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Optics Express

Optics Express

  • Vol. 16, Iss. 16 — Aug. 4, 2008
  • pp: 12124–12138
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Power dependent soliton location and stability in complex photonic structures

Y. Kominis and K. Hizanidis  »View Author Affiliations


Optics Express, Vol. 16, Issue 16, pp. 12124-12138 (2008)
http://dx.doi.org/10.1364/OE.16.012124


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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

Electromagnetic waves propagating in periodically structured dielectric materials, such as Photonic Crystals (PCs), share many properties with electron waves in ordinary semiconductors [1

1. J.D. Joannopoulos, P.R. Villeneuve, and S. Fan, ”Photonic crystals: putting a new twist on light,” Nature 386, 143149 (1997). [CrossRef]

, 2

2. P. Russel, ”Photonic crystal fibers,” Science 299, 358–362 (2003). [CrossRef]

] and matter-wave realizations of a Bose-Einstein Condensate (BEC) in optical lattices [3

3. B.P. Anderson and M.A. Kasevich, ”Macroscopic quantum interference from atomic tunnel arrays,” Science 282, 1686–1689 (1998). [CrossRef] [PubMed]

, 4

4. A. Trombettoni and A. Smerzi, ”Discrete solitons and breathers with dilute BoseEinstein condensates,” Phys. Rev. Lett. 86, 2353–2356 (2001). [CrossRef] [PubMed]

]. The periodicity leads to the appearance of a band structure of the respective underlying linear system. This structure along with a corroborating nonlinear response is responsible for a multitude of effects such as modulation instability, Bloch oscillations and soliton generation, to name a few. Spatially localized structures in optical lattices is a subject of intense theoretical and experimental efforts in our days [5

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

].

Periodicity in the structure implies periodicity in the material properties, linear and nonlinear. In the PCs the effective linear refractive index is a function of both the wavelength and the structural parameters of the unit lattice. The light is linearly guided by the differences in the effective refractive index of the lattice elements. The linear material properties can be controlled either by choosing a suitable configuration for the lattice or dynamically in the sense of optically induced waveguide arrays and lattices [6

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

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]

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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]

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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]

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

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

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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]

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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]

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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]

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

, 16

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]

]. Along this direction, lattice solitons in photonic structures with periodically varying linear refractive index have been extensively investigated either by studying the original continuous model, consisting of the NonLinear Schrodinger Equation (NLSE) describing soliton propagation [17

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

], or by utilizing approximate discrete models [17

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

, 18

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]

]. On the other hand, recent advances in fabrication techniques have made possible the fabrication of photonic structures with rapidly varying nonlinear refractive index [19

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

], as well. Soliton formation and propagation in such structures has been studied mostly for cases where the nonlinear refractive index varies along the propagation distance, and, as an analog of dispersion management, is usually referred as nonlinearity management [20

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

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]

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

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

]. However, the effect of a transversely varying nonlinear refractive index has been only recently studied theoretically for the case of a homogeneous [24

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]

] as well as a spatially modulated [25

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

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]

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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]

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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]

] linear refractive index. Cases where both linear and nonlinear refractive index are spatially modulated have been also studied for waveguide arrays consisting of interlaced linear and nonlinear waveguides, and analytical soliton solutions have been found [29

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

, 30

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

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

]. The emphasis of the theoretical research has been mainly focused on achieving high localization and investigating the mobility of localized waves [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]

, 32

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

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]

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

, 35

35. R.A. Vicencio and M. Johansson ”Discrete soliton mobility in two-dimensional waveguide arrays with saturable nonlinearity,” Phys. Rev. E 73046602 (2006). [CrossRef]

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

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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]

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

, 39

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

]. While localization and mobility is a common goal in all the diverse physical systems exhibiting lattice solitons, not enough emphasis has been given so far to the location of the localized states within the periodic spatial structure and its controllability.

In this work we study lattice soliton formation and stability in a configuration of planar geometry where both the linear and the nonlinear refractive index are inhomogeneous with respect to the transverse dimension, with their spatial dependence being of a generic form describing either periodic or quasiperiodic structures [40

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

]. It is well known that the presence of a transverse inhomogeneity results in the breaking of the translational invariance of the system. Therefore, localized states can be no longer located in any position within the photonic structure meaning that there exist only specific positions within the structure where solitons can be formed. The simple case of a lattice with a sinusoidally varying linear refractive index, is well studied and solitons have been shown to exist in the high- and low- index positions, refereed as on-site and off-site solitons, respectively [41

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

, 42

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]

], with the former being stable and the latter being unstable for all values of the propagation constants [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]

].

