Spatiotemporal surface solitons in two-dimensional ... - CiteSeerX
November 1, 2007 / Vol. 32, No. 21 / OPTICS LETTERS
3173
Spatiotemporal surface solitons in two-dimensional photonic lattices Dumitru Mihalache,1 Dumitru Mazilu,1 Falk Lederer,2 and Yuri S. Kivshar3,* 1
Horia Hulubei National Institute for Physics and Nuclear Engineering (IFIN-HH), 407 Atomistilor, Magurele-Bucharest 077125, Romania 2 Institute of Solid State Theory and Theoretical Optics, Friedrich-Schiller Universität Jena, Max-Wien-Platz 1, D-077743 Jena, Germany 3 Nonlinear Physics Center, Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia *Corresponding author: [email protected] Received August 15, 2007; accepted September 4, 2007; posted September 28, 2007 (Doc. ID 86408); published October 25, 2007 We analyze spatiotemporal light localization in truncated two-dimensional photonic lattices and demonstrate the existence of two-dimensional surface light bullets localized in the lattice corners or the edges. We study the families of the spatiotemporal surface solitons and their properties such as bistability and compare them with the modes located deep inside the photonic lattice. © 2007 Optical Society of America OCIS codes: 190.4420, 190.5530, 190.5940.
Theoretical studies of discrete surface solitons localized in the corners or at the edges of two-dimensional photonic lattices [1–3] and recent observations of two-dimensional surface solitons in optically induced photonic lattices [4] and laser-written waveguide arrays in fused silica [5] demonstrated novel features of these nonlinear surface modes in comparison with their counterparts in one-dimensional waveguide arrays [6–8]. In particular, in a sharp contrast with one-dimensional discrete surface solitons, the mode threshold is lower at the surface than in a bulk making the mode excitation easier [2]. Recently, we have studied spatiotemporal discrete localization near surfaces and suggested the concept of spatiotemporal surface solitons [9]. These solitons provide an important extension of bulk spatiotemporal optical solitons [10], often referred to as light bullets in the three-dimensional case. The study of the properties of light bullets attracted the attention of many research groups as an unique opportunity to create self-supporting fully localized objects in both space and time [10]. In this Letter, we extend these important concepts to the case of two-dimensional photonic lattices, demonstrate the existence of twodimensional surface light bullets localized in the corners or at the edges of the lattice, and describe their properties such as bistability. We consider light propagation in two-dimensional photonic lattices applying the coupled-mode theory. In the discrete model the electric field is decomposed into modes of the identical waveguides with the mode profiles e共x , y , 0兲 and the normalized (with respect to the coupling constant) propagation constant kz at the center frequency of the pulse 0 as E共x , y , z , t兲 = 兺n,mEn,m共z , t兲e共x , y , 0兲exp关i共kzz − 0t兲兴 + c.c., where z is normalized with respect to the coupling constant, which has assumed to be equal, and t is the time normalized with respect to the ratio of group velocity dispersion and coupling constant. The slowly varying normalized envelope in the waveguide 共n , m兲 is described by the equation 0146-9592/07/213173-3/$15.00
i
En,m z
−␥
2En,m t2
+ 共Vn + Vm + 兩En,m兩2兲En,m = 0, 共1兲
where the lattice indices n , m = 0 , 1 , . . ., and E−1,m = En,−1 ⬅ 0 due to the lattice termination [Figs. 1(a)–1(c)]. Here ␥ is the dispersion coefficient, and = ± 1 is for either focusing or defocusing nonlinearity. We define the lattice couplings as VnEn,m = E1,m, for n = 0, m 艌 0 and VnEn,m = En+1,m + En−1,m, for n ⬎ 0, respectively, VmEn,m = En,1 for m = 0, n 艌 0 and VmEn,m = En,m+1 + En,m−1, for m ⬎ 0. We are looking for spatiotemporal soliton solutions of this nonlinear model in the form En,m共t ; z兲 = En,m共t兲exp共iz兲, where  is the nonlinearity-induced shift of the propagation constant (soliton family parameter), and the envelope En,m共t兲 describes the temporal shape of the solitonlike pulse at the 共n , m兲 lattice site. We find different families of localized surface solitons by solving the stationary version of Eq. (1) by means of a standard band-matrix algo-
Fig. 1. (Color online) Top: (a)–(c) Examples of the modes localized in the lattice corner, at the edge, and in the center of the lattice. Bottom: Spatial cross sections of the corresponding stable spatiotemporal surface solitons. The modes correspond to points (a)–(c) in Fig. 3(c) for (a)  = 5, P = 9.5; (b)  = 5.5, P = 10.09; and (c)  = 6, P = 10.66. © 2007 Optical Society of America
3174
OPTICS LETTERS / Vol. 32, No. 21 / November 1, 2007
rithm [9] applied to the corresponding two-point boundary-value problem
␥
d2En,m dt2
= 共−  + Vn + Vm + 兩En,m兩2兲En,m .
