Abstract

I study vector solitons involving two incoherently coupled field components in periodic parity-time (PT)-symmetric optical lattices. The specific symmetry of the lattice imposes restrictions on the symmetry of available vector soliton states. While all configurations with asymmetric intensity distributions are prohibited, such lattices support multihump solitons with an equal number of “in-phase” or “out-of-phase” spots in two components, residing on neighboring lattice channels. In the focusing medium, only the solitons containing out-of-phase spots in at least one component can be stable, while in the defocusing medium stability is achieved for structures consisting of in-phase spots. Mixed-gap vector solitons with components emerging from different gaps in the lattice spectrum also exist and can be stable in the PT-symmetric lattice.

© 2013 Optical Society of America

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2012 (6)

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. A 85, 063837 (2012).
[CrossRef]

D. A. Zezyulin and V. V. Konotop, Phys. Rev. Lett. 108, 213906 (2012).
[CrossRef]

Y. He, D. Mihalache, X. Zhu, L. Guo, and Y. V. Kartashov, Opt. Lett. 37, 2526 (2012).
[CrossRef]

S. Nixon, L. Ge, and J. Yang, Phys. Rev. A 85, 023822 (2012).
[CrossRef]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, Phys. Rev. A 85, 013831 (2012).
[CrossRef]

C. Li, C. Huang, H. Liu, and L. Dong, Opt. Lett. 37, 4543 (2012).
[CrossRef]

2011 (9)

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, Phys. Rev. A 83, 041805(R) (2011).

D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, Europhys. Lett. 96, 64003 (2011).
[CrossRef]

A. E. Miroshnichenko, B. A. Malomed, and Y. S. Kivshar, Phys. Rev. A 84, 012123 (2011).
[CrossRef]

F. K. Abdullaev, V. V. Konotop, M. Ögren, and M. P. Sørensen, Opt. Lett. 36, 4566 (2011).
[CrossRef]

R. Driben and B. A. Malomed, Europhys. Lett. 96, 51001 (2011).
[CrossRef]

X. Zhu, H. Wang, L.-X. Zheng, H. Li, and Y. He, Opt. Lett. 36, 2680 (2011).
[CrossRef]

S. V. Suchkov, B. A. Malomed, S. V. Dmitriev, and Y. S. Kivshar, Phys. Rev. E 84, 046609 (2011).
[CrossRef]

K. Li and P. G. Kevrekidis, Phys. Rev. E 83, 066608 (2011).
[CrossRef]

R. Driben and B. A. Malomed, Opt. Lett. 36, 4323 (2011).
[CrossRef]

2010 (4)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

A. A. Sukhorukov, Z. Hu, and Y. S. Kivshar, Phys. Rev. A 82, 043818 (2010).
[CrossRef]

S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, Opt. Lett. 35, 2976 (2010).
[CrossRef]

2009 (1)

S. Longhi, Phys. Rev. Lett. 103, 123601 (2009).
[CrossRef]

2008 (2)

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

2007 (1)

2004 (1)

Y. V. Kartashov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, Phys. Rev. E 70, 066623 (2004).
[CrossRef]

2003 (3)

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, Phys. Rev. Lett. 91, 113902 (2003).
[CrossRef]

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef]

1999 (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, J. Math. Phys. 40, 2201 (1999).
[CrossRef]

1998 (2)

C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[CrossRef]

S. Darmanyan, A. Kobyakov, E. Schmidt, and F. Lederer, Phys. Rev. E 57, 3520 (1998).
[CrossRef]

1988 (1)

Abdullaev, F. K.

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, Phys. Rev. A 83, 041805(R) (2011).

F. K. Abdullaev, V. V. Konotop, M. Ögren, and M. P. Sørensen, Opt. Lett. 36, 4566 (2011).
[CrossRef]

Aitchison, J. S.

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef]

Alexeeva, N. V.

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. A 85, 063837 (2012).
[CrossRef]

Barashenkov, I. V.

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. A 85, 063837 (2012).
[CrossRef]

Bender, C. M.

C. M. Bender, S. Boettcher, and P. N. Meisinger, J. Math. Phys. 40, 2201 (1999).
[CrossRef]

C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[CrossRef]

Boettcher, S.

C. M. Bender, S. Boettcher, and P. N. Meisinger, J. Math. Phys. 40, 2201 (1999).
[CrossRef]

C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[CrossRef]

Chen, Z.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, Phys. Rev. A 85, 013831 (2012).
[CrossRef]

Christodoulides, D.

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef]

Christodoulides, D. N.

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
[CrossRef]

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef]

D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 53 (1988).
[CrossRef]

Cohen, O.

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef]

Darmanyan, S.

