Abstract

Stable discrete compactons in interconnected three-line waveguide arrays are found in linear and nonlinear limits in conservative and in parity-time (PT)-symmetric models. The compactons result from the interference of the fields in the two lines of waveguides ensuring that the third (middle) line caries no energy. PT-symmetric compactons require not only the presence of gain and losses in the two lines of the waveguides but also complex coupling, i.e., gain and losses in the coupling between the lines carrying the energy and the third line with zero field. The obtained compactons can be stable and their branches can cross the branches of the dissipative solitons. Unusual bifurcations of branches of solitons from linear compactons are described.

© 2013 Optical Society of America

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2013

P. G. Kevrekidis, D. E. Pelinovsky, and D. Y. Tyugin, SIAM J. Appl. Dyn. Syst. 12, 1210 (2013).
[CrossRef]

2011

M. Heinrich, R. Keil, F. Dreisow, A. Tunnermann, A. Szameit, and S. Nolte, Appl. Phys. B 104, 469 (2011).
[CrossRef]

2010

F. Kh. Abdullaev, P. G. Kevrekidis, and M. Salerno, Phys. Rev. Lett. 105, 113901 (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]

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

2008

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

2007

2006

T. M. Monro and H. Ebendorff-Heidepriem, Annu. Rev. Mater. Res. 36, 467 (2006).
[CrossRef]

2005

2002

P. G. Kevrekidis and V. V. Konotop, Phys. Rev. E 65, 066614 (2002).
[CrossRef]

P. G. Kevrekidis, V. V. Konotop, and S. Takeno, Phys. Lett. A 299, 166 (2002).
[CrossRef]

2000

E. Coquet, M. Remoissenet, and P. T. Dinda, Phys. Rev. E 62, 5767 (2000).
[CrossRef]

M. Eleftheriou, B. Dey, and G. P. Tsironis, Phys. Rev. E 62, 7540 (2000).
[CrossRef]

1999

V. V. Konotop and S. Takeno, Phys. Rev. E 60, 1001 (1999).
[CrossRef]

1994

P. Rosenau, Phys. Rev. Lett. 73, 1737 (1994).
[CrossRef]

1993

P. Rosenau and J. M. Hyman, Phys. Rev. Lett. 70, 564 (1993).
[CrossRef]

Abdullaev, F. Kh.

F. Kh. Abdullaev, P. G. Kevrekidis, and M. Salerno, Phys. Rev. Lett. 105, 113901 (2010).
[CrossRef]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Azaña, J.

Bélanger, N.

Christodoulides, D.

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

Christodoulides, D. N.

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

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

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

Coquet, E.

E. Coquet, M. Remoissenet, and P. T. Dinda, Phys. Rev. E 62, 5767 (2000).
[CrossRef]

Delgado, F.

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

Dey, B.

M. Eleftheriou, B. Dey, and G. P. Tsironis, Phys. Rev. E 62, 7540 (2000).
[CrossRef]

Dinda, P. T.

E. Coquet, M. Remoissenet, and P. T. Dinda, Phys. Rev. E 62, 5767 (2000).
[CrossRef]

Dreisow, F.

M. Heinrich, R. Keil, F. Dreisow, A. Tunnermann, A. Szameit, and S. Nolte, Appl. Phys. B 104, 469 (2011).
[CrossRef]

Ebendorff-Heidepriem, H.

T. M. Monro and H. Ebendorff-Heidepriem, Annu. Rev. Mater. Res. 36, 467 (2006).
[CrossRef]

Eleftheriou, M.

M. Eleftheriou, B. Dey, and G. P. Tsironis, Phys. Rev. E 62, 7540 (2000).
[CrossRef]

El-Ganainy, R.

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. Christodoulides, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

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

Heinrich, M.

M. Heinrich, R. Keil, F. Dreisow, A. Tunnermann, A. Szameit, and S. Nolte, Appl. Phys. B 104, 469 (2011).
[CrossRef]

Hyman, J. M.

P. Rosenau and J. M. Hyman, Phys. Rev. Lett. 70, 564 (1993).
[CrossRef]

Jalali, B.

Keil, R.

M. Heinrich, R. Keil, F. Dreisow, A. Tunnermann, A. Szameit, and S. Nolte, Appl. Phys. B 104, 469 (2011).
[CrossRef]

Kevrekidis, P. G.