共2兲
The stationary solutions of Eq. (2) become less localized near the minimum (cutoff) value of the propagation constant 共co = 4兲. Therefore, depending on the value of the propagation constant, we use up to 301 discretization points in the continuous time interval 关0 , tmax兴, and up to 35⫻ 35 grid points for the discrete spatial coordinates. Figures 1 and 2 show typical examples of spatiotemporal continuous-discrete surface localized states located in the corners or at the edges of the lattice, together with the central mode representing a spatiotemporal discrete soliton in an infinite two-dimensional lattice, for the focusing nonlinearity 共 = + 1兲. To make a preliminary conclusion about the linear stability of the surface states found numerically, we calculate the total mode power [Figs. 3(a) and 3(d)], P共兲 =
兺
n,m
再
冕
+⬁
兩En,m共兲兩2dt,
−⬁
and the system Hamiltonian H共P兲 [Fig. 3(b)], H=−
兺 n,m
+␥
冕
+⬁
−⬁
* * 关En,m共En+1,m + En,m+1 兲 + c.c.兴
冏 冏 En,m t
2
1
冎
+ 兩En,m兩4 dt. 2
Figures 3(a)–3(d) present several one-parameter families of spatiotemporal continuous-discrete surface solitons found numerically for the modes localized in the corner, at the edge, and in the center of a two-dimensional lattice. The families are characterized by the dependencies P = P共兲 and H = H共兲, as well as the dependence of the peak amplitude max兩En,m兩 on the total power. Similar to the previous studies of continuous-discrete spatial solitons in similar systems [9,11], we expect that stable spatiotemporal solitons should correspond to the lower branch of the dependence H = H共P兲. The typical single-cusp behavior of the dependence H = H共P兲 is shown in Fig. 3(b), where the lower (blue) branches correspond to the stable surface modes. This observation is fully confirmed by direct simulations of the
Fig. 2. (Color online) Temporal cross sections of spatiotemporal surface solitons localized in the corner, at the edge, and in the center of the lattice, respectively. Shown are the profiles of stable (higher curves) and unstable (lower curves) spatiotemporal solitons corresponding to points (a)–(f) in Fig. 3(c).
Fig. 3. (Color online) Families of spatiotemporal surface solitons in two-dimensional lattices for the modes localized in the corner (co), at the edge (ed), and in the center (ce), respectively. (a), (d) Power versus propagation constant for lower and higher powers, respectively, (b) Hamiltonian versus power, (c) peak amplitude versus power. Points (a)–(f) mark the examples shown in Figs. 1 and 2.
