Abstract

It is commonly known that stable bright solitons in periodic potentials, which represent gratings in photonics/plasmonics, or optical lattices in quantum gases, exist either in the spectral semi-infinite gap (SIG) or in finite bandgaps. Using numerical methods, we demonstrate that, under the action of the cubic self-focusing nonlinearity, defects in the form of “holes” in two-dimensional (2D) lattices support continuous families of 2D solitons embedded into the first two Bloch bands of the respective linear spectrum, where solitons normally do not exist. The two families of the embedded defect solitons (EDSs) are found to be continuously linked by the branch of gap defect solitons (GDSs) populating the first finite bandgap. Further, the EDS branch traversing the first band links the GDS family with the branch of regular defect-supported solitons populating the SIG. Thus, we construct a continuous chain of regular, embedded, and gap-mode solitons (“superfamily”) threading the entire bandgap structure considered here. The EDSs are stable in the first Bloch band, and partly stable in the second. They exist with the norm exceeding a minimum value; hence they do not originate from linear defect modes. Further, we demonstrate that double, triple, and quadruple lattice defects support stable dipole-mode solitons and vortices.

© 2013 Optical Society of America

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2013

F. C. Moreira, V. V. Konotop, and B. A. Malomed, “Solitons in PT-symmetric periodic systems with the quadratic nonlinearity,” Phys. Rev. A 87, 013832 (2013).
[CrossRef]

S. Hu and W. Hu, “Defect solitons in parity-time symmetric optical lattices with self-defocusing nonlinearity,” Opt. Commun. 294, 311–324 (2013).
[CrossRef]

2012

J. Zeng and Y. Lan, “Two-dimensional solitons in PT linear lattice potentials,” Phys. Rev. E 85, 047601 (2012).
[CrossRef]

V. A. Brazhnyi and B. A. Malomed, “Symmetric and asymmetric localized modes in linear lattices with an embedded pair of χ(2)-nonlinear sites,” Phys. Rev. A 86, 013829 (2012).
[CrossRef]

F. C. Moreira, F. K. Abdullaev, V. V. Konotop, and A. V. Yulin, “Localized modes in χ(2) media with PT—symmetric localized potential,” Phys. Rev. A 86, 053815 (2012).
[CrossRef]

J. Zeng and B. A. Malomed, “Stabilization of one-dimensional solitons against the critical collapse by quintic nonlinear lattices,” Phys. Rev. A 85, 023824 (2012).
[CrossRef]

J. Zeng and B. A. Malomed, “Two-dimensional solitons and vortices in media with incommensurate linear and nonlinear lattice potentials,” Phys. Scr. T149, 014035 (2012).
[CrossRef]

I. Dolev, I. Epstein, and A. Arie, “Surface-plasmon holographic beam shaping,” Phys. Rev. Lett. 109, 203903 (2012).
[CrossRef]

L. Dal Negro and S. V. Boriskina, “Deterministic aperiodic nanostructures for photonics and plasmonics applications,” Laser Photon. Rev. 6, 178–218 (2012).
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2011

G. D’Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. A. Vincenti, and A. Alú, “Transmission resonances in plasmonic metallic gratings,” J. Opt. Soc. Am. B 28, 253–264 (2011).
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D. de Ceglia, M. A. Vincenti, M. Scalora, N. Akozbek, and M. J. Bloemer, “Plasmonic band edge effects on the transmission properties of metal gratings,” AIP Adv. 1, 032151 (2011).
[CrossRef]

F. H. Bennet, T. J. Alexander, F. Haslinger, A. Mitchell, D. N. Neshev, and Y. S. Kivshar, “Observation of nonlinear self-trapping of broad beams in defocusing waveguide arrays,” Phys. Rev. Lett. 106, 093901 (2011).
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E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical coupling in a two-dimensional photonic crystal defect cavity,” Phys. Rev. Lett. 106, 203902 (2011).
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M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, “Amorphous photonic lattices: band gaps, effective mass, and suppressed transport,” Phys. Rev. Lett. 106, 193904 (2011).
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E. Bulgakov, K. Pichugin, and A. Sadreev, “Symmetry breaking for transmission in a photonic waveguide coupled with two off-channel nonlinear defects,” Phys. Rev. B 83, 045109 (2011).
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V. A. Brazhnyi and B. A. Malomed, “Localization and delocalization of two-dimensional discrete solitons pinned to linear and nonlinear defects,” Phys. Rev. E 83, 016604 (2011).
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V. A. Brazhnyi and B. A. Malomed, “Dragging two-dimensional discrete solitons by moving linear defects,” Phys. Rev. E 84, 016608 (2011).
[CrossRef]

