Abstract

We address shock waves generated upon the interaction of tilted plane waves with negative refractive index defects in defocusing media with linear gain and two-photon absorption. We found that, in contrast to conservative media where one-dimensional dispersive shock waves usually exist only as nonstationary objects expanding away from a defect or generating beam, the competition between gain and two-photon absorption in a dissipative medium results in the formation of localized stationary dispersive shock waves, whose transverse extent may considerably exceed that of the refractive index defect. One-dimensional dispersive shock waves are stable if the defect strength does not exceed a certain critical value.

© 2012 Optical Society of America

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    [CrossRef]
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  6. A. M. Kamchatnov, R. A. Kraenkel, and B. A. Umarov, Phys. Rev. E 66, 036609 (2002).
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  7. C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 102, 083902 (2009).
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  8. S. Malaguti, A. Corli, and S. Trillo, Opt. Lett. 35, 4217(2010).
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    [CrossRef]
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    [CrossRef]
  11. A. M. Leszczyszyn, G. A. El, Y. G. Gladush, and A. M. Kamchatnov, Phys. Rev. A 79, 063608 (2009).
    [CrossRef]
  12. W. Wan, S. Muenzel, and J. W. Fleischer, Phys. Rev. Lett. 104, 073903 (2010).
    [CrossRef]
  13. M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, Phys. Rev. A 74, 023623 (2006).
    [CrossRef]
  14. P. Engels and C. Atherton, Phys. Rev. Lett. 99, 160405 (2007).
    [CrossRef]
  15. G. A. El, A. M. Kamchatnov, V. V. Khodorovskii, E. S. Annibale, and A. Gammal, Phys. Rev. E 80, 043317 (2009).
    [CrossRef]
  16. M. A. Hoefer and B. Ilan, arXiv: 1105.3702 [nlin.PS] (2011).
  17. O. Katz, Y. Lahini, and Y. Silberberg, Opt. Lett. 33, 2830 (2008).
    [CrossRef]
  18. M. J. Connelly, Semiconductor Optical Amplifiers (Springer, 2002).
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    [CrossRef]
  20. R. S. Johnson, J. Fluid Mech. 42, 49 (1970).
    [CrossRef]
  21. R. S. Johnson, Phys. Fluids 15, 1693 (1972).
    [CrossRef]
  22. A. M. Kamchatnov, Physica D 188, 247 (2004).
    [CrossRef]

2010 (2)

W. Wan, S. Muenzel, and J. W. Fleischer, Phys. Rev. Lett. 104, 073903 (2010).
[CrossRef]

S. Malaguti, A. Corli, and S. Trillo, Opt. Lett. 35, 4217(2010).
[CrossRef]

2009 (3)

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef]

A. M. Leszczyszyn, G. A. El, Y. G. Gladush, and A. M. Kamchatnov, Phys. Rev. A 79, 063608 (2009).
[CrossRef]

G. A. El, A. M. Kamchatnov, V. V. Khodorovskii, E. S. Annibale, and A. Gammal, Phys. Rev. E 80, 043317 (2009).
[CrossRef]

2008 (1)

2007 (4)

C. Barsi, W. Wan, C. Sun, and J. W. Fleischer, Opt. Lett. 32, 2930 (2007).
[CrossRef]

P. Engels and C. Atherton, Phys. Rev. Lett. 99, 160405 (2007).
[CrossRef]

W. Wan, S. Jia, and J. W. Fleischer, Nat. Phys. 3, 46 (2007).
[CrossRef]

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 99, 043903 (2007).
[CrossRef]

2006 (1)

M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, Phys. Rev. A 74, 023623 (2006).
[CrossRef]

2004 (1)

A. M. Kamchatnov, Physica D 188, 247 (2004).
[CrossRef]

2002 (1)

A. M. Kamchatnov, R. A. Kraenkel, and B. A. Umarov, Phys. Rev. E 66, 036609 (2002).
[CrossRef]

1992 (2)

1989 (1)

J. E. Rothenberg and D. Grischkowsky, Phys. Rev. Lett. 62, 531 (1989).
[CrossRef]

1985 (1)

1972 (1)

R. S. Johnson, Phys. Fluids 15, 1693 (1972).
[CrossRef]

1970 (1)

R. S. Johnson, J. Fluid Mech. 42, 49 (1970).
[CrossRef]

1954 (1)

T. B. Benjamin and M. J. Lighthill, Proc. R. Soc. A 224, 448 (1954).
[CrossRef]

Ablowitz, M. J.