propagation of the stationary solitons perturbed by a white input noise, as discussed below. In addition, we find higher-order spatiotemporal solitons [see Fig. 3(d)], which are all unstable. Figure 3(c) shows the power dependence of the peak amplitude of the stationary spatiotemporal solitons. As expected, for a fixed power there exist two stationary solitons, stable and unstable ones, the stable soliton having the peak amplitude larger than the unstable one and, correspondingly, a smaller width. The threshold power Pth for the spatiotemporal surface soliton generated in a two-dimensional photonic lattice is smaller than that corresponding to the spatiotemporal soliton located far away from the lattice edges, with the corner surface soliton having the smallest threshold energy: Pth = 9.317 (corner), Pth = 9.933 (edge), and Pth = 10.553 (central). This conclusion is similar to the case of two-dimensional discrete solitons [1,2]. These numerical results should be compared with the value of the threshold power of discrete surface light bullets in one-dimensional photonic lattices [9] Pth ⯝ 7.55. The stability results following directly from the analysis of the dependence H = H共P兲 shown in Fig. 3(b), have been cross-checked by direct numerical simulations of the dynamic equation (1) carried out by means of the Crank–Nicholson scheme; transparent boundary conditions were implemented in order to permit the escape of radiation from the computation window. The system of nonlinear finitedifference equations is solved first by means of the Picard iteration method (see details in [9]), and the resulting linear system is treated using the Gauss– Seidel iterative scheme. For a good convergence we
November 1, 2007 / Vol. 32, No. 21 / OPTICS LETTERS
Fig. 4. (Color online) Evolution of the soliton amplitude versus propagation distance for (a) stable corner (co) 共 = 5.0兲, and edge (ed) 共 = 5.5兲 spatiotemporal surface solitons perturbed by a white input noise, and (b) unstable, unperturbed corner (co) 共 = 4.1兲 and edge (ed) 共 = 4.3兲 spatiotemporal surface solitons.
3175
This reshaping to the stable modes is clearly observed in Figs. 5(a) and 5(b) where we show the hopping of the unstable flat-top-like (0,0) corner soliton from the upper “co” branch in Fig. 3(d) (for  = 5 and E = 28.77) to a stable (1,1) corner soliton of the lower branch. Figures 5(c) and 5(d) show an example of reshaping of an unstable edge soliton of the “ed” branch of Fig. 3(d) (for  = 5.5 and E = 18.31) into a stable edge soliton of the same branch. In conclusion, we have predicted theoretically the existence of spatiotemporal surface solitons localized in the corners and at the edges of two-dimensional photonic lattices. Such continuous-discrete localized states are similar to the spatiotemporal solitons localized far away from the lattice edges, and they differ substantially from the properties of onedimensional discrete surface solitons. This work was supported by the Australian Research Council and the German Federal Ministry of Education and Research (FKZ 13N8340/1) and the Deutsche Forschungsgemeinschaft (Priority Program SPP1113, FOR557 and Research Unit 532). References
Fig. 5. (Color online) (a), (b) Evolution of the unstable flattop corner soliton into a stable corner soliton. (c), (d) Reshaping of an unstable edge soliton into a stable edge soliton.
typically need five Picard iterations and six Gauss– Seidel iterations. We employ a temporal grid with the step length ⌬t = 0.05 and use a typical longitudinal step size of ⌬z = 0.001. The results of the direct propagation simulations are found to be in agreement with the predictions of the power diagram 共H , P兲. Thus, we find that stable spatiotemporal surface solitons resist a 10% input white noise, see Fig. 4(a), whereas unstable solitons reshape and transform into stable solitons pertaining to the lower (stable) branch of the soliton family, by increasing their peak amplitude during this process, Fig. 4(b).
1. K. G. Makris, J. Hudock, D. N. Christodoulides, G. Stegeman, O. Manela, and M. Segev, Opt. Lett. 31, 2774 (2006). 2. R. A. Vicencio, S. Flach, M. I. Molina, and Yu. S. Kivshar, Phys. Lett. A 364, 274 (2007). 3. H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Franzeskakis, Phys. Rev. E 75, 056605 (2007). 4. X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, Phys. Rev. Lett. 98, 123903 (2007). 5. A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and L. Torner, Phys. Rev. Lett. 98, 173903 (2007). 6. K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Haché, Opt. Lett. 30, 2466 (2005). 7. S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Haché, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, Phys. Rev. Lett. 96, 063901 (2006). 8. M. Molina, R. Vicencio, and Yu. S. Kivshar, Opt. Lett. 31, 1693 (2006). 9. D. Mihalache, D. Mazilu, F. Lederer, and Yu. S. Kivshar, Opt. Express 15, 589 (2007). 10. See the review paper: B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, J. Opt. B: Quantum Semiclassical Opt. 7, R53 (2005). 11. E. W. Laedke, K. H. Spatschek, S. K. Turitsyn, and V. K. Mezentsev, Phys. Rev. E 52, 5549 (1995).
