Abstract

We study wave propagation in a two-dimensional photonic lattice with focusing Kerr nonlinearity, and report on the existence of various nonlinear localized structures in the form of fundamental, dipole, and vortex solitons. First, the linear bandgap structure induced by the two-dimensional photonic crystal is determined, and solitons are found to exist in the photonic bandgap. Next, structures of these solitons and their stability properties are analyzed in detail. When the propagation constant is not close to the edge of the bandgap, the fundamental soliton is largely confined to one lattice site; the dipole soliton consists of two π-out-of-phase, Gaussian-like humps, whereas the vortex comprises four fundamental modes superimposed in a square configuration with a phase structure that is topologically equivalent to the conventional homogeneous-bulk vortex. At high lattice potential, all these soliton states are stable against small perturbations. However, among the three states, the fundamental solitons are the most robust, whereas vortices are the least. If the propagation constant is close to the edge of the bandgap, then all three soliton states spread over many lattice sites and become linearly unstable as a result of the Vakhitov–Kolokolov instability.

© 2004 Optical Society of America

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    [CrossRef]
  25. J. J. Garcia-Ripoll, V. M. Perez-Garcia, E. A. Ostrovskaya, and Y. S. Kivshar, “Dipole-mode vector solitons,” Phys. Rev. Lett. 85, 82–85 (2000).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2003 (6)

J. Fleischer, T. Carmon, M. Segev, N. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

J. Fleischer, M. Segev, N. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef] [PubMed]

J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. 28, 2094–2096 (2003).
[CrossRef] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

J. Yang and D. E. Pelinovsky, “Stable vortex and dipole vector solitons in a saturable nonlinear medium,” Phys. Rev. E 67, 016608 (2003).
[CrossRef]

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multi-dimensional solitons in periodic potentials,” Europhys. Lett. 63, 642–648 (2003).
[CrossRef]

2002 (2)

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

F. Lederer and Y. Silberberg, “Discrete solitons,” Opt. Photon. News 13, 48–53 (2002).
[CrossRef]

2001 (2)

B. A. Malomed and P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E 64, 026601 (2001).
[CrossRef]

A. A. Sukhorukov and Yu. S. Kivshar, “Nonlinear localized waves in a periodic medium,” Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

2000 (3)

Z. H. Musslimani, M. Segev, D. N. Christodoulides, and M. Soljacic, “Composite multihump vector solitons carrying topological charge,” Phys. Rev. Lett. 84, 1164–1167 (2000).
[CrossRef] [PubMed]

T. Carmon, C. Anastassiou, S. Lan, D. Kip, Z. H. Musslimani, M. Segev, and D. N. Christodoulides, “Observation of two-dimensional multimode solitons,” Opt. Lett. 25, 1113–1115 (2000).
[CrossRef]

J. J. Garcia-Ripoll, V. M. Perez-Garcia, E. A. Ostrovskaya, and Y. S. Kivshar, “Dipole-mode vector solitons,” Phys. Rev. Lett. 85, 82–85 (2000).
[CrossRef] [PubMed]

1999 (4)

R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillation,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[CrossRef]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

G. Lenz, I. Talanina, and C. Martijn de Sterke, “Bloch oscillations in an array of curved optical waveguides,” Phys. Rev. Lett. 83, 963–966 (1999).
[CrossRef]

1998 (1)

H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

1997 (1)

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
[CrossRef]

1996 (3)

L. P. Pitaevskii, “Dynamics of collapse of a confined Bose gas,” Phys. Lett. A 221, 14–18 (1996).
[CrossRef]

D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, “Nonlinear theory of oscillating, decaying, and collapsing solitons in the generalized nonlinear Schrödinger equation,” Phys. Rev. E 53, 1940–1953 (1996).
[CrossRef]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

1988 (1)

1976 (1)

V. I. Petviashvili, Plasma Phys. 2, 469 (1976).

1973 (1)

N. G. Vakhitov and A. A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973).
[CrossRef]

1972 (1)

V. E. Zakharov, “Collapse of Langmuir waves,” Sov. Phys. JETP 35, 908 (1972).

1965 (1)

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

Aceves, A. B.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Afanasjev, V. V.

D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, “Nonlinear theory of oscillating, decaying, and collapsing solitons in the generalized nonlinear Schrödinger equation,” Phys. Rev. E 53, 1940–1953 (1996).
[CrossRef]

Aitchison, J.

R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillation,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[CrossRef]

H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Aitchison, J. S.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

Anastassiou, C.

Baizakov, B. B.

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multi-dimensional solitons in periodic potentials,” Europhys. Lett. 63, 642–648 (2003).
[CrossRef]

Boyd, A.

H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Bräuer, A.

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

Carmon, T.