M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, Phys. Rev. A 74, 023623 (2006).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal and C. Headley, Phys. Rev. A 46, 1573 (1992).
[CrossRef]

Anderson, D.

Annibale, E. S.

G. A. El, A. M. Kamchatnov, V. V. Khodorovskii, E. S. Annibale, and A. Gammal, Phys. Rev. E 80, 043317 (2009).
[CrossRef]

Atherton, C.

P. Engels and C. Atherton, Phys. Rev. Lett. 99, 160405 (2007).
[CrossRef]

Barsi, C.

Benjamin, T. B.

T. B. Benjamin and M. J. Lighthill, Proc. R. Soc. A 224, 448 (1954).
[CrossRef]

Coddington, I.

M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, Phys. Rev. A 74, 023623 (2006).
[CrossRef]

Connelly, M. J.

M. J. Connelly, Semiconductor Optical Amplifiers (Springer, 2002).

Conti, C.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef]

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 99, 043903 (2007).
[CrossRef]

Corli, A.

Cornell, E. A.

M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, Phys. Rev. A 74, 023623 (2006).
[CrossRef]

Desaix, M.

El, G. A.

G. A. El, A. M. Kamchatnov, V. V. Khodorovskii, E. S. Annibale, and A. Gammal, Phys. Rev. E 80, 043317 (2009).
[CrossRef]

A. M. Leszczyszyn, G. A. El, Y. G. Gladush, and A. M. Kamchatnov, Phys. Rev. A 79, 063608 (2009).
[CrossRef]

Engels, P.

P. Engels and C. Atherton, Phys. Rev. Lett. 99, 160405 (2007).
[CrossRef]

M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, Phys. Rev. A 74, 023623 (2006).
[CrossRef]

Fleischer, J. W.

W. Wan, S. Muenzel, and J. W. Fleischer, Phys. Rev. Lett. 104, 073903 (2010).
[CrossRef]

W. Wan, S. Jia, and J. W. Fleischer, Nat. Phys. 3, 46 (2007).
[CrossRef]

C. Barsi, W. Wan, C. Sun, and J. W. Fleischer, Opt. Lett. 32, 2930 (2007).
[CrossRef]

Fratalocchi, A.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef]

Gammal, A.

G. A. El, A. M. Kamchatnov, V. V. Khodorovskii, E. S. Annibale, and A. Gammal, Phys. Rev. E 80, 043317 (2009).
[CrossRef]

Ghofraniha, N.

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 99, 043903 (2007).
[CrossRef]

Gladush, Y. G.

A. M. Leszczyszyn, G. A. El, Y. G. Gladush, and A. M. Kamchatnov, Phys. Rev. A 79, 063608 (2009).
[CrossRef]

Grischkowsky, D.

J. E. Rothenberg and D. Grischkowsky, Phys. Rev. Lett. 62, 531 (1989).
[CrossRef]

Headley, C.

G. P. Agrawal and C. Headley, Phys. Rev. A 46, 1573 (1992).
[CrossRef]

Hoefer, M. A.

M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, Phys. Rev. A 74, 023623 (2006).
[CrossRef]

M. A. Hoefer and B. Ilan, arXiv: 1105.3702 [nlin.PS] (2011).

Ilan, B.

M. A. Hoefer and B. Ilan, arXiv: 1105.3702 [nlin.PS] (2011).

Jia, S.

W. Wan, S. Jia, and J. W. Fleischer, Nat. Phys. 3, 46 (2007).
[CrossRef]

Johnson, A. M.

Johnson, R. S.

R. S. Johnson, Phys. Fluids 15, 1693 (1972).
[CrossRef]

R. S. Johnson, J. Fluid Mech. 42, 49 (1970).
[CrossRef]

Kamchatnov, A. M.

A. M. Leszczyszyn, G. A. El, Y. G. Gladush, and A. M. Kamchatnov, Phys. Rev. A 79, 063608 (2009).
[CrossRef]

G. A. El, A. M. Kamchatnov, V. V. Khodorovskii, E. S. Annibale, and A. Gammal, Phys. Rev. E 80, 043317 (2009).
[CrossRef]

A. M. Kamchatnov, Physica D 188, 247 (2004).
[CrossRef]

A. M. Kamchatnov, R. A. Kraenkel, and B. A. Umarov, Phys. Rev. E 66, 036609 (2002).
[CrossRef]

Katz, O.