N. Dror and B. A. Malomed, “Solitons supported by localized nonlinearities in periodic media,” Phys. Rev. A 83, 033828 (2011).
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T. Mayteevarunyoo, B. A. Malomed, and A. Reoksabutr, “Spontaneous symmetry breaking of photonic and matter waves in two-dimensional pseudopotentials,” J. Mod. Opt. 58, 1977–1989 (2011).
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Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
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S. Longhi, “Dynamic reflectionless defects in tight-binding lattices,” Phys. Rev. B 84, 193105 (2011).
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A. Shapira, N. Voloch-Bloch, B. A. Malomed, and A. Arie, “Spatial quadratic solitons guided by narrow layers of a nonlinear material,” J. Opt. Soc. Am. B 28, 1481–1489 (2011).
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2010

L. Sanchez-Palencia and M. Lewenstein, “Disordered quantum gases under control,” Nat. Phys. 6, 87–95 (2010).
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Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and L. Torner, “Dissipative defect modes in periodic structures,” Opt. Lett. 35, 1638–1640 (2010).
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S. Longhi, “Bloch oscillations in tight-binding lattices with defects,” Phys. Rev. B 81, 195118 (2010).
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J. Yang, “Fully localized two-dimensional embedded solitons,” Phys. Rev. A 82, 053828 (2010).
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P. Genevet, J. P. Tetienne, E. Gatzogiannis, R. Blanchard, M. A. Kats, M. O. Scully, and F. Capasso, “Large enhancement of nonlinear optical phenomena by plasmonic nanocavity gratings,” Nano Lett. 10, 4880–4883 (2010).
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2009

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Progress in Optics 52, 63–148 (2009).
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A. Abdelrahman, P. Hannaford, and K. Alameh, “Adiabatically induced coherent Josephson oscillations of ultracold atoms in an asymmetric two-dimensional magnetic lattice,” Opt. Express 17, 24358–24370 (2009).
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H. Sakaguchi and B. A. Malomed, “Two-dimensional dissipative gap solitons,” Phys. Rev. E 80, 026606 (2009).
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2008

H. Sakaguchi and B. A. Malomed, “Gap solitons in Ginzburg-Landau media,” Phys. Rev. E 77, 056606 (2008).
[CrossRef]

R. Driben and B. A. Malomed, “Stabilization of two-dimensional solitons and vortices against supercritical collapse by lattice potentials,” Eur. Phys. J. D 50, 317–323 (2008).
[CrossRef]

S. K. Adhikari and B. A. Malomed, “Symbiotic gap and semigap solitons in Bose-Einstein condensates,” Phys. Rev. A 77, 023607 (2008).
[CrossRef]

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, “Direct observation of Anderson localization of matter waves in a controlled disorder,” Nature 453, 891–894 (2008).
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G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, “Anderson localization of a non-interacting Bose–Einstein condensate,” Nature 453, 895–898 (2008).
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Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
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J. Park, H. Kim, and B. Lee, “High order plasmonic Bragg reflection in the metal-insulator-metal waveguide Bragg grating,” Opt. Express 16, 413–425 (2008).
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T. Mayteevarunyoo, B. A. Malomed, and G. Dong, “Spontaneous symmetry breaking in a nonlinear double-well structure,” Phys. Rev. A 78, 053601 (2008).
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2007

S. A. Rinne, F. Garca-Santamara, and P. V. Braun, “Embedded cavities and waveguides in three-dimensional silicon photonic crystals,” Nat. Photonics 2, 52–56 (2007).
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Z. Han, E. Forsberg, and S. He, “Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides,” IEEE Photon. Technol. Lett. 19, 91–93 (2007).
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A. Gubeskys and B. A. Malomed, “Spontaneous soliton symmetry breaking in two-dimensional coupled Bose-Einstein condensates supported by optical lattices,” Phys. Rev. A 76, 043623 (2007).
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T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
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Z. Shi and J. Yang, “Solitary waves bifurcated from Bloch-band edges in two-dimensional periodic media,” Phys. Rev. E 75, 056602 (2007).
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M. Matuszewski, B. A. Malomed, and M. Trippenbach, “Competition between attractive and repulsive interactions in two-component Bose-Einstein condensates trapped in an optical lattice,” Phys. Rev. A 76, 043826 (2007).
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2006