J. Fleischer, T. Carmon, M. Segev, N. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

T. Carmon, C. Anastassiou, S. Lan, D. Kip, Z. H. Musslimani, M. Segev, and D. N. Christodoulides, “Observation of two-dimensional multimode solitons,” Opt. Lett. 25, 1113–1115 (2000).
[CrossRef]

Christodoulides, D. N.

J. Fleischer, T. Carmon, M. Segev, N. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

J. Fleischer, M. Segev, N. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef] [PubMed]

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

T. Carmon, C. Anastassiou, S. Lan, D. Kip, Z. H. Musslimani, M. Segev, and D. N. Christodoulides, “Observation of two-dimensional multimode solitons,” Opt. Lett. 25, 1113–1115 (2000).
[CrossRef]

Z. H. Musslimani, M. Segev, D. N. Christodoulides, and M. Soljacic, “Composite multihump vector solitons carrying topological charge,” Phys. Rev. Lett. 84, 1164–1167 (2000).
[CrossRef] [PubMed]

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

Dannberg, P.

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

De Angelis, C.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Efremidis, N.

J. Fleischer, T. Carmon, M. Segev, N. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

J. Fleischer, M. Segev, N. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef] [PubMed]

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

Eisenberg, H.

R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillation,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[CrossRef]

R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Eisenberg, H. S.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

Elflein, W.

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

Firth, W. J.

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
[CrossRef]

Fleischer, J.

J. Fleischer, M. Segev, N. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef] [PubMed]

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

Garcia-Ripoll, J. J.

J. J. Garcia-Ripoll, V. M. Perez-Garcia, E. A. Ostrovskaya, and Y. S. Kivshar, “Dipole-mode vector solitons,” Phys. Rev. Lett. 85, 82–85 (2000).
[CrossRef] [PubMed]

Joseph, R. J.

Kelley, P. L.

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

Kevrekidis, P. G.

B. A. Malomed and P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E 64, 026601 (2001).
[CrossRef]

Kip, D.

Kivshar, Y. S.

J. J. Garcia-Ripoll, V. M. Perez-Garcia, E. A. Ostrovskaya, and Y. S. Kivshar, “Dipole-mode vector solitons,” Phys. Rev. Lett. 85, 82–85 (2000).
[CrossRef] [PubMed]

Kivshar, Yu. S.

A. A. Sukhorukov and Yu. S. Kivshar, “Nonlinear localized waves in a periodic medium,” Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, “Nonlinear theory of oscillating, decaying, and collapsing solitons in the generalized nonlinear Schrödinger equation,” Phys. Rev. E 53, 1940–1953 (1996).
[CrossRef]

Kolokolov, A. A.

N. G. Vakhitov and A. A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973).
[CrossRef]

Lan, S.

Lederer, F.

F. Lederer and Y. Silberberg, “Discrete solitons,” Opt. Photon. News 13, 48–53 (2002).
[CrossRef]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Lenz, G.

G. Lenz, I. Talanina, and C. Martijn de Sterke, “Bloch oscillations in an array of curved optical waveguides,” Phys. Rev. Lett. 83, 963–966 (1999).
[CrossRef]

Malomed, B. A.

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multi-dimensional solitons in periodic potentials,” Europhys. Lett. 63, 642–648 (2003).
[CrossRef]

B. A. Malomed and P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E 64, 026601 (2001).
[CrossRef]

Mandelik, D.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

Martijn de Sterke, C.

G. Lenz, I. Talanina, and C. Martijn de Sterke, “Bloch oscillations in an array of curved optical waveguides,” Phys. Rev. Lett. 83, 963–966 (1999).
[CrossRef]

Morandotti, R.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillation,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[CrossRef]

R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Muschall, R.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Musslimani, Z. H.

Ostrovskaya, E. A.

J. J. Garcia-Ripoll, V. M. Perez-Garcia, E. A. Ostrovskaya, and Y. S. Kivshar, “Dipole-mode vector solitons,” Phys. Rev. Lett. 85, 82–85 (2000).
[CrossRef] [PubMed]

Pelinovsky, D. E.

J. Yang and D. E. Pelinovsky, “Stable vortex and dipole vector solitons in a saturable nonlinear medium,” Phys. Rev. E 67, 016608 (2003).
[CrossRef]

D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, “Nonlinear theory of oscillating, decaying, and collapsing solitons in the generalized nonlinear Schrödinger equation,” Phys. Rev. E 53, 1940–1953 (1996).
[CrossRef]

Perez-Garcia, V. M.

J. J. Garcia-Ripoll, V. M. Perez-Garcia, E. A. Ostrovskaya, and Y. S. Kivshar, “Dipole-mode vector solitons,” Phys. Rev. Lett. 85, 82–85 (2000).
[CrossRef] [PubMed]

Pertsch, T.

