Abstract

We study the interaction of a soliton in a parity-time (PT) symmetric coupler which has local perturbation of the coupling constant. This defect does not change the PT-symmetry of the system, but locally can achieve the exceptional point. We found that the symmetric solitons after interaction with the defect either transform into breathers or blow up. The dynamics of antisymmetric solitons are more complex, showing domains of successive broadening of the beam and of the beam splitting in two outward propagating solitons, in addition to the single breather generation and blowup. All the effects are preserved when the coupling strength in the center of the defect deviates from the exceptional point. If the coupling is strong enough, the only observable outcome of the soliton-defect interaction is the generation of the breather.

© 2014 Optical Society of America

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  1. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
    [CrossRef]
  2. H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. A 82, 043803 (2010).
    [CrossRef]
  3. R. Driben and B. A. Malomed, Opt. Lett. 36, 4323 (2011).
    [CrossRef]
  4. R. Driben and B. A. Malomed, Europhys. Lett. 96, 51001 (2011).
  5. N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. A 85, 063837 (2012).
    [CrossRef]
  6. Y. V. Bludov, V. V. Konotop, and B. A. Malomed, Phys. Rev. A 87, 013816 (2013).
    [CrossRef]
  7. I. V. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, Phys. Rev. A 86, 053809 (2012).
    [CrossRef]
  8. Y. V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, J. Opt. 15, 064010 (2013).
    [CrossRef]
  9. W. D. Heiss, J. Phys. A 45, 44016 (2012).
  10. F. K. Abdullaev, V. V. Konotop, M. Ögren, and M. P. Sørensen, Opt. Lett. 36, 4566 (2011).
    [CrossRef]
  11. P. L. Chu, B. A. Malomed, G. D. Peng, and I. M. Skinner, Phys. Rev. E 49, 5763 (1994).
    [CrossRef]
  12. I. M. Skinner, G. D. Peng, B. A. Malomed, and P. L. Chu, Opt. Commun. 113, 493 (1995).
    [CrossRef]

2013 (2)

Y. V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, J. Opt. 15, 064010 (2013).
[CrossRef]

Y. V. Bludov, V. V. Konotop, and B. A. Malomed, Phys. Rev. A 87, 013816 (2013).
[CrossRef]

2012 (3)

I. V. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, Phys. Rev. A 86, 053809 (2012).
[CrossRef]

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

W. D. Heiss, J. Phys. A 45, 44016 (2012).

2011 (3)

2010 (2)

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]

1995 (1)

I. M. Skinner, G. D. Peng, B. A. Malomed, and P. L. Chu, Opt. Commun. 113, 493 (1995).
[CrossRef]

1994 (1)

P. L. Chu, B. A. Malomed, G. D. Peng, and I. M. Skinner, Phys. Rev. E 49, 5763 (1994).
[CrossRef]

Abdullaev, F. K.

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]

I. V. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, Phys. Rev. A 86, 053809 (2012).
[CrossRef]

Bludov, Y. V.

Y. V. Bludov, V. V. Konotop, and B. A. Malomed, Phys. Rev. A 87, 013816 (2013).
[CrossRef]

Y. V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, J. Opt. 15, 064010 (2013).
[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]

Chu, P. L.

I. M. Skinner, G. D. Peng, B. A. Malomed, and P. L. Chu, Opt. Commun. 113, 493 (1995).
[CrossRef]

P. L. Chu, B. A. Malomed, G. D. Peng, and I. M. Skinner, Phys. Rev. E 49, 5763 (1994).
[CrossRef]

Dmitriev, S. V.

I. V. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, Phys. Rev. A 86, 053809 (2012).
[CrossRef]

Driben, R.

Y. V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, J. Opt. 15, 064010 (2013).
[CrossRef]

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

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

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]

Heiss, W. D.

W. D. Heiss, J. Phys. A 45, 44016 (2012).

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]

I. V. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, Phys. Rev. A 86, 053809 (2012).
[CrossRef]

Konotop, V. V.

Y. V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, J. Opt. 15, 064010 (2013).
[CrossRef]

Y. V. Bludov, V. V. Konotop, and B. A. Malomed, Phys. Rev. A 87, 013816 (2013).
[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]

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]

Malomed, B. A.

Y. V. Bludov, V. V. Konotop, and B. A. Malomed, Phys. Rev. A 87, 013816 (2013).
[CrossRef]

Y. V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, J. Opt. 15, 064010 (2013).
[CrossRef]

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

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

I. M. Skinner, G. D. Peng, B. A. Malomed, and P. L. Chu, Opt. Commun. 113, 493 (1995).
[CrossRef]

P. L. Chu, B. A. Malomed, G. D. Peng, and I. M. Skinner, Phys. Rev. E 49, 5763 (1994).
[CrossRef]

Ögren, M.

Peng, G. D.