S. Ghanbari, T. D. Kieu, A. Sidorov, and P. Hannaford, “Permanent magnetic lattices for ultracold atoms and quantum degenerate gases,” J. Phys. B 39, 847–860 (2006).
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M. Storzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
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T. J. Alexander, E. A. Ostrovskaya, and Y. S. Kivshar, “Self-trapped nonlinear matter waves in periodic potentials,” Phys. Rev. Lett. 96, 040401 (2006).
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A. Boltasseva, S. I. Bozhevolnyi, T. Nikolajsen, and K. Leosson, “Compact Bragg gratings for long-range surface plasmon polaritons,” J. Lightwave Technol. 24, 912–918 (2006).
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O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys. 78, 179–215 (2006).
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J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E 73, 026609 (2006).
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I. Makasyuk, Z. Chen, and J. Yang, “Band-gap guidance in optically induced photonic lattices with a negative defect,” Phys. Rev. Lett. 96, 223903 (2006).
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B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440, 1166–1169 (2006).
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V. A. Brazhnyi, V. V. Konotop, and V. M. Pérez-Garca, “Defect modes of a Bose-Einstein condensate in an optical lattice with a localized impurity,” Phys. Rev. A 74, 023614 (2006).
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V. A. Brazhnyi, V. V. Konotop, and V. M. Pérez-Garca, “Driving defect modes of Bose-Einstein condensates in optical lattices,” Phys. Rev. Lett. 96, 060403 (2006).
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2005

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vučković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904 (2005).
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B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).
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T. Anker, M. Albiez, R. Gati, S. Hunsmann, B. Eiermann, A. Trombettoni, and M. K. Oberthaler, “Nonlinear self-trapping of matter waves in periodic potentials,” Phys. Rev. Lett. 94, 020403 (2005).
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U. Gavish and Y. Castin, “Matter-wave localization in disordered cold atom lattices,” Phys. Rev. Lett. 95, 020401 (2005).
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H. Sakaguchi and B. A. Malomed, “Higher-order vortex solitons, multipoles, and supervortices on a square optical lattice,” Europhys. Lett. 72, 698–704 (2005).
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2004

E. A. Ostrovskaya and Y. S. Kivshar, “Photonic crystals for matter waves: Bose-Einstein condensates in optical lattices,” Opt. Express 12, 19–29 (2004).
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H. Sakaguchi and B. A. Malomed, “Dynamics of positive- and negative-mass solitons in optical lattices and inverted traps,” J. Phys. B 37, 1443–1459 (2004).
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H. Sakaguchi and B. A. Malomed, “Two-dimensional loosely and tightly bound solitons in optical lattices and inverted traps,” J. Phys. B 37, 2225–2239 (2004).
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G. Van de Walle and J. Neugebauer, “First-principles calculations for defects and impurities: applications to III-nitrides,” J. Appl. Phys. 95, 3851–3879 (2004).
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V. A. Brazhnyi and V. V. Konotop, “Theory of nonlinear matter waves in optical lattices,” Mod. Phys. Lett. B 18, 627–651 (2004).
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K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, “Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes,” Phys. Rev. Lett. 92, 183901 (2004).
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J. G. Rivas, M. Kuttge, P. H. Bolivar, H. Kurz, and J. A. Sanchez-Gil, “Propagation of surface plasmon polaritons on semiconductor gratings,” Phys. Rev. Lett. 93, 256804 (2004).
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W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Symmetric and asymmetric solitons in linearly coupled Bragg gratings,” Phys. Rev. E 69, 066610 (2004).
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S. Ogawa, M. Imada, S. Yoshimoto, M. Okano, and S. Noda, “Control of light emission by 3D photonic crystals,” Science 305, 227–229 (2004).
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M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, “A three-dimensional optical photonic crystal with designed point defects,” Nature 429, 538–542 (2004).
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2003