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

Peschel, T.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Peschel, U.

R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillation,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[CrossRef]

Petviashvili, V. I.

V. I. Petviashvili, Plasma Phys. 2, 469 (1976).

Pitaevskii, L. P.

L. P. Pitaevskii, “Dynamics of collapse of a confined Bose gas,” Phys. Lett. A 221, 14–18 (1996).
[CrossRef]

Salerno, M.

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multi-dimensional solitons in periodic potentials,” Europhys. Lett. 63, 642–648 (2003).
[CrossRef]

Sears, S.

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[CrossRef]

Segev, M.

J. Fleischer, T. Carmon, M. Segev, N. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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

T. Carmon, C. Anastassiou, S. Lan, D. Kip, Z. H. Musslimani, M. Segev, and D. N. Christodoulides, “Observation of two-dimensional multimode solitons,” Opt. Lett. 25, 1113–1115 (2000).
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Z. H. Musslimani, M. Segev, D. N. Christodoulides, and M. Soljacic, “Composite multihump vector solitons carrying topological charge,” Phys. Rev. Lett. 84, 1164–1167 (2000).
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D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
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F. Lederer and Y. Silberberg, “Discrete solitons,” Opt. Photon. News 13, 48–53 (2002).
[CrossRef]

R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillation,” Phys. Rev. Lett. 83, 4756–4759 (1999).
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H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

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W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
[CrossRef]

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Z. H. Musslimani, M. Segev, D. N. Christodoulides, and M. Soljacic, “Composite multihump vector solitons carrying topological charge,” Phys. Rev. Lett. 84, 1164–1167 (2000).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef]

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A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

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N. G. Vakhitov and A. A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973).
[CrossRef]

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A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

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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).
[CrossRef] [PubMed]

J. Yang and D. E. Pelinovsky, “Stable vortex and dipole vector solitons in a saturable nonlinear medium,” Phys. Rev. E 67, 016608 (2003).
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Europhys. Lett. (1)

B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multi-dimensional solitons in periodic potentials,” Europhys. Lett. 63, 642–648 (2003).
[CrossRef]

Nature (1)

J. Fleischer, M. Segev, N. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef] [PubMed]

Opt. Lett. (3)

Opt. Photon. News (1)

F. Lederer and Y. Silberberg, “Discrete solitons,” Opt. Photon. News 13, 48–53 (2002).
[CrossRef]

Phys. Lett. A (1)

L. P. Pitaevskii, “Dynamics of collapse of a confined Bose gas,” Phys. Lett. A 221, 14–18 (1996).
[CrossRef]

Phys. Rev. E (5)

D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, “Nonlinear theory of oscillating, decaying, and collapsing solitons in the generalized nonlinear Schrödinger equation,” Phys. Rev. E 53, 1940–1953 (1996).
[CrossRef]

B. A. Malomed and P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E 64, 026601 (2001).
[CrossRef]

J. Yang and D. E. Pelinovsky, “Stable vortex and dipole vector solitons in a saturable nonlinear medium,” Phys. Rev. E 67, 016608 (2003).
[CrossRef]

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

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Phys. Rev. Lett. (12)

H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillation,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[CrossRef]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

G. Lenz, I. Talanina, and C. Martijn de Sterke, “Bloch oscillations in an array of curved optical waveguides,” Phys. Rev. Lett. 83, 963–966 (1999).
[CrossRef]

J. Fleischer, T. Carmon, M. Segev, N. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

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[CrossRef]

A. A. Sukhorukov and Yu. S. Kivshar, “Nonlinear localized waves in a periodic medium,” Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
[CrossRef]

Z. H. Musslimani, M. Segev, D. N. Christodoulides, and M. Soljacic, “Composite multihump vector solitons carrying topological charge,” Phys. Rev. Lett. 84, 1164–1167 (2000).
[CrossRef] [PubMed]

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Radiophys. Quantum Electron. (1)

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[CrossRef]

Sov. Phys. JETP (1)

V. E. Zakharov, “Collapse of Langmuir waves,” Sov. Phys. JETP 35, 908 (1972).

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R. Grimshaw, Nonlinear Ordinary Differential Equations (CRC Press, Boca Raton, Fla., 1993).

Z. Chen, H. Martin, E. D. Eugenieva, and D. N. Christodoulides, “Soliton-induced dislocations and discrete solitons in partially-coherent photonic lattices” (unpublished).

F. Lederer, S. Darmanyan, and A. Kobyakov, “Discrete solitons,” in Spatial Solitons, S. Trillo and W. Torruellas, eds. (Springer, New York, 2001), p. 269.