I. M. Skinner, G. D. Peng, B. A. Malomed, and P. L. Chu, Opt. Commun. 113, 493 (1995).
[CrossRef]

P. L. Chu, B. A. Malomed, G. D. Peng, and I. M. Skinner, Phys. Rev. E 49, 5763 (1994).
[CrossRef]

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]

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]

Skinner, I. M.

I. M. Skinner, G. D. Peng, B. A. Malomed, and P. L. Chu, Opt. Commun. 113, 493 (1995).
[CrossRef]

P. L. Chu, B. A. Malomed, G. D. Peng, and I. M. Skinner, Phys. Rev. E 49, 5763 (1994).
[CrossRef]

Sørensen, M. P.

Suchkov, S. V.

I. V. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, Phys. Rev. A 86, 053809 (2012).
[CrossRef]

Sukhorukov, A. A.

I. V. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, Phys. Rev. A 86, 053809 (2012).
[CrossRef]

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

Europhys. Lett. (1)

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

J. Opt. (1)

Y. V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, J. Opt. 15, 064010 (2013).
[CrossRef]

J. Phys. A (1)

W. D. Heiss, J. Phys. A 45, 44016 (2012).

Nat. Phys. (1)

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

Opt. Commun. (1)

I. M. Skinner, G. D. Peng, B. A. Malomed, and P. L. Chu, Opt. Commun. 113, 493 (1995).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (4)

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

Y. V. Bludov, V. V. Konotop, and B. A. Malomed, Phys. Rev. A 87, 013816 (2013).
[CrossRef]

I. V. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, Phys. Rev. A 86, 053809 (2012).
[CrossRef]

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

Phys. Rev. E (1)

P. L. Chu, B. A. Malomed, G. D. Peng, and I. M. Skinner, Phys. Rev. E 49, 5763 (1994).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Upper panels: field intensities with (a) η=0.15 and (b) η=0.5 interacting with defect at z=0. The coupling κmin=1, (a) κ0=2 and (b) 4. Lower panels: respective evolution of the total energy flow S0 for =1 (thick solid lines) and soliton amplitudes |q1| and |q2| (thin solid and dashed lines, respectively). The thick dotted line in (b) shows blowup at =1.1. The local maxima (minima) of S0 [the vertical lines in the lower panel (a)] happen where the powers in the waveguides are equal: |q1|2dx=|q2|2dx. The simulations for bounded solutions have been performed between zini=10 and zfin=100 and on the grid 40<x<40.

Fig. 2.
Fig. 2.

(a) S0 (solid line) and S3 (dashed line) versus z, (b) S2 (solid line) and S1 (dashed line) versus z, and (c)”θ versus z. The parameters are the same with those used in Fig. 1(a).

Fig. 3.
Fig. 3.

Dependencies of cr versus κmin for η=0.5 (a) and versus η for κmin=1 (b). In both panels κ0=4. If κmin>κmin*1.5 (a) and η<η*0.1 (b) (the gray domains) no blowup is found under the given values of the parameters. Insets show the dynamics of Stokes components (a) for a soliton interacting with a strong coupling defect κmin=1.5 and (b) for a small amplitude soliton (η=0.1) interacting with the exceptional point defect, where a sufficiently long defect, =10, results in the excitation of a breather.

Fig. 4.
Fig. 4.

Upper panels: The dynamics of soliton-defect interactions for (a) η=0.5 and =1.1, (b) 2.2, (c) 2.7, and (d) 3.2, respectively, for the coupling κ0=4 and κmin=1. In(a) and (b), the broadening is repeated along the propagation distance with the period 10. Lower panels: The total energy flow S0 (thick solid lines) and soliton amplitudes |q1| and |q2| (thin solid and dashed lines, respectively) for each solution. The thick dotted line in (d) corresponds to the blow up happening at =3.4.

Fig. 5.
Fig. 5.

(a) cr versus κmin for η=0.5 and (b) cr versus η for κmin=1. In both panels κ0=4. If (a) κmin>κmin*1 and (b) η<η*0.2 (the gray domains), no blowup occurs for the given parameters. The insets show the Stokes components for the defect with (a) κmin=1.1 and (b) for the small amplitude soliton (η=0.1) interacting with the exceptional point defect, where the defect of the length =10 results in the excitation of breathers.

Fig. 6.
Fig. 6.

Soliton interaction with super-Gaussian defect for (a) η=0.25, κ0=4, κmin=1 and =0.2; (b) =2.0; (c) =2.1; and (d) =2.2.

Equations (3)

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

iq1,z=q1,xx+iγq1κ(z)q2|q1|2q1,iq2,z=q2,xxiγq2κ(z)q1|q2|2q2,
q1(σ)=2ηexp[i(η2+σκ0cosδ)z]cosh(ηx)=σq2(σ)eiσδ,
dS0dz=2γS3,dS2dz=2κ(z)S3+s1s3dx,dS1dz=s2s3dx,dS3dz=2γS0+2κ(z)S2.

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