J. Yang and T. R. Akylas, “Continuous families of embedded solitons in the third-order nonlinear Schrödinger equation,” Stud. Appl. Math. 111, 359–375 (2003).
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J. Yang, “Stable embedded solitons,” Phys. Rev. Lett. 91, 143903 (2003).
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A. Espinosa-Cerón, J. Fujioka, and A. Gómez-Rodrguez, “Embedded solitons: four-frequency radiation, front propagation and radiation inhibition,” Phys. Scr. 67, 314–324 (2003).
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R. F. Rodrguez, J. A. Reyes, A. Espinosa-Cerón, J. Fujioka, and B. A. Malomed, “Standard and embedded solitons in nematic optical fibers,” Phys. Rev. E 68, 036606 (2003).
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S. A. Darmanyan and A. V. Zayats, “Light tunneling via resonant surface plasmon polariton states and the enhanced transmission of periodically nanostructured metal films: an analytical study,” Phys. Rev. B 67, 035424 (2003).
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E. A. Ostrovskaya and Y. S. Kivshar, “Matter-wave gap solitons in atomic band-gap structures,” Phys. Rev. Lett. 90, 160407 (2003).
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J. Hukriede, D. Runde, and D. Kip, “Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides,” J. Phys. D 36, R1–R16 (2003).
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A. Khelif, A. Choujaa, B. Djafari-Rouhani, M. Wilm, S. Ballandras, and V. Laude, “Trapping and guiding of acoustic waves by defect modes in a full-band-gap ultrasonic crystal,” Phys. Rev. B 68, 214301 (2003).
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B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials,” Europhys. Lett. 63, 642–648 (2003).
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J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. 28, 2094–2096 (2003).
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2002

A. A. Sukhorukov and Y. S. Kivshar, “Spatial optical solitons in nonlinear photonic crystals,” Phys. Rev. E 65, 036609 (2002).
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F. J. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66, 155412 (2002).
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D. E. Pelinovsky and J. Yang, “A normal form for nonlinear resonance of embedded solitons,” Proc. R. Soc. A 458, 1469–1497 (2002).
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2001

J. Yang, “Dynamics of embedded solitons in the extended Korteweg–de Vries equations,” Stud. Appl. Math. 106, 337–365 (2001).
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A. R. Champneys, B. A. Malomed, J. Yang, and D. J. Kaup, “Embedded solitons: solitary waves in resonance with the linear spectrum,” Physica D 152, 340–354 (2001).
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J. Atai and B. A. Malomed, “Solitary waves in systems with separated Bragg grating and nonlinearity,” Phys. Rev. E 64, 066617 (2001).
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V. Yannopapas, N. Stefanou, and A. Modinos, “Effect of stacking faults on the optical properties of inverted opals,” Phys. Rev. Lett. 86, 4811–4814 (2001).
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A. A. Sukhorukov and Y. S. Kivshar, “Nonlinear localized waves in a periodic medium,” Phys. Rev. Lett. 87, 083901 (2001).
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M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
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C. Vanneste and P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. 87, 183903 (2001).
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2000

H. Cao, J. Y. Xu, E. W. Seelig, and R. P. H. Chang, “Microlaser made of disordered media,” Appl. Phys. Lett. 76, 2997–2999 (2000).
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G. Kaczmarczyk, A. Kaschner, A. Hoffmann, and C. Thomsen, “Impurity-induced modes of Mg, As, Si, and C in hexagonal and cubic GaN,” Phys. Rev. B 61, 5353–5357 (2000).
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M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. Lett. 84, 2140–2143 (2000).
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S. Noda, A. Chutinan, and M. Imada, “Trapping, and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608–610 (2000).
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1999

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
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J. Yang, B. A. Malomed, and D. J. Kaup, “Embedded solitons in second-harmonic-generating systems,” Phys. Rev. Lett. 83, 1958–1961 (1999).
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A. R. Champneys and B. A. Malomed, “Moving embedded solitons,” J. Phys. A 32, L547–L553 (1999).
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J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999).
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O. Painter, J. Vučković, and A. Scherer, “Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab,” J. Opt. Soc. Am. B 16, 275–285 (1999).
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M. C. W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313–371 (1999).
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Y. A. Vlasov, M. A. Kaliteevski, and V. V. Nikolaev, “Different regimes of light localization in a disordered photonic crystal,” Phys. Rev. B 60, 1555–1562 (1999).
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1998

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998).
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S. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
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Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. 80, 956–959 (1998).
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1997

