view of practical applications, where the propagation distances of interest can be too small for the appearance of the instability. For the case where the linear refractive index is modulated by a large number of wavenumbers, we can always find a nonlinear refractive index profile with parameters (

*A*

^{(2)}

_{m},

*K*

^{(2)}

_{m},

*ϕ*

^{(2)}

_{m}) for which the Melnikov function vanishes for all

*x*

_{0}, for a specific propagation constant β. However, it is worth emphasizing that such a case corresponds to a bifurcation point and is structurally unstable in the parameter space, therefore any deviation of the parameters from their bifurcation values results in drastic and qualitatively different features. This case has been previously studied with respect to the enhancement of the mobility of a solitary wave [26

26. Z. Rapti, P.G. Kevrekidis, V.V. Konotop, and C.K.R.T. Jones, ”Solitary waves under the competition of linear and nonlinear periodic potentials,” J. Phys. A: Math. Theor. **40**, 14151–14163 (2007). [CrossRef]

*A*

^{(0)}

_{1}= 1,

*K*

^{(0)}

_{1}= 1,

*A*

^{(2)}

_{1}= 0.5,2,

*K*

^{(2)}

_{1}=

*π*/2 and

*ϕ*

^{(0)}

_{1}=

*ϕ*

^{(2)}

_{1}= 0 the positions of the stationary solutions with respect to the linear and the nonlinear refractive index profiles are shown in Fig. 12, for β = 1. It is shown that for

*A*

^{(2)}

_{1}= 0.5 the positions with respect to the linear refractive index profile are close to its extrema, while the positions with respect to the nonlinear refractive index profile appear irregularly distributed (Fig. 12(a)). For stronger modulation (

*A*

^{(2)}

_{1}= 2) of the nonlinear refractive index the positions of the stationary solutions are irregularly distributed with respect to both the linear and the nonlinear refractive index profiles (Fig. 12(b)). In general, it can be easily shown that depending on the ratio

*x*– axis, if reduced within the same interval T (

*x*

_{0}mod

*T*), can densely fill the interval of one period (

*T*) of either the linear or the nonlinear refractive index profiles. This means that for an infinite lattice there is always a part of the lattice where a stationary solution can be found with any location relative to the underlying linear or nonlinear refractive index profiles. Note that the absolute position of this location can be controlled by the appropriate choice of the phases (

*ϕ*

^{(0)}

_{1},

*ϕ*

^{(2)}

_{1}) so that the position can always be located within the first period (with respect to the origin) of the linear or the nonlinear refractive index profile. This mechanism of controlling the position of the soliton can be proved useful in applications where one of the index profiles

*n*

_{0}(

*x*) or

*n*

_{2}(

*x*) is determined by the geometrical structure of the configuration, while the other is dynamically induced by an optical control wave.

## 5. Summary and conclusions

## References and links

1. | J.D. Joannopoulos, P.R. Villeneuve, and S. Fan, ”Photonic crystals: putting a new twist on light,” Nature |

2. | P. Russel, ”Photonic crystal fibers,” Science |

3. | B.P. Anderson and M.A. Kasevich, ”Macroscopic quantum interference from atomic tunnel arrays,” Science |

4. | A. Trombettoni and A. Smerzi, ”Discrete solitons and breathers with dilute BoseEinstein condensates,” Phys. Rev. Lett. |

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

6. | N.K. Efremidis, S. Sears, D.N. Christodoulides, J.W. Fleischer, and M. Segev, ”Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E |

7. | J.W. Fleischer, T. Carmon, M. Segev, N.K. Efremidis, and D.N. Christodoulides, ”Observation of Discrete Solitons in Optically Induced Real Time Waveguide Arrays,” Phys. Rev. Lett. |

8. | D. Neshev, E. Ostrovskaya, Y.S. Kivshar, and W. Krolikowski, ”Spatial solitons in optically induced gratings,” Opt. Lett. |

9. | D. Neshev, A.A. Sukhorukov, Y.S. Kivshar, and W. Krolikowski, ”Observation of transverse instabilities in optically induced lattices,” Opt. Lett. |

10. | Y. Kominis and K. Hizanidis, ”Continuous-wave-controlled steering of spatial solitons,” J. Opt. Soc. Am. B |

11. | Y. Kominis and K. Hizanidis, ”Optimal multidimensional solitary wave steering,” J. Opt. Soc. Am. B |

12. | Z. Chen, H. Martin, E. Eugenieva, J. Xu, and J. Yang, ”Formation of discrete solitons in light-induced photonic lattices,” Opt. Express |

13. | C.R. Rosberg, D.N. Neshev, A.A. Sukhorukov, Y.S. Kivshar, and W. Krolikowski, ”Tunable positive and negative refraction in optically induced photonic lattices,” Opt. Lett. |

14. | T. Song, S.M. Liu, R. Guo, Z.H. Liu, N. Zhu, and Y.M. Gao, ”Observation of composite gap solitons in optically induced nonlinear lattices in LiNbO3:Fe crystal,” Opt. Express , |

15. | I. Tsopelas, Y. Kominis, and K. Hizanidis, ”Soliton dynamics and interactions in dynamically photoinduced lattices,” Phys. Rev. E |

16. | I. Tsopelas, Y. Kominis, and K. Hizanidis, ”Dark soliton dynamics and interactions in continuous-wave-induced lattices,” Phys. Rev. E |