S. Darmanyan, A. Kobyakov, E. Schmidt, and F. Lederer, Phys. Rev. E 57, 3520 (1998).
[CrossRef]

Dmitriev, S. V.

S. V. Suchkov, B. A. Malomed, S. V. Dmitriev, and Y. S. Kivshar, Phys. Rev. E 84, 046609 (2011).
[CrossRef]

S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, Opt. Lett. 35, 2976 (2010).
[CrossRef]

Dong, L.

Driben, R.

R. Driben and B. A. Malomed, Europhys. Lett. 96, 51001 (2011).
[CrossRef]

R. Driben and B. A. Malomed, Opt. Lett. 36, 4323 (2011).
[CrossRef]

El-Ganainy, R.

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
[CrossRef]

Fleischer, J. W.

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef]

Ge, L.

S. Nixon, L. Ge, and J. Yang, Phys. Rev. A 85, 023822 (2012).
[CrossRef]

Guo, L.

He, Y.

Hu, Z.

A. A. Sukhorukov, Z. Hu, and Y. S. Kivshar, Phys. Rev. A 82, 043818 (2010).
[CrossRef]

Huang, C.

Hudock, J.

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef]

Joseph, R. I.

Kartashov, Y. V.

Y. He, D. Mihalache, X. Zhu, L. Guo, and Y. V. Kartashov, Opt. Lett. 37, 2526 (2012).
[CrossRef]

D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, Europhys. Lett. 96, 64003 (2011).
[CrossRef]

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, Phys. Rev. A 83, 041805(R) (2011).

Y. V. Kartashov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, Phys. Rev. E 70, 066623 (2004).
[CrossRef]

Kevrekidis, P. G.

K. Li and P. G. Kevrekidis, Phys. Rev. E 83, 066608 (2011).
[CrossRef]

Kip, D.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Kivshar, Y. S.

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. A 85, 063837 (2012).
[CrossRef]

A. E. Miroshnichenko, B. A. Malomed, and Y. S. Kivshar, Phys. Rev. A 84, 012123 (2011).
[CrossRef]

S. V. Suchkov, B. A. Malomed, S. V. Dmitriev, and Y. S. Kivshar, Phys. Rev. E 84, 046609 (2011).
[CrossRef]

S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, Opt. Lett. 35, 2976 (2010).
[CrossRef]

A. A. Sukhorukov, Z. Hu, and Y. S. Kivshar, Phys. Rev. A 82, 043818 (2010).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, Phys. Rev. Lett. 91, 113902 (2003).
[CrossRef]

Kobyakov, A.

S. Darmanyan, A. Kobyakov, E. Schmidt, and F. Lederer, Phys. Rev. E 57, 3520 (1998).
[CrossRef]

Konotop, V. V.

D. A. Zezyulin and V. V. Konotop, Phys. Rev. Lett. 108, 213906 (2012).
[CrossRef]

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, Phys. Rev. A 83, 041805(R) (2011).

D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, Europhys. Lett. 96, 64003 (2011).
[CrossRef]

F. K. Abdullaev, V. V. Konotop, M. Ögren, and M. P. Sørensen, Opt. Lett. 36, 4566 (2011).
[CrossRef]

Kottos, T.

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

Lederer, F.

S. Darmanyan, A. Kobyakov, E. Schmidt, and F. Lederer, Phys. Rev. E 57, 3520 (1998).
[CrossRef]

Li, C.

Li, H.

Li, K.

K. Li and P. G. Kevrekidis, Phys. Rev. E 83, 066608 (2011).
[CrossRef]

Liu, H.

Liu, J.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, Phys. Rev. A 85, 013831 (2012).
[CrossRef]

Longhi, S.

S. Longhi, Phys. Rev. Lett. 103, 123601 (2009).
[CrossRef]

Makris, K. G.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
[CrossRef]

Malomed, B. A.

R. Driben and B. A. Malomed, Opt. Lett. 36, 4323 (2011).
[CrossRef]

R. Driben and B. A. Malomed, Europhys. Lett. 96, 51001 (2011).
[CrossRef]

S. V. Suchkov, B. A. Malomed, S. V. Dmitriev, and Y. S. Kivshar, Phys. Rev. E 84, 046609 (2011).
[CrossRef]

A. E. Miroshnichenko, B. A. Malomed, and Y. S. Kivshar, Phys. Rev. A 84, 012123 (2011).
[CrossRef]

Meier, J.

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef]

Meisinger, P. N.

C. M. Bender, S. Boettcher, and P. N. Meisinger, J. Math. Phys. 40, 2201 (1999).
[CrossRef]

Mihalache, D.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, Phys. Rev. A 85, 013831 (2012).
[CrossRef]

Y. He, D. Mihalache, X. Zhu, L. Guo, and Y. V. Kartashov, Opt. Lett. 37, 2526 (2012).
[CrossRef]

Miroshnichenko, A. E.