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Figures (13)

Fig. 1
Fig. 1

Bandgap structure of the linear periodic problem of Eqs. (9) and (10). Solid curves, the numerically computed bandgap boundaries; dashed curves, analytic approximations of Eqs. (14)–(16) for the same boundaries.

Fig. 2
Fig. 2

Profiles of fundamental solitons at (a) μ=1 and (b) μ=1.72 with ν0=2.

Fig. 3
Fig. 3

(a) Normalized power P of fundamental solitons versus μ for ν0=1 and ν0=2; (b) unstable eigenvalues σ of these solitons.

Fig. 4
Fig. 4

Stable evolution of the fundamental soliton with μ=1 and ν0=2 under strong perturbations =0.2 in Eq. (26).

Fig. 5
Fig. 5

Intensity (left) and phase (right) profiles of dipole solitons at μ=1 (top row) and μ=1.70 (bottom row) with ν0=2.

Fig. 6
Fig. 6

(a) Normalized power P of dipole solitons versus μ for ν0=1 and ν0=2; (b) unstable eigenvalues σ of dipole solitons with ν0=1 and ν0=2; Re(σ), solid curve; Im(σ), dashed curve.

Fig. 7
Fig. 7

Breakup of a linearly unstable dipole soliton into a fundamental soliton under weak perturbations. Here the dipole soliton has ν0=2, μ=1.3, and the perturbation [Eq. (26)] has =0.01.

Fig. 8
Fig. 8

Stable evolution of a linearly stable dipole soliton under weak perturbations. Here the dipole soliton has ν0=2, μ=1, and the perturbation [Eq. (26)] has =0.05.

Fig. 9
Fig. 9

Symmetry breaking of a linearly stable dipole soliton under stronger perturbations. Here the dipole soliton is the same as in Fig. 8 (i.e., with ν0=2, μ=1), but =0.1 in the perturbation [Eq. (26)] now.

Fig. 10
Fig. 10

Intensity (left) and phase (right) profiles of vortex solitons at μ=1 (top row) and μ=1.69 (bottom row) with ν0=2.

Fig. 11
Fig. 11

(a) Normalized power P of vortex solitons versus μ for ν0=1 and ν0=2; (b) unstable eigenvalues σ of vortex solitons with ν0=1 and ν0=2; Re(σ), solid curve; Im(σ), dashed curve.

Fig. 12
Fig. 12

Stable evolution of a linearly stable vortex soliton under weak perturbations. Here the vortex soliton has ν0=2, μ=1, and =0.005 in the perturbation [Eq. (26)]. The lower right figure shows the peak-intensity evolution of the first-quadrant hump of the vortex soliton. Intensity evolutions of the other three humps are similar.

Fig. 13
Fig. 13

Unstable evolution of a linearly stable vortex soliton under stronger perturbations. Here the vortex soliton is the same as in Fig. 12 (i.e., with ν0=2, μ=1), but =0.02 in the perturbation [Eq. (26)] now.

Equations (26)

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iUz+2X2+2Y2U-VU+|U|2U=0,
V=V0(cos2 X+cos2 Y).
P=--|U|2dXdY,
E=--|ψ|2-12|U|4+V|U|2dXdY.
U(X, Y, z)=exp(-iμz)u(X, Y),
2uX2+2uY2-Vu+|u|2u=-μu.
2X2+2Y2u-V0(cos2 X+cos2 Y)u=-μu.
u(X, Y)=u1(X)u2(Y),
2u1(X)X2-V0 cos2 Xu1(X)=-μ1u1(X),
2u2(Y)Y2-V0 cos2 Yu2(Y)=-μ2u2(Y),
μ=μ1+μ2.
u1(X)=ϕk(X)exp(ikX),
2u1(X)X2-V02[1+cos(2X)]u1(X)=-μ1u1(X).
μ11/2V0-1/32V02,
μ11+1/4V0,
μ11+3/4V0.
μmax=2μ1,maxV0-V02/16.
F(f)=fˆ(k)=dxf(x)exp(-ik·x),
f(x)=F-1(fˆ)=12π dkfˆ(k)exp(+ik·x).
uˆ=-1|k|2-μF(Vu)+1|k|2-μF(|u|2u).
uˆm+1=1c+|k|2P(ηL)P(ηN)1/2[(μ+c)uˆm-F(Vum)]+1c+|k|2P(ηL)P(ηN)3/2F(|um|2um),
P(ϕˆ)dkuˆ(k)ϕˆ(k),
ηL=(|k|2-μ)uˆ+F(Vu),
ηN=F(|u|2u).
iU˜z+2U˜X2+2U˜Y2+(μ-V+2|u|2)U˜+u2U˜*=0.
U(X, Y, z=0)=u(X, Y)1++ tanh rˆj=110 sinjθˆ.

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