B. C. Gupta and K. Kundu, “Formation of stationary localized states due to nonlinear impurities using the discrete nonlinear Schrödinger equation,” Phys. Rev. B 55, 894–905 (1997).
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B. C. Gupta and K. Kundu, “Stationary localized states due to a nonlinear dimeric impurity embedded in a perfect one-dimensional chain,” Phys. Rev. B 55, 11033–11036 (1997).
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J. Fujioka and A. Espinosa, “Soliton-like solution of an extended NLS equation existing in resonance with linear dispersive waves,” J. Phys. Soc. Jpn. 66, 2601–2607 (1997).
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1996

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
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1993

M. I. Molina and G. Tsironis, “Nonlinear impurities in a linear chain,” Phys. Rev. B 47, 15330–15333 (1993).
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1991

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S. R. Bickham and A. J. Sievers, “Intrinsic localized modes in a monatomic lattice with weakly anharmonic nearest-neighbor force constants,” Phys. Rev. B 43, 2339–2346 (1991).
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1973

M. Vakhitov and A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783–789 (1973).
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Adhikari, S. K.

S. K. Adhikari and B. A. Malomed, “Symbiotic gap and semigap solitons in Bose-Einstein condensates,” Phys. Rev. A 77, 023607 (2008).
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M. Storzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
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D. de Ceglia, M. A. Vincenti, M. Scalora, N. Akozbek, and M. J. Bloemer, “Plasmonic band edge effects on the transmission properties of metal gratings,” AIP Adv. 1, 032151 (2011).
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J. Yang and T. R. Akylas, “Continuous families of embedded solitons in the third-order nonlinear Schrödinger equation,” Stud. Appl. Math. 111, 359–375 (2003).
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T. Anker, M. Albiez, R. Gati, S. Hunsmann, B. Eiermann, A. Trombettoni, and M. K. Oberthaler, “Nonlinear self-trapping of matter waves in periodic potentials,” Phys. Rev. Lett. 94, 020403 (2005).
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F. H. Bennet, T. J. Alexander, F. Haslinger, A. Mitchell, D. N. Neshev, and Y. S. Kivshar, “Observation of nonlinear self-trapping of broad beams in defocusing waveguide arrays,” Phys. Rev. Lett. 106, 093901 (2011).
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T. Anker, M. Albiez, R. Gati, S. Hunsmann, B. Eiermann, A. Trombettoni, and M. K. Oberthaler, “Nonlinear self-trapping of matter waves in periodic potentials,” Phys. Rev. Lett. 94, 020403 (2005).
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D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vučković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904 (2005).
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E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical coupling in a two-dimensional photonic crystal defect cavity,” Phys. Rev. Lett. 106, 203902 (2011).
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J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, “Direct observation of Anderson localization of matter waves in a controlled disorder,” Nature 453, 891–894 (2008).
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J. Atai and B. A. Malomed, “Solitary waves in systems with separated Bragg grating and nonlinearity,” Phys. Rev. E 64, 066617 (2001).
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Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
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B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials,” Europhys. Lett. 63, 642–648 (2003).
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Ballandras, S.

A. Khelif, A. Choujaa, B. Djafari-Rouhani, M. Wilm, S. Ballandras, and V. Laude, “Trapping and guiding of acoustic waves by defect modes in a full-band-gap ultrasonic crystal,” Phys. Rev. B 68, 214301 (2003).
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T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
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B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440, 1166–1169 (2006).
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D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
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M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. Lett. 84, 2140–2143 (2000).
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F. H. Bennet, T. J. Alexander, F. Haslinger, A. Mitchell, D. N. Neshev, and Y. S. Kivshar, “Observation of nonlinear self-trapping of broad beams in defocusing waveguide arrays,” Phys. Rev. Lett. 106, 093901 (2011).
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V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998).
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J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, “Direct observation of Anderson localization of matter waves in a controlled disorder,” Nature 453, 891–894 (2008).
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Eur. Phys. J. D

R. Driben and B. A. Malomed, “Stabilization of two-dimensional solitons and vortices against supercritical collapse by lattice potentials,” Eur. Phys. J. D 50, 317–323 (2008).
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Europhys. Lett.

H. Sakaguchi and B. A. Malomed, “Higher-order vortex solitons, multipoles, and supervortices on a square optical lattice,” Europhys. Lett. 72, 698–704 (2005).
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B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials,” Europhys. Lett. 63, 642–648 (2003).
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IEEE Photon. Technol. Lett.