3173
Spatiotemporal surface solitons in two-dimensional photonic lattices Dumitru Mihalache,1 Dumitru Mazilu,1 Falk Lederer,2 and Yuri S. Kivshar3,* 1
Horia Hulubei National Institute for Physics and Nuclear Engineering (IFIN-HH), 407 Atomistilor, Magurele-Bucharest 077125, Romania 2 Institute of Solid State Theory and Theoretical Optics, Friedrich-Schiller Universität Jena, Max-Wien-Platz 1, D-077743 Jena, Germany 3 Nonlinear Physics Center, Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia *Corresponding author: [email protected] Received August 15, 2007; accepted September 4, 2007; posted September 28, 2007 (Doc. ID 86408); published October 25, 2007 We analyze spatiotemporal light localization in truncated two-dimensional photonic lattices and demonstrate the existence of two-dimensional surface light bullets localized in the lattice corners or the edges. We study the families of the spatiotemporal surface solitons and their properties such as bistability and compare them with the modes located deep inside the photonic lattice. © 2007 Optical Society of America OCIS codes: 190.4420, 190.5530, 190.5940.
Theoretical studies of discrete surface solitons localized in the corners or at the edges of two-dimensional photonic lattices [1–3] and recent observations of two-dimensional surface solitons in optically induced photonic lattices [4] and laser-written waveguide arrays in fused silica [5] demonstrated novel features of these nonlinear surface modes in comparison with their counterparts in one-dimensional waveguide arrays [6–8]. In particular, in a sharp contrast with one-dimensional discrete surface solitons, the mode threshold is lower at the surface than in a bulk making the mode excitation easier [2]. Recently, we have studied spatiotemporal discrete localization near surfaces and suggested the concept of spatiotemporal surface solitons [9]. These solitons provide an important extension of bulk spatiotemporal optical solitons [10], often referred to as light bullets in the three-dimensional case. The study of the properties of light bullets attracted the attention of many research groups as an unique opportunity to create self-supporting fully localized objects in both space and time [10]. In this Letter, we extend these important concepts to the case of two-dimensional photonic lattices, demonstrate the existence of twodimensional surface light bullets localized in the corners or at the edges of the lattice, and describe their properties such as bistability. We consider light propagation in two-dimensional photonic lattices applying the coupled-mode theory. In the discrete model the electric field is decomposed into modes of the identical waveguides with the mode profiles e共x , y , 0兲 and the normalized (with respect to the coupling constant) propagation constant kz at the center frequency of the pulse 0 as E共x , y , z , t兲 = 兺n,mEn,m共z , t兲e共x , y , 0兲exp关i共kzz − 0t兲兴 + c.c., where z is normalized with respect to the coupling constant, which has assumed to be equal, and t is the time normalized with respect to the ratio of group velocity dispersion and coupling constant. The slowly varying normalized envelope in the waveguide 共n , m兲 is described by the equation 0146-9592/07/213173-3/$15.00
i
En,m z
−␥
2En,m t2
+ 共Vn + Vm + 兩En,m兩2兲En,m = 0, 共1兲
where the lattice indices n , m = 0 , 1 , . . ., and E−1,m = En,−1 ⬅ 0 due to the lattice termination [Figs. 1(a)–1(c)]. Here ␥ is the dispersion coefficient, and = ± 1 is for either focusing or defocusing nonlinearity. We define the lattice couplings as VnEn,m = E1,m, for n = 0, m 艌 0 and VnEn,m = En+1,m + En−1,m, for n ⬎ 0, respectively, VmEn,m = En,1 for m = 0, n 艌 0 and VmEn,m = En,m+1 + En,m−1, for m ⬎ 0. We are looking for spatiotemporal soliton solutions of this nonlinear model in the form En,m共t ; z兲 = En,m共t兲exp共iz兲, where  is the nonlinearity-induced shift of the propagation constant (soliton family parameter), and the envelope En,m共t兲 describes the temporal shape of the solitonlike pulse at the 共n , m兲 lattice site. We find different families of localized surface solitons by solving the stationary version of Eq. (1) by means of a standard band-matrix algo-
Fig. 1. (Color online) Top: (a)–(c) Examples of the modes localized in the lattice corner, at the edge, and in the center of the lattice. Bottom: Spatial cross sections of the corresponding stable spatiotemporal surface solitons. The modes correspond to points (a)–(c) in Fig. 3(c) for (a)  = 5, P = 9.5; (b)  = 5.5, P = 10.09; and (c)  = 6, P = 10.66. © 2007 Optical Society of America
3174
OPTICS LETTERS / Vol. 32, No. 21 / November 1, 2007
rithm [9] applied to the corresponding two-point boundary-value problem
␥
d2En,m dt2
= 共−  + Vn + Vm + 兩En,m兩2兲En,m .