A. E. Miroshnichenko, B. A. Malomed, and Y. S. Kivshar, Phys. Rev. A 84, 012123 (2011).
[CrossRef]

Morandotti, R.

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
[CrossRef]

Nixon, S.

S. Nixon, L. Ge, and J. Yang, Phys. Rev. A 85, 023822 (2012).
[CrossRef]

Ögren, M.

Ramezani, H.

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

Rüter, C. E.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Schmidt, E.

S. Darmanyan, A. Kobyakov, E. Schmidt, and F. Lederer, Phys. Rev. E 57, 3520 (1998).
[CrossRef]

Schwartz, T.

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef]

Segev, M.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef]

Silberberg, Y.

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef]

Sørensen, M. P.

Stegeman, G.

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef]

Suchkov, S. V.

S. V. Suchkov, B. A. Malomed, S. V. Dmitriev, and Y. S. Kivshar, Phys. Rev. E 84, 046609 (2011).
[CrossRef]

Sukhorukov, A. A.

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. A 85, 063837 (2012).
[CrossRef]

A. A. Sukhorukov, Z. Hu, and Y. S. Kivshar, Phys. Rev. A 82, 043818 (2010).
[CrossRef]

S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, Opt. Lett. 35, 2976 (2010).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, Phys. Rev. Lett. 91, 113902 (2003).
[CrossRef]

Torner, L.

Y. V. Kartashov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, Phys. Rev. E 70, 066623 (2004).
[CrossRef]

Vysloukh, V. A.

Y. V. Kartashov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, Phys. Rev. E 70, 066623 (2004).
[CrossRef]

Wang, H.

Yang, J.

S. Nixon, L. Ge, and J. Yang, Phys. Rev. A 85, 023822 (2012).
[CrossRef]

Zelenina, A. S.

Y. V. Kartashov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, Phys. Rev. E 70, 066623 (2004).
[CrossRef]

Zezyulin, D. A.

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Figures (7)

Fig. 1.
Fig. 1.

Real part br (solid curves) and imaginary part bi (dashed curves) of the propagation constant of Floquet–Bloch waves from the first and second allowed bands versus Bloch momentum kb at (a) pi=0.5 and (b) pi=2. In both cases, pr=1.

Fig. 2.
Fig. 2.

(a) U versus pi for odd–odd solitons in the focusing medium at b1=b2=1 and pr=1. (b) U versus pi for odd–odd solitons in the defocusing medium at b1=b2=2 (1) and b1=b2=2.3 (2), for pr=3. Panels (c) and (d) show solitons corresponding to circles in (a) and (b), respectively. Cyan line shows the refractive index shape. Only the first component is shown.

Fig. 3.
Fig. 3.

(a) Energy flow and (b) energy sharing between components of the twisted-even vector soliton in the focusing medium versus pi at b1=2.62 and b2=3. Circles correspond to the soliton shown in Fig. 4(a). (c) δr versus pi at b1=2.72 and b2=3. (d) Domains of existence and stability on the plane (b1,pi) at b2=3. Dashed line shows the critical value pipr. In all cases, pr=1.

Fig. 4.
Fig. 4.

Profiles of (a) twisted-even soliton in the focusing medium at pi=2, pr=1, b1=2.62, and b2=3. (b) Twisted-even soliton in the defocusing medium at pi=1.5, pr=3, b1=2.66, and b2=2. (c) odd–odd soliton in the focusing medium at pi=1.8, pr=3, b1=2.9, and b2=1.5.

Fig. 5.
Fig. 5.

(a) Energy flow, (b) energy sharing, and (c) perturbation growth rate versus pi for the twisted-even soliton in the defocusing medium at b1=2.66 and b2=2. Circles correspond to the soliton from Fig. 4(b). (d) Domain of existence and stability on the plane (b1,pi) at b2=2. Dashed line indicates the critical value pipr. In all cases, pr=3.

Fig. 6.
Fig. 6.

(a) Decay of unstable twisted-even soliton at pi=1.1 and (b) its stable propagation at pi=1.5 in the defocusing medium. In both cases pr=3, b1=2.66, and b2=2.

Fig. 7.
Fig. 7.

(a) U(pi) curve for mixed-gap soliton in the focusing medium at b1=2.9, b2=1.5, and pr=3. Circle corresponds to soliton in Fig. 4(c). (b) Domain of existence and stability on the plane (b1,pi) at b2=1.5 and pr=3. Dashed line corresponds to pipr.

Equations (1)

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iq1,2ξ=122q1,2η2+σq1,2(|q1|2+|q2|2)[prRr(η)+ipiRi(η)]q1,2.

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