P. G. Kevrekidis, D. E. Pelinovsky, and D. Y. Tyugin, SIAM J. Appl. Dyn. Syst. 12, 1210 (2013).
[CrossRef]

F. Kh. Abdullaev, P. G. Kevrekidis, and M. Salerno, Phys. Rev. Lett. 105, 113901 (2010).
[CrossRef]

P. G. Kevrekidis and V. V. Konotop, Phys. Rev. E 65, 066614 (2002).
[CrossRef]

P. G. Kevrekidis, V. V. Konotop, and S. Takeno, Phys. Lett. A 299, 166 (2002).
[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]

Konotop, V. V.

P. G. Kevrekidis and V. V. Konotop, Phys. Rev. E 65, 066614 (2002).
[CrossRef]

P. G. Kevrekidis, V. V. Konotop, and S. Takeno, Phys. Lett. A 299, 166 (2002).
[CrossRef]

V. V. Konotop and S. Takeno, Phys. Rev. E 60, 1001 (1999).
[CrossRef]

Koonath, P.

Kottos, T.

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

Kulishov, M.

Laniel, J.

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[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]

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

Monro, T. M.

T. M. Monro and H. Ebendorff-Heidepriem, Annu. Rev. Mater. Res. 36, 467 (2006).
[CrossRef]

Muga, J. G.

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

Musslimani, Z. H.

Nolte, S.

M. Heinrich, R. Keil, F. Dreisow, A. Tunnermann, A. Szameit, and S. Nolte, Appl. Phys. B 104, 469 (2011).
[CrossRef]

Pelinovsky, D. E.

P. G. Kevrekidis, D. E. Pelinovsky, and D. Y. Tyugin, SIAM J. Appl. Dyn. Syst. 12, 1210 (2013).
[CrossRef]

Plant, D.

Ramezani, H.

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

Remoissenet, M.

E. Coquet, M. Remoissenet, and P. T. Dinda, Phys. Rev. E 62, 5767 (2000).
[CrossRef]

Rosenau, P.

P. Rosenau, Phys. Rev. Lett. 73, 1737 (1994).
[CrossRef]

P. Rosenau and J. M. Hyman, Phys. Rev. Lett. 70, 564 (1993).
[CrossRef]

Ruschhaupt, A.

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[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]

Salerno, M.

F. Kh. Abdullaev, P. G. Kevrekidis, and M. Salerno, Phys. Rev. Lett. 105, 113901 (2010).
[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]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Szameit, A.

M. Heinrich, R. Keil, F. Dreisow, A. Tunnermann, A. Szameit, and S. Nolte, Appl. Phys. B 104, 469 (2011).
[CrossRef]

Takeno, S.

P. G. Kevrekidis, V. V. Konotop, and S. Takeno, Phys. Lett. A 299, 166 (2002).
[CrossRef]

V. V. Konotop and S. Takeno, Phys. Rev. E 60, 1001 (1999).
[CrossRef]

Tsironis, G. P.

M. Eleftheriou, B. Dey, and G. P. Tsironis, Phys. Rev. E 62, 7540 (2000).
[CrossRef]

Tunnermann, A.

M. Heinrich, R. Keil, F. Dreisow, A. Tunnermann, A. Szameit, and S. Nolte, Appl. Phys. B 104, 469 (2011).
[CrossRef]

Tyugin, D. Y.

P. G. Kevrekidis, D. E. Pelinovsky, and D. Y. Tyugin, SIAM J. Appl. Dyn. Syst. 12, 1210 (2013).
[CrossRef]

Annu. Rev. Mater. Res.

T. M. Monro and H. Ebendorff-Heidepriem, Annu. Rev. Mater. Res. 36, 467 (2006).
[CrossRef]

Appl. Phys. B

M. Heinrich, R. Keil, F. Dreisow, A. Tunnermann, A. Szameit, and S. Nolte, Appl. Phys. B 104, 469 (2011).
[CrossRef]

J. Phys. A

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

Nat. Phys.

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

Opt. Express

Opt. Lett.

Phys. Lett. A

P. G. Kevrekidis, V. V. Konotop, and S. Takeno, Phys. Lett. A 299, 166 (2002).
[CrossRef]

Phys. Rep.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Phys. Rev. A

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

Phys. Rev. E

V. V. Konotop and S. Takeno, Phys. Rev. E 60, 1001 (1999).
[CrossRef]

E. Coquet, M. Remoissenet, and P. T. Dinda, Phys. Rev. E 62, 5767 (2000).
[CrossRef]

M. Eleftheriou, B. Dey, and G. P. Tsironis, Phys. Rev. E 62, 7540 (2000).
[CrossRef]

P. G. Kevrekidis and V. V. Konotop, Phys. Rev. E 65, 066614 (2002).
[CrossRef]

Phys. Rev. Lett.