However, it is shown that, for the case of a superlattice (where solely the linear refractive index is modulated by several wavenumbers) the determination of the position of a localized state with respect to the underlying structure as well as its stability is not a trivial task. The case of incommensurate wavenumbers, describing quasi periodic lattices, is also more complex. More importantly, both the position and the stability type of the localized states depend on the respective propagation constant and the power. Furthermore, it is shown that the appropriate modulation of the nonlinear refractive index can be used for altering either the stability of the localized states or their positions, in comparison to a case where only the linear refractive index is modulated. In all cases investigated in this work it is shown that additional complexity of the photonic structure results in additional functionality allowing for soliton discrimination with respect to their characteristics, due to underlying power (or width / propagation constant) dependent bifurcations. The aforementioned features are investigated by utilizing the Melnikov’s theory [44

44. J. Guckenheimer and P. Holmes, ”Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,” Applied Mathematical Series 42, Springer (New York, Berlin) (1983).

, 45

45. S. Wiggins, ”Introduction to Applied Nonlinear Dynamical Systems and Chaos,” Texts in Applied Mathematics 2, Springer (New York, Berlin) (1990).

] resulting in an analytic expression from which both the position and stability type of a localized state are determined. It is worth mentioning that such an expression allows for the consideration of a large variety of different quite complex configurations and it can be used for providing physical intuition for the designing of specific structures possessing desirable properties.

The paper is organized as follows: The model and its stationary solutions along with the application of the Melnikov’s method are presented in Section II. The cases of a periodically and a quasiperiodically modulated linear refractive index is investigated in Section III, while the case where both the linear and the nonlinear refractive indices are spatially modulated is investigated in Section IV. The summary of the results and the conclusions are given in Section V.

2. Model and stationary solutions

Soliton propagation in transversely inhomogeneous planar media is described by a one-dimensional NLSE with coefficients depending on the transverse coordinate

iψz+2ψx2+2ψ2ψ+ε[n0(x)ψ+n2(x)ψ2ψ]=0
(1)

where x is the transverse coordinate normalized to x 0, z is the propagation distance normalized to z 0 = 2kx 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 ni(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 10.7-24.3mm, 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 = 2-3μm.

The stationary solutions of Eq. (1) have the form

ψxz=u(x)eiβz
(2)

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

d2udx2βu+2u3+ε[n0(x)u+n2(x)u3]=0
(3)

This equation corresponds to a one-degree of freedom dynamical system with Hamiltonian

H=p22βq22+q42+ε[n0(x)q22+n2(x)q42]
(4)

The unperturbed part of the system described by Eq.(4) has the homoclinic solution (for β > 0)

(q0(x),p0(x))=(±βsech[β(xx0)],βsech[β(xx0)]tanh[β(xx0)])
(5)

M(x0)=ε+p0(x)[n0(x)q0(x)+n2(x)q0(x)3]dx
(6)

corresponding to the Melnikov function. Alternatively, M(x 0) can be written as

M(x0)=ε+[n'0(x+x0)q02(x)2+n'2(x+x0)q04(x)2]dx
(7)

which is sometimes more appropriate for calculations (primes denote differentiation with respect to x).

Fig. 1. Dependence of the functions F(K (0) m ,β) and G(K (2) m ,β ) on the wavenumber Km for β = 0.1 (a) and β = 1 (b)

In the following we investigate the information provided by the zeros of the function 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

ni(x)=mAm(i)cos(Km(i)x+ϕm(i)),i=0,2
(8)

corresponding to Fourier series for the case of periodic modulations, generalized Fourier series for quasi-periodic modulations, or finite trigonometric sums. The corresponding Melnikov function is

M(x0)=επβ2mAm(0)FKm(0)βsin(Km(0)x0+ϕm(0))+επβ2mAm(2)GKm(2)βsin(Km(2)x0+ϕm(2))
(9)

where

FKm(0)β=(Km(0))2sinh(πKm(0)2β)
(10)
GKm(2)β=(Km(2))2[(Km(2))2+4β]12sinh(πKm(2)2β)
(11)

Due to the fact that the Melnikov function is linear in the Fourier components, without loss of generality, we can consider one or a few Fourier components of the expansion, while the results can be directly extended for the case where all terms of the Fourier expansion are taken into account. A wide variety of configurations can be investigated with the utilization of Eq. (9). In the following we focus on specific cases having essential qualitatively different properties. We consider the case where ε = 0.1 so that the perturbative character of our approach is relevant; however, most of the results also hold even for stronger perturbations.