17. | Discrete Solitons, edited by S. Trillo and W. TorruellasSpringer-Verlag, Berlin,2001. |

18. | D.N. Christodoulides and R.I. Joseph, ”Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. |

19. | D.C. Hutchings, ”Theory of Ultrafast Nonlinear Refraction in Semiconductor Superlattices,” IEEE J. Sel. Top. Quantum Electron. |

20. | L. Berge, V.K. Mezentsev, J.J. Rasmussen, P.L. Christiansen, and Y.B. Gaididei, ”Self-guiding light in layered nonlinear media,” Opt. Lett. |

21. | D.E. Pelinovsky, P.G. Kevrekidis, and D.J. Frantzeskakis, ”Averaging for Solitons with Nonlinearity Management,” Phys. Rev. Lett. |

22. | H. Sakaguchi and B.A. Malomed, ”Resonant nonlinearity management for nonlinear Schrodinger solitons,” Phys. Rev. E |

23. | Y.V. Kartashov and V.A. Vysloukh, ”Resonant phenomena in nonlinearly managed lattice solitons,” Phys. Rev. E |

24. | G. Fibich, Y. Sivan, and M.I. Weinstein, ”Bound states of nonlinear Schrodinger equations with a periodic nonlinear microstructure,” Physica D |

25. | F. Abdullaev, A. Abdumalikov, and R. Galimzyanov, ”Gap solitons in Bose-Einstein condensates in linear and nonlinear optical lattices,” Phys. Lett. A |

26. | Z. Rapti, P.G. Kevrekidis, V.V. Konotop, and C.K.R.T. Jones, ”Solitary waves under the competition of linear and nonlinear periodic potentials,” J. Phys. A: Math. Theor. |

27. | R. Hao, R. Yang, L. Li, and G. Zhou, ”Solutions for the propagation of light in nonlinear optical media with spatially inhomogeneous nonlinearities,” Opt. Commun. |

28. | J. Belmonte-Beitia, V.M. Perez-Garcia, V. Vekslerchik, and P.J. Torres, ”Lie Symmetries and Solitons in Nonlinear Systems with Spatially Inhomogeneous Nonlinearities,” Phys. Rev. Lett. |

29. | Y. Kominis, ”Analytical solitary wave solutions of the nonlinear Kronig-Penney model in photonic structures,” Phys. Rev. E |

30. | Y. Kominis and K. Hizanidis, ”Lattice solitons in self-defocusing optical media: Analytical solutions of the nonlinear Kronig-Penney model,” Opt. Lett. |

31. | Y. Kominis, A. Papadopoulos, and K. Hizanidis, ”Surface solitons in waveguide arrays: Analytical solutions,” Opt. Express |

32. | R. Morandotti, U. Peschel, J.S. Aitchison, H.S. Eisenberg, and Y. Silberberg, ”Dynamics of Discrete Solitons in Optical Waveguide Arrays,” Phys. Rev. Lett. |

33. | A.A. Sukhorukov and Y.S. Kivshar, ”Soliton control and Bloch-wave filtering in periodic photonic lattices,” Opt. Lett. |

34. | Z. Xu, Y.V. Kartashov, and L. Torner, ”Soliton Mobility in Nonlocal Optical Lattices,” Phys. Rev. Lett. |

35. | R.A. Vicencio and M. Johansson ”Discrete soliton mobility in two-dimensional waveguide arrays with saturable nonlinearity,” Phys. Rev. E |

36. | A.A. Sukhorukov, ”Enhanced soliton transport in quasiperiodic lattices with introduced aperiodicity,” Phys. Rev. Lett. |

37. | T.R.O. Melvin, A.R. Champneys, P.G. Kevrekidis, and J. Cuevas, ”Radiationless Traveling Waves in Saturable Nonlinear Schrodinger Lattices,” Phys. Rev. Lett. |

38. | D.E. Pelinovsky, ”Translationally invariant nonlinear Schrodinge lattices,” Nonlinearity |

39. | Y.V. Kartashov, V.A. Vysloukh, and L. Torner, ”Soliton percolation in random optical lattices,” Opt. Express |

40. | H. Sakaguchi and B.A. Malomed, ”Gap solitons in quasiperiodic optical lattices,” Phys. Rev. E |

41. | N.K. Efremidis and D.N. Christodoulides, ”Lattice solitons in Bose-Einstein condensates,” Phys. Rev. A |

42. | P.J.Y. Louis, E.A. Ostrovskaya, C.M. Savage, and Y.S. Kivshar, ”Bose-Einstein condensates in optical lattices: Band-gap structure and solitons,” Phys. Rev. A |

43. | D.E. Pelinovsky, A.A. Sukhorukov, and Y.S. Kivshar, ”Bifurcations and stability of gap solitons in periodic potentials,” Phys. Rev. E |

44. | J. Guckenheimer and P. Holmes, ”Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,” Applied Mathematical Series |

45. | S. Wiggins, ”Introduction to Applied Nonlinear Dynamical Systems and Chaos,” Texts in Applied Mathematics |

46. | T. Kapitula, ”Stability of waves in perturbed Hamiltonian systems,” 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: May 21, 2008

Revised Manuscript: July 9, 2008

Manuscript Accepted: July 25, 2008

Published: July 29, 2008

**Citation**

Y. Kominis and K. Hizanidis, "Power dependent soliton location and
stability in complex photonic structures," Opt. Express **16**, 12124-12138 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-16-12124

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