Z. Han, E. Forsberg, and S. He, “Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides,” IEEE Photon. Technol. Lett. 19, 91–93 (2007).
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J. Appl. Phys.

G. Van de Walle and J. Neugebauer, “First-principles calculations for defects and impurities: applications to III-nitrides,” J. Appl. Phys. 95, 3851–3879 (2004).
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J. Lightwave Technol.

J. Mod. Opt.

T. Mayteevarunyoo, B. A. Malomed, and A. Reoksabutr, “Spontaneous symmetry breaking of photonic and matter waves in two-dimensional pseudopotentials,” J. Mod. Opt. 58, 1977–1989 (2011).
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J. Opt. B

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B 7, R53–R72 (2005).
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J. Opt. Soc. Am. B

J. Phys. A

A. R. Champneys and B. A. Malomed, “Moving embedded solitons,” J. Phys. A 32, L547–L553 (1999).
[CrossRef]

J. Phys. B

H. Sakaguchi and B. A. Malomed, “Dynamics of positive- and negative-mass solitons in optical lattices and inverted traps,” J. Phys. B 37, 1443–1459 (2004).
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H. Sakaguchi and B. A. Malomed, “Two-dimensional loosely and tightly bound solitons in optical lattices and inverted traps,” J. Phys. B 37, 2225–2239 (2004).
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S. Ghanbari, T. D. Kieu, A. Sidorov, and P. Hannaford, “Permanent magnetic lattices for ultracold atoms and quantum degenerate gases,” J. Phys. B 39, 847–860 (2006).
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J. Phys. D

J. Hukriede, D. Runde, and D. Kip, “Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides,” J. Phys. D 36, R1–R16 (2003).
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J. Phys. Soc. Jpn.

J. Fujioka and A. Espinosa, “Soliton-like solution of an extended NLS equation existing in resonance with linear dispersive waves,” J. Phys. Soc. Jpn. 66, 2601–2607 (1997).
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Laser Photon. Rev.

L. Dal Negro and S. V. Boriskina, “Deterministic aperiodic nanostructures for photonics and plasmonics applications,” Laser Photon. Rev. 6, 178–218 (2012).
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Mod. Phys. Lett. B

V. A. Brazhnyi and V. V. Konotop, “Theory of nonlinear matter waves in optical lattices,” Mod. Phys. Lett. B 18, 627–651 (2004).
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Nano Lett.

P. Genevet, J. P. Tetienne, E. Gatzogiannis, R. Blanchard, M. A. Kats, M. O. Scully, and F. Capasso, “Large enhancement of nonlinear optical phenomena by plasmonic nanocavity gratings,” Nano Lett. 10, 4880–4883 (2010).
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Nat. Photonics

S. A. Rinne, F. Garca-Santamara, and P. V. Braun, “Embedded cavities and waveguides in three-dimensional silicon photonic crystals,” Nat. Photonics 2, 52–56 (2007).
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Nat. Phys.

L. Sanchez-Palencia and M. Lewenstein, “Disordered quantum gases under control,” Nat. Phys. 6, 87–95 (2010).
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Nature

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, “Direct observation of Anderson localization of matter waves in a controlled disorder,” Nature 453, 891–894 (2008).
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G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, “Anderson localization of a non-interacting Bose–Einstein condensate,” Nature 453, 895–898 (2008).
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T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
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D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
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B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440, 1166–1169 (2006).
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Opt. Commun.

S. Hu and W. Hu, “Defect solitons in parity-time symmetric optical lattices with self-defocusing nonlinearity,” Opt. Commun. 294, 311–324 (2013).
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Opt. Express

Opt. Lett.