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

Firstly, let us consider the case where only the linear refractive index is spatially modulated, so that 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)

M(x0)=A1(0)FK1(0)βsin(K1(0)x0+ϕ1(0))
(12)

Let us now consider the more general case where the linear refractive index is periodic but has two Fourier components 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]

]. According to Eq. (9), the stationary solutions correspond to the zeros of the Melnikov function given by the equation

M(x0)=F1βsin(x0)+A2(0)F2βsin(2x0+ϕ2(0))=0
(13)

The spatial profile of the linear refractive index along with the positions of the corresponding zeros of the Melnikov function are shown in Fig. 2 for the case 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 Ḿ(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.

Fig. 2. Linear refractive index profile along with the corresponding position of the zeros of the Melnikov function for the case where the linear refractive index is modulated by two commensurate wavenumbers (n 0(x) = cos(x)+cos (2x), n 2(x) = 0). Circles correspond to β = 0.1 and asterisks to β = 1.
Fig. 3. Profiles of stable (solid line) and unstable (dashed line) stationary solutions for the case where the linear refractive index is modulated by two commensurate wavenumbers (n 0(x) = cos(x)+cos(2x), n 2(x) = 0). The propagation constant is β = 0.1 (a) and β = 1 (b).

The stationary solutions for each case are shown in Fig. 3 (due to the symmetry of the linear refractive index profile, only solutions in [0,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.

Fig. 4. Propagation of the stationary solutions shown in Figs. 3. (a) β = 0.1, x 0 = 0, (b) β = 0.1, x 0 = π, (c) β = 1, x 0 = 0, (d) β = 1, x 0 = 2.25, (e) β = 1, x 0 = π.

Fig. 5. Linear refractive index profile along with the corresponding position of the zeros of the Melnikov function for the case where the linear refractive index is modulated by two incommensurate wavenumbers (n 0(x) = cos(x)+cos (πx/2), n 2(x) = 0). Circles correspond to β = 0.1 and asterisks to β = 1.
Fig. 6. Profiles of stable (solid line) and unstable (dashed line) stationary solutions for the case where the linear refractive index is modulated two incommensurate wavenumbers (n 0(x) = cos(x)+cos(πx/2), n 2(x) = 0). The propagation constant is β = 0.1 (a) and β = 1 (b).

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

Fig. 7. Propagation of the stationary solutions shown in Figs. 6. (a) β = 0.1, x 0 = 0, (b) β = 0.1, x 0 = 3, (c) β = 0.1, x 0 = 6.36, (d) β = 1, x 0 = 0, (e) β = 1, x 0 = 2.45, (f) β = 1, x 0 = 4.96.
M(x0)=F1βsin(x0)+A1(2)G1βsin(x0+ϕ1(2))=0
(14)

More specifically, if ϕ (2) 1 = 0, the Melnikov function is

M(x0)=[F1β+A1(2)G1β]sin(x0)
(15)

F1β+A1(2)G1β
(16)

is not zero, (i) The number of the stationary solutions within a period of the lattice as well as and their positions (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.

It is obvious that, in comparison with the case where only the linear refractive index is modulated, the stability type of the two solutions can be interchanged depending on β i.e. for β = 1 the soliton located at 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.

Fig. 8. Profiles of stable (solid line) and unstable (dashed line) stationary solutions for the case where n 0(x) = cos(x) and n 2(x) = -4.8cos(x). The propagation constant is β = 0.1 (a) and β = 1 (b).
Fig. 9. Propagation of the stationary solutions shown in Figs. 8. (a) β = 0.1, x 0 = 0, (b) β = 0.1, x 0 = π, (c) β = 1, x 0 = 0, (d) β = 1, x 0 = π.
Fig. 10. Profiles of stationary solutions for the case where the quantity (16) is zero for β = 0.1, (n 0(x) = cos(x), n 2(x) = -8.57cos(x)).
Fig. 11. Propagation of the stationary solutions shown in Fig. 10.

