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

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2008 (1)

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]

2007 (6)

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]

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

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]

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]

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

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

2006 (10)

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

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

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

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]

D. E. Pelinovsky, "Translationally invariant nonlinear Schrodinge lattices," Nonlinearity 19, 2695-2716 (2006).
[CrossRef]

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]

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

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]

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

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]

2005 (5)

2004 (6)

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]

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

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

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

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

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]

2003 (7)

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

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]

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

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

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

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]

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

2002 (1)

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]

2001 (2)

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

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

2000 (1)

1999 (1)

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]

1998 (1)

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

1997 (1)

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

1988 (1)

Abdullaev, F.

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]

Abdumalikov, A.

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]

Aitchison, J. S.

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]

Anderson, B.P.

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

Belmonte-Beitia, J.

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]

Berge, L.

Carmon, T.

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]

Champneys, A. R.

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]

Chen, Z.

Christiansen, P. L.

Christodoulides, D. N.

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

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

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]

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]

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

Cuevas, J.

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]

Efremidis, N. K.

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]

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

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]

Eisenberg, H. S.

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]

Eugenieva, E.

Fan, S.

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

Fibich, G.

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]

Fleischer, J. W.

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]

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]

Frantzeskakis, D. J.

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

Gaididei, Y. B.

Galimzyanov, R.

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]

Gao, Y. M.

Guo, R.

Hao, R.

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]

Hizanidis, K.

Hutchings, D. C.

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

Joannopoulos, J. D.

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

Johansson, M.

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

Jones, C. K. R. T.

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]

Joseph, R. I.

Kapitula, T.

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

Kartashov, Y. V.

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

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

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

Kasevich, M. A.

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

Kevrekidis, P. G.

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]

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]

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

Kivshar, Y. S.

Kominis, Y.

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

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

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

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

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]

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

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

Konotop, V. V.

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]

Krolikowski, W.

Lederer, F.

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

Li, L.

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]

Liu, S. M.

Liu, Z. H.

Louis, P. J. Y.

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]

Malomed, B. A.

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

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

Martin, H.

Melvin, T. R. O.

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]

Mezentsev, V. K.

Morandotti, R.

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]

Neshev, D.

Neshev, D. N.

Ostrovskaya, E.

Ostrovskaya, E. A.

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]

Papadopoulos, A.

Pelinovsky, D. E.

D. E. Pelinovsky, "Translationally invariant nonlinear Schrodinge lattices," Nonlinearity 19, 2695-2716 (2006).
[CrossRef]

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]

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

Perez-Garcia, V. M.

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]

Peschel, U.

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]

Rapti, Z.

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]

Rasmussen, J. J.

Rosberg, C. R.

Russel, P.

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

Sakaguchi, H.

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

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

Savage, C. M.

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]

Sears, S.

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]

Segev, M.

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]

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]

Silberberg, Y.

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

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]

Sivan, Y.

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]

Smerzi, A.

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

Song, T.

Sukhorukov, A. A.

Torner, L.

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

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

Torres, P. J.

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]

Trombettoni, A.

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

Tsopelas, I.

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

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

Vekslerchik, V.

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]

Vicencio, R. A.

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

Villeneuve, P. R.

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

Vysloukh, V. A.

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

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

Weinstein, M. I.

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]

Xu, J.

Xu, Z.

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

Yang, J.

Yang, R.

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]

Zhou, G.

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]

Zhu, N.

IEEE J. Sel. Top. Quantum Electron. (1)

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

J. Opt. Soc. Am. B (2)

J. Phys. A: Math. Theor. (1)

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]

Nature (2)

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

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

Nonlinearity (1)

D. E. Pelinovsky, "Translationally invariant nonlinear Schrodinge lattices," Nonlinearity 19, 2695-2716 (2006).
[CrossRef]

Opt. Commun. (1)

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]

Opt. Express (4)

Opt. Lett. (7)

Phys. Lett. A (1)

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]

Phys. Rev. A (2)

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

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]

Phys. Rev. E (9)

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]

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

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

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

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

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

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]

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

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

Phys. Rev. Lett. (8)

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

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]

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

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]

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

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]

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

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]

Physica D (2)

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]

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

Science (2)

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

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

Other (3)

S. Trillo and W. Torruellas, Discrete Solitons, (Springer-Verlag, Berlin, 2001).

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

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

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Figures (12)

Fig. 1.
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)

Fig. 2.
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.
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).

Fig. 4.
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.
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.
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).

Fig. 7.
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.

Fig. 8.
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.
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.
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.
Fig. 11.

Propagation of the stationary solutions shown in Fig. 10.

Fig. 12.
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.

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

i ψ z + 2 ψ x 2 + 2 ψ 2 ψ + ε [ n 0 ( x ) ψ + n 2 ( x ) ψ 2 ψ ] = 0
ψ x z = u ( x ) e iβz
d 2 u dx 2 βu + 2 u 3 + ε [ n 0 ( x ) u + n 2 ( x ) u 3 ] = 0
H = p 2 2 β q 2 2 + q 4 2 + ε [ n 0 ( x ) q 2 2 + n 2 ( x ) q 4 2 ]
( q 0 ( x ) , p 0 ( x ) ) = ( ± β sech [ β ( x x 0 ) ] , β sech [ β ( x x 0 ) ] tanh [ β ( x x 0 ) ] )
M ( x 0 ) = ε + p 0 ( x ) [ n 0 ( x ) q 0 ( x ) + n 2 ( x ) q 0 ( x ) 3 ] dx
M ( x 0 ) = ε + [ n' 0 ( x + x 0 ) q 0 2 ( x ) 2 + n' 2 ( x + x 0 ) q 0 4 ( x ) 2 ] dx
n i ( x ) = m A m ( i ) cos ( K m ( i ) x + ϕ m ( i ) ) , i = 0,2
M ( x 0 ) = ε π β 2 m A m ( 0 ) F K m ( 0 ) β sin ( K m ( 0 ) x 0 + ϕ m ( 0 ) ) + ε π β 2 m A m ( 2 ) G K m ( 2 ) β sin ( K m ( 2 ) x 0 + ϕ m ( 2 ) )
F K m ( 0 ) β = ( K m ( 0 ) ) 2 sinh ( π K m ( 0 ) 2 β )
G K m ( 2 ) β = ( K m ( 2 ) ) 2 [ ( K m ( 2 ) ) 2 + 4 β ] 12 sinh ( π K m ( 2 ) 2 β )
M ( x 0 ) = A 1 ( 0 ) F K 1 ( 0 ) β sin ( K 1 ( 0 ) x 0 + ϕ 1 ( 0 ) )
M ( x 0 ) = F 1 β sin ( x 0 ) + A 2 ( 0 ) F 2 β sin ( 2 x 0 + ϕ 2 ( 0 ) ) = 0
M ( x 0 ) = F 1 β sin ( x 0 ) + A 1 ( 2 ) G 1 β sin ( x 0 + ϕ 1 ( 2 ) ) = 0
M ( x 0 ) = [ F 1 β + A 1 ( 2 ) G 1 β ] sin ( x 0 )
F 1 β + A 1 ( 2 ) G 1 β
a = A 1 ( 0 ) F K 1 ( 0 ) β A 1 ( 2 ) G K 1 ( 2 ) β

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