共2兲
The stationary solutions of Eq. (2) become less localized near the minimum (cutoff) value of the propagation constant 共co = 4兲. Therefore, depending on the value of the propagation constant, we use up to 301 discretization points in the continuous time interval 关0 , tmax兴, and up to 35⫻ 35 grid points for the discrete spatial coordinates. Figures 1 and 2 show typical examples of spatiotemporal continuous-discrete surface localized states located in the corners or at the edges of the lattice, together with the central mode representing a spatiotemporal discrete soliton in an infinite two-dimensional lattice, for the focusing nonlinearity 共 = + 1兲. To make a preliminary conclusion about the linear stability of the surface states found numerically, we calculate the total mode power [Figs. 3(a) and 3(d)], P共兲 =
兺
n,m
再
冕
+⬁
兩En,m共兲兩2dt,
−⬁
and the system Hamiltonian H共P兲 [Fig. 3(b)], H=−
兺 n,m
+␥
冕
+⬁
−⬁
* * 关En,m共En+1,m + En,m+1 兲 + c.c.兴
冏 冏 En,m t
2
1
冎
+ 兩En,m兩4 dt. 2
Figures 3(a)–3(d) present several one-parameter families of spatiotemporal continuous-discrete surface solitons found numerically for the modes localized in the corner, at the edge, and in the center of a two-dimensional lattice. The families are characterized by the dependencies P = P共兲 and H = H共兲, as well as the dependence of the peak amplitude max兩En,m兩 on the total power. Similar to the previous studies of continuous-discrete spatial solitons in similar systems [9,11], we expect that stable spatiotemporal solitons should correspond to the lower branch of the dependence H = H共P兲. The typical single-cusp behavior of the dependence H = H共P兲 is shown in Fig. 3(b), where the lower (blue) branches correspond to the stable surface modes. This observation is fully confirmed by direct simulations of the
Fig. 2. (Color online) Temporal cross sections of spatiotemporal surface solitons localized in the corner, at the edge, and in the center of the lattice, respectively. Shown are the profiles of stable (higher curves) and unstable (lower curves) spatiotemporal solitons corresponding to points (a)–(f) in Fig. 3(c).
Fig. 3. (Color online) Families of spatiotemporal surface solitons in two-dimensional lattices for the modes localized in the corner (co), at the edge (ed), and in the center (ce), respectively. (a), (d) Power versus propagation constant for lower and higher powers, respectively, (b) Hamiltonian versus power, (c) peak amplitude versus power. Points (a)–(f) mark the examples shown in Figs. 1 and 2.