Fig. 12. Linear (solid line) and nonlinear (dashed line) refractive index profiles along with the corresponding relative position of the zeros of the Melnikov function (circles: position relative to the linear refractive index profile, rhombs: position relative to the nonlinear refractive index profile) for the case where the linear and the nonlinear refractive indices are modulated by two incommensurate wave (n 0(x) = cos(x), n 2(x) = A (2) 1 cos(πx/2)). (a)A (2) 1 = 0.5, (b) A (2) 1 = 2.

Finally, it is interesting to consider the case where the wavenumbers modulating the linear and nonlinear refractive indices are incommensurate. In this case, even though each index profile is periodic, the position of the stationary solitary solutions within the lattice appear irregularly distributed with respect to either the linear or the nonlinear refractive index profile. For example in the case where 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

a=A1(0)FK1(0)βA1(2)GK1(2)β
(17)

the infinite number of zeros of the respective Melnikov function in the x– axis, if reduced within the same interval T (x 0 modT), 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

The results of the method are shown to be applicable to a large variety of different configurations. Characteristic cases having qualitatively different features were investigated, and the predictions of the theory were successfully tested by numerical simulations. Among these interesting features we can refer to the power (or propagation constant) dependence of the position and the stability type of the respective localized states, for the cases where either the linear refractive index is spatially modulated by more than one wavenumbers or both linear and nonlinear refractive are spatially modulated. It is shown that, in contrast to cases of monochromatic modulation of solely the linear or the nonlinear refractive index, more complex spatial modulations result in the capability of the photonic structure for discriminating solitons having different characteristics. Moreover, it is worth mentioning the case where incommensurate wavenumbers modulate the linear and/or nonlinear refractive index which result a plethora of asymmetric localized states located in irregularly distributed positions within the photonic structures. The aforementioned features along with the capability of controlling the spatial profiles of the linear and the nonlinear refractive indices, either in the fabrication process or dynamically by using optical control signals, opens the possibility of potential applications having technological interest. The method and results presented in this work are expected to be proved useful for the designing of appropriately engineered photonic structures having the desirable properties.

References and links

1.

J.D. Joannopoulos, P.R. Villeneuve, and S. Fan, ”Photonic crystals: putting a new twist on light,” Nature 386, 143149 (1997). [CrossRef]

2.

P. Russel, ”Photonic crystal fibers,” Science 299, 358–362 (2003). [CrossRef]

3.

B.P. Anderson and M.A. Kasevich, ”Macroscopic quantum interference from atomic tunnel arrays,” Science 282, 1686–1689 (1998). [CrossRef] [PubMed]

4.

A. Trombettoni and A. Smerzi, ”Discrete solitons and breathers with dilute BoseEinstein condensates,” Phys. Rev. Lett. 86, 2353–2356 (2001). [CrossRef] [PubMed]

5.

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

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]

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. 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]

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 73046602 (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]

44.

J. Guckenheimer and P. Holmes, ”Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,” Applied Mathematical Series 42, Springer (New York, Berlin) (1983).

45.

S. Wiggins, ”Introduction to Applied Nonlinear Dynamical Systems and Chaos,” Texts in Applied Mathematics 2, Springer (New York, Berlin) (1990).

46.

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

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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References

  1. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature 386, 143149 (1997). [CrossRef]
  2. P. Russel, "Photonic crystal fibers," Science 299, 358-362 (2003). [CrossRef]
  3. B.P. Anderson and M. A. Kasevich, "Macroscopic quantum interference from atomic tunnel arrays," Science 282, 1686-1689 (1998). [CrossRef] [PubMed]
  4. A. Trombettoni and A. Smerzi, "Discrete solitons and breathers with dilute BoseEinstein condensates," Phys. Rev. Lett. 86, 2353-2356 (2001). [CrossRef] [PubMed]
  5. D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003). [CrossRef] [PubMed]
  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]
  17. S. Trillo and W. Torruellas, Discrete Solitons, (Springer-Verlag, Berlin, 2001).
  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]
  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, 2695-2716 (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]
  44. J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields," Applied Mathematical Series 42, (Springer, New York, Berlin, 1983).
  45. S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," Texts in Applied Mathematics 2, (Springer, New York, Berlin, 1990).
  46. T. Kapitula, "Stability of waves in perturbed Hamiltonian systems," Physica D 156, 186-200 (2001). [CrossRef]

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