Phys. Rev. A

M. Matuszewski, B. A. Malomed, and M. Trippenbach, “Competition between attractive and repulsive interactions in two-component Bose-Einstein condensates trapped in an optical lattice,” Phys. Rev. A 76, 043826 (2007).
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S. K. Adhikari and B. A. Malomed, “Symbiotic gap and semigap solitons in Bose-Einstein condensates,” Phys. Rev. A 77, 023607 (2008).
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J. Zeng and B. A. Malomed, “Stabilization of one-dimensional solitons against the critical collapse by quintic nonlinear lattices,” Phys. Rev. A 85, 023824 (2012).
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V. A. Brazhnyi and B. A. Malomed, “Symmetric and asymmetric localized modes in linear lattices with an embedded pair of χ(2)-nonlinear sites,” Phys. Rev. A 86, 013829 (2012).
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F. C. Moreira, F. K. Abdullaev, V. V. Konotop, and A. V. Yulin, “Localized modes in χ(2) media with PT—symmetric localized potential,” Phys. Rev. A 86, 053815 (2012).
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V. A. Brazhnyi, V. V. Konotop, and V. M. Pérez-Garca, “Defect modes of a Bose-Einstein condensate in an optical lattice with a localized impurity,” Phys. Rev. A 74, 023614 (2006).
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T. Mayteevarunyoo, B. A. Malomed, and G. Dong, “Spontaneous symmetry breaking in a nonlinear double-well structure,” Phys. Rev. A 78, 053601 (2008).
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N. Dror and B. A. Malomed, “Solitons supported by localized nonlinearities in periodic media,” Phys. Rev. A 83, 033828 (2011).
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F. C. Moreira, V. V. Konotop, and B. A. Malomed, “Solitons in PT-symmetric periodic systems with the quadratic nonlinearity,” Phys. Rev. A 87, 013832 (2013).
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J. Yang, “Fully localized two-dimensional embedded solitons,” Phys. Rev. A 82, 053828 (2010).
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A. Gubeskys and B. A. Malomed, “Spontaneous soliton symmetry breaking in two-dimensional coupled Bose-Einstein condensates supported by optical lattices,” Phys. Rev. A 76, 043623 (2007).
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Phys. Rev. B

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Phys. Rev. E

Z. Shi and J. Yang, “Solitary waves bifurcated from Bloch-band edges in two-dimensional periodic media,” Phys. Rev. E 75, 056602 (2007).
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W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Symmetric and asymmetric solitons in linearly coupled Bragg gratings,” Phys. Rev. E 69, 066610 (2004).
[CrossRef]

H. Sakaguchi and B. A. Malomed, “Gap solitons in Ginzburg-Landau media,” Phys. Rev. E 77, 056606 (2008).
[CrossRef]

H. Sakaguchi and B. A. Malomed, “Two-dimensional dissipative gap solitons,” Phys. Rev. E 80, 026606 (2009).
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J. Zeng and Y. Lan, “Two-dimensional solitons in PT linear lattice potentials,” Phys. Rev. E 85, 047601 (2012).
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V. A. Brazhnyi and B. A. Malomed, “Localization and delocalization of two-dimensional discrete solitons pinned to linear and nonlinear defects,” Phys. Rev. E 83, 016604 (2011).
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V. A. Brazhnyi and B. A. Malomed, “Dragging two-dimensional discrete solitons by moving linear defects,” Phys. Rev. E 84, 016608 (2011).
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J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E 73, 026609 (2006).
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R. F. Rodrguez, J. A. Reyes, A. Espinosa-Cerón, J. Fujioka, and B. A. Malomed, “Standard and embedded solitons in nematic optical fibers,” Phys. Rev. E 68, 036606 (2003).
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J. Atai and B. A. Malomed, “Solitary waves in systems with separated Bragg grating and nonlinearity,” Phys. Rev. E 64, 066617 (2001).
[CrossRef]

Phys. Rev. Lett.

J. Yang, “Stable embedded solitons,” Phys. Rev. Lett. 91, 143903 (2003).
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J. Yang, B. A. Malomed, and D. J. Kaup, “Embedded solitons in second-harmonic-generating systems,” Phys. Rev. Lett. 83, 1958–1961 (1999).
[CrossRef]

T. Anker, M. Albiez, R. Gati, S. Hunsmann, B. Eiermann, A. Trombettoni, and M. K. Oberthaler, “Nonlinear self-trapping of matter waves in periodic potentials,” Phys. Rev. Lett. 94, 020403 (2005).
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T. J. Alexander, E. A. Ostrovskaya, and Y. S. Kivshar, “Self-trapped nonlinear matter waves in periodic potentials,” Phys. Rev. Lett. 96, 040401 (2006).
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F. H. Bennet, T. J. Alexander, F. Haslinger, A. Mitchell, D. N. Neshev, and Y. S. Kivshar, “Observation of nonlinear self-trapping of broad beams in defocusing waveguide arrays,” Phys. Rev. Lett. 106, 093901 (2011).
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K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, “Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes,” Phys. Rev. Lett. 92, 183901 (2004).
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J. G. Rivas, M. Kuttge, P. H. Bolivar, H. Kurz, and J. A. Sanchez-Gil, “Propagation of surface plasmon polaritons on semiconductor gratings,” Phys. Rev. Lett. 93, 256804 (2004).
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A. A. Sukhorukov and Y. S. Kivshar, “Nonlinear localized waves in a periodic medium,” Phys. Rev. Lett. 87, 083901 (2001).
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D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vučković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904 (2005).
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Phys. Scr.