propagation of the stationary solitons perturbed by a white input noise, as discussed below. In addition, we find higher-order spatiotemporal solitons [see Fig. 3(d)], which are all unstable. Figure 3(c) shows the power dependence of the peak amplitude of the stationary spatiotemporal solitons. As expected, for a fixed power there exist two stationary solitons, stable and unstable ones, the stable soliton having the peak amplitude larger than the unstable one and, correspondingly, a smaller width. The threshold power Pth for the spatiotemporal surface soliton generated in a two-dimensional photonic lattice is smaller than that corresponding to the spatiotemporal soliton located far away from the lattice edges, with the corner surface soliton having the smallest threshold energy: Pth = 9.317 (corner), Pth = 9.933 (edge), and Pth = 10.553 (central). This conclusion is similar to the case of two-dimensional discrete solitons [1,2]. These numerical results should be compared with the value of the threshold power of discrete surface light bullets in one-dimensional photonic lattices [9] Pth ⯝ 7.55. The stability results following directly from the analysis of the dependence H = H共P兲 shown in Fig. 3(b), have been cross-checked by direct numerical simulations of the dynamic equation (1) carried out by means of the Crank–Nicholson scheme; transparent boundary conditions were implemented in order to permit the escape of radiation from the computation window. The system of nonlinear finitedifference equations is solved first by means of the Picard iteration method (see details in [9]), and the resulting linear system is treated using the Gauss– Seidel iterative scheme. For a good convergence we
November 1, 2007 / Vol. 32, No. 21 / OPTICS LETTERS
Fig. 4. (Color online) Evolution of the soliton amplitude versus propagation distance for (a) stable corner (co) 共 = 5.0兲, and edge (ed) 共 = 5.5兲 spatiotemporal surface solitons perturbed by a white input noise, and (b) unstable, unperturbed corner (co) 共 = 4.1兲 and edge (ed) 共 = 4.3兲 spatiotemporal surface solitons.
3175
This reshaping to the stable modes is clearly observed in Figs. 5(a) and 5(b) where we show the hopping of the unstable flat-top-like (0,0) corner soliton from the upper “co” branch in Fig. 3(d) (for  = 5 and E = 28.77) to a stable (1,1) corner soliton of the lower branch. Figures 5(c) and 5(d) show an example of reshaping of an unstable edge soliton of the “ed” branch of Fig. 3(d) (for  = 5.5 and E = 18.31) into a stable edge soliton of the same branch. In conclusion, we have predicted theoretically the existence of spatiotemporal surface solitons localized in the corners and at the edges of two-dimensional photonic lattices. Such continuous-discrete localized states are similar to the spatiotemporal solitons localized far away from the lattice edges, and they differ substantially from the properties of onedimensional discrete surface solitons. This work was supported by the Australian Research Council and the German Federal Ministry of Education and Research (FKZ 13N8340/1) and the Deutsche Forschungsgemeinschaft (Priority Program SPP1113, FOR557 and Research Unit 532). References
Fig. 5. (Color online) (a), (b) Evolution of the unstable flattop corner soliton into a stable corner soliton. (c), (d) Reshaping of an unstable edge soliton into a stable edge soliton.
typically need five Picard iterations and six Gauss– Seidel iterations. We employ a temporal grid with the step length ⌬t = 0.05 and use a typical longitudinal step size of ⌬z = 0.001. The results of the direct propagation simulations are found to be in agreement with the predictions of the power diagram 共H , P兲. Thus, we find that stable spatiotemporal surface solitons resist a 10% input white noise, see Fig. 4(a), whereas unstable solitons reshape and transform into stable solitons pertaining to the lower (stable) branch of the soliton family, by increasing their peak amplitude during this process, Fig. 4(b).
1. K. G. Makris, J. Hudock, D. N. Christodoulides, G. Stegeman, O. Manela, and M. Segev, Opt. Lett. 31, 2774 (2006). 2. R. A. Vicencio, S. Flach, M. I. Molina, and Yu. S. Kivshar, Phys. Lett. A 364, 274 (2007). 3. H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Franzeskakis, Phys. Rev. E 75, 056605 (2007). 4. X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, Phys. Rev. Lett. 98, 123903 (2007). 5. A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and L. Torner, Phys. Rev. Lett. 98, 173903 (2007). 6. K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Haché, Opt. Lett. 30, 2466 (2005). 7. S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Haché, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, Phys. Rev. Lett. 96, 063901 (2006). 8. M. Molina, R. Vicencio, and Yu. S. Kivshar, Opt. Lett. 31, 1693 (2006). 9. D. Mihalache, D. Mazilu, F. Lederer, and Yu. S. Kivshar, Opt. Express 15, 589 (2007). 10. See the review paper: B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, J. Opt. B: Quantum Semiclassical Opt. 7, R53 (2005). 11. E. W. Laedke, K. H. Spatschek, S. K. Turitsyn, and V. K. Mezentsev, Phys. Rev. E 52, 5549 (1995).