P. Rosenau and J. M. Hyman, Phys. Rev. Lett. 70, 564 (1993).
[CrossRef]

P. Rosenau, Phys. Rev. Lett. 73, 1737 (1994).
[CrossRef]

F. Kh. Abdullaev, P. G. Kevrekidis, and M. Salerno, Phys. Rev. Lett. 105, 113901 (2010).
[CrossRef]

SIAM J. Appl. Dyn. Syst.

P. G. Kevrekidis, D. E. Pelinovsky, and D. Y. Tyugin, SIAM J. Appl. Dyn. Syst. 12, 1210 (2013).
[CrossRef]

Other

N. V. Alexeeva, I. V. Barashenkov, K. Rayanov, and S. Flach, http://xxx.tau.ac.il/abs/1308.5862 .

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Figures (4)

Fig. 1.
Fig. 1.

Schematic representation of the array with the notations used in the text. Black dots and lines designate the waveguides and links among them.

Fig. 2.
Fig. 2.

(a) Dispersion characteristics for κ1=0.4, κ=1.2. The horizontal dashed line indicates b0, while the solid green lines show the dispersion curves b±. The thick short magenta line marks the position of the dimer frequency. The black dots show the frequencies of the linear excitations in the presence of an excited dipole mode. (b) The dynamics of the unstable modes for κ=0.6 (upper panel) and κ1=1.0 (lower panel). (c) The same as panel (a) but for κ1=0.4, κ=0.4. (d) The dynamics of the unstable mode for κ1=0.6.

Fig. 3.
Fig. 3.

Dependencies of the maximum absolute value of the field un on the dissipation γ for the solitons bifurcating from the compacton at γ=0 are shown in panels (a) and (d) for structures with the propagation constants bs=0.44 and bs=2.44 correspondingly. The bifurcation points are marked by the squares. Panel (b) shows the distribution of the fields un (red circles) and wn (black circles) for the point marked on the bifurcation curve on panel (a) as (B). Panel (c) shows the distribution of the mutual phase of the fields un and vn defined as θ=arg(vnun*). The distributions marked as 1 and 2 correspond to the points C1 and C2 in panel (a). Panels (e) and (f) show the distributions of the fields un and wn in the solitons corresponding to the points (E) and (F) on the bifurcation curve shown in panel (d). The parameters are g=1, κ=1, κ1=0.25.

Fig. 4.
Fig. 4.

Thinner black curve in panel (a) shows the trajectory in γImκ1 plane along which the compact solution can exist. The thicker blue (bs=0.44) and thinner magenta lines (bs=0.36) show the bifurcation diagrams of the compacton. The bifurcation diagram of the soliton is shown by the green line. In panel (b) the bifurcation diagrams of the compactons and the soliton are shown as functions of γ. Panels (c) and (d) show the same but for the negative nonlinearity g=1 and the propagation constant bs=1.25. The other parameters are κ=1, κ1=0.25.

Equations (15)

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

iu˙n=κvniγun+κ1(wn1+wn)+g|un|2un,
iv˙n=κun+iγvn+κ1*(wn1+wn)+g|vn|2vn,
iw˙n=κ1(un+un+1)+κ1*(vn+vn+1)+g˜|wn|2wn.
b3(κ2γ2)b8κ˜2[bcosϕ+γsinϕ+κ]cos2k2=0.
b0=0ifγ=κ/sinϕ,
b0=κcosϕifγ=κsinϕ.
u=κ˜sinϕκcos2ϕ(1+eik)(ieiϕ/2+eiϕ/2sinϕ)w,
v=κ˜sinϕκcos2ϕ(1+eik)(ieiϕ/2+eiϕ/2sinϕ)w.
udip=αvdip,α=κ2κ˜2(1+icosk)κ˜2(1+icosk)κ2cosϕeiϕ,
b0=κ,b±=κ2±12κ2+16κ12(1+cosk).
u±=v±=α±w,α±=κ2+16κ12(1+cosk)±κ4κ˜(1+eik),
iη˙0=gρ2(η0+η0*),
iξ˙n=2κξn+δn0gρ2(ξn+ξn*)+2κ1(wn1+wn),
iw˙n=κ1(ξn+ξn+1)
2κ12<κ2,γ2<κ2+4κ123(2κκ12)2/3.

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