J. Zeng and B. A. Malomed, “Two-dimensional solitons and vortices in media with incommensurate linear and nonlinear lattice potentials,” Phys. Scr. T149, 014035 (2012).
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A. Espinosa-Cerón, J. Fujioka, and A. Gómez-Rodrguez, “Embedded solitons: four-frequency radiation, front propagation and radiation inhibition,” Phys. Scr. 67, 314–324 (2003).
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Physica D

A. R. Champneys, B. A. Malomed, J. Yang, and D. J. Kaup, “Embedded solitons: solitary waves in resonance with the linear spectrum,” Physica D 152, 340–354 (2001).
[CrossRef]

Proc. R. Soc. A

D. E. Pelinovsky and J. Yang, “A normal form for nonlinear resonance of embedded solitons,” Proc. R. Soc. A 458, 1469–1497 (2002).
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Progress in Optics

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Progress in Optics 52, 63–148 (2009).
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Rev. Mod. Phys.

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
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H. T. C. Stoof, K. B. Gubbels, and D. B. M. Dickerscheid, Ultracold Quantum Fields (Springer, 2009).

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J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

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Figures (9)

Fig. 1.
Fig. 1.

Contour plots of the lattice potential with (a) single, (b) double, (c) triple, and (d) quadruple defects, at ε=2.

Fig. 2.
Fig. 2.

(a) Linear spectra of the perfect OL, and OLs with (b) single, (c) double, and (d) triple defects displayed in Fig. 1. Isolated points emerging in the second finite bandgap represent the corresponding linear defect modes which are not related to solitons considered in this work.

Fig. 3.
Fig. 3.

Propagation constant (μ) versus the total power (norm), for the families of solitons pinned to the single defect [it is shown in Fig. 1(a)], which were found inside the first and second Bloch bands, as well as in the first finite bandgap between them. The solid and dashed curves indicate stable and unstable solutions, respectively. Typical examples of the pinned solitons, corresponding to the marked points, are shown below in Figs. 4 and 5. Some irregularity of the curves is explained by the fact that they were plotted with interval Δμ=0.1.

Fig. 4.
Fig. 4.

Examples of the intraband (embedded) defect solitons, EDSs, found in (first (a, b) and (c) second Bloch bands, which correspond to points A and B, and C, respectively, in Fig. 3. The shapes of the modes are shown by means of 3D images (left) and contour plots (right). The modes in panels (a) and (b) are stable, while the one in (c) is unstable.

Fig. 5.
Fig. 5.

Example of a stable gap soliton pinned to the single defect, found in the first bandgap (at point D marked in Fig. 3) under self-focusing nonlinearity.

Fig. 6.
Fig. 6.

Examples of stable dipole-mode solitons supported by the double defect, shown in Fig. 1(b), in the first Bloch band (left, with μ=1.4, N=3.9), and in the first finite bandgap (right, with μ=0.3, N=5.4).

Fig. 7.
Fig. 7.

Examples of stable dipole-mode solitons supported by the triple triangular defect, shown in Fig. 1(c), in the first Bloch band (left, μ=1.45, N=6.3), and in the first finite bandgap (right, μ=0.5, N=6.9).

Fig. 8.
Fig. 8.

Examples of stable vortices with topological charge 1, supported by the quadruple defect inside the first Bloch band (top, μ=1.5 and N=8.8), and in the first bandgap (bottom, μ=0.8 and N=7.2). The contour plots show the phase distribution in the vortices.

Fig. 9.
Fig. 9.

Curve μ(N) for the vortices with topological charge 1, pinned to the quadrupole lattice defect. The solid and dashed curves show stable and unstable solutions, respectively.

Equations (2)

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iψz=(1/2)2ψ[1δ˜(x,y)]VOLψ|ψ|2ψ
μϕ=(1/2)2ϕ[1δ(x,y)]VOL(x,y)ϕ|ϕ|2ϕ.

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