Khodorovskii, V. V.

G. A. El, A. M. Kamchatnov, V. V. Khodorovskii, E. S. Annibale, and A. Gammal, Phys. Rev. E 80, 043317 (2009).
[CrossRef]

Kraenkel, R. A.

A. M. Kamchatnov, R. A. Kraenkel, and B. A. Umarov, Phys. Rev. E 66, 036609 (2002).
[CrossRef]

Lahini, Y.

Leszczyszyn, A. M.

A. M. Leszczyszyn, G. A. El, Y. G. Gladush, and A. M. Kamchatnov, Phys. Rev. A 79, 063608 (2009).
[CrossRef]

Lighthill, M. J.

T. B. Benjamin and M. J. Lighthill, Proc. R. Soc. A 224, 448 (1954).
[CrossRef]

Lisak, M.

Malaguti, S.

Muenzel, S.

W. Wan, S. Muenzel, and J. W. Fleischer, Phys. Rev. Lett. 104, 073903 (2010).
[CrossRef]

Peccianti, M.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef]

Quiroga-Teixeiro, M. L.

Rothenberg, J. E.

J. E. Rothenberg and D. Grischkowsky, Phys. Rev. Lett. 62, 531 (1989).
[CrossRef]

Ruocco, G.

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef]

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 99, 043903 (2007).
[CrossRef]

Schweikhard, V.

M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, Phys. Rev. A 74, 023623 (2006).
[CrossRef]

Silberberg, Y.

Stolen, R. H.

Sun, C.

Tomlinson, W. J.

Trillo, S.

S. Malaguti, A. Corli, and S. Trillo, Opt. Lett. 35, 4217(2010).
[CrossRef]

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef]

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 99, 043903 (2007).
[CrossRef]

Umarov, B. A.

A. M. Kamchatnov, R. A. Kraenkel, and B. A. Umarov, Phys. Rev. E 66, 036609 (2002).
[CrossRef]

Wan, W.

W. Wan, S. Muenzel, and J. W. Fleischer, Phys. Rev. Lett. 104, 073903 (2010).
[CrossRef]

W. Wan, S. Jia, and J. W. Fleischer, Nat. Phys. 3, 46 (2007).
[CrossRef]

C. Barsi, W. Wan, C. Sun, and J. W. Fleischer, Opt. Lett. 32, 2930 (2007).
[CrossRef]

J. Fluid Mech. (1)

R. S. Johnson, J. Fluid Mech. 42, 49 (1970).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nat. Phys. (1)

W. Wan, S. Jia, and J. W. Fleischer, Nat. Phys. 3, 46 (2007).
[CrossRef]

Opt. Lett. (4)

Phys. Fluids (1)

R. S. Johnson, Phys. Fluids 15, 1693 (1972).
[CrossRef]

Phys. Rev. A (3)

M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, Phys. Rev. A 74, 023623 (2006).
[CrossRef]

G. P. Agrawal and C. Headley, Phys. Rev. A 46, 1573 (1992).
[CrossRef]

A. M. Leszczyszyn, G. A. El, Y. G. Gladush, and A. M. Kamchatnov, Phys. Rev. A 79, 063608 (2009).
[CrossRef]

Phys. Rev. E (2)

A. M. Kamchatnov, R. A. Kraenkel, and B. A. Umarov, Phys. Rev. E 66, 036609 (2002).
[CrossRef]

G. A. El, A. M. Kamchatnov, V. V. Khodorovskii, E. S. Annibale, and A. Gammal, Phys. Rev. E 80, 043317 (2009).
[CrossRef]

Phys. Rev. Lett. (5)

P. Engels and C. Atherton, Phys. Rev. Lett. 99, 160405 (2007).
[CrossRef]

C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 102, 083902 (2009).
[CrossRef]

J. E. Rothenberg and D. Grischkowsky, Phys. Rev. Lett. 62, 531 (1989).
[CrossRef]

W. Wan, S. Muenzel, and J. W. Fleischer, Phys. Rev. Lett. 104, 073903 (2010).
[CrossRef]

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 99, 043903 (2007).
[CrossRef]

Physica D (1)

A. M. Kamchatnov, Physica D 188, 247 (2004).
[CrossRef]

Proc. R. Soc. A (1)

T. B. Benjamin and M. J. Lighthill, Proc. R. Soc. A 224, 448 (1954).
[CrossRef]

Other (2)

M. A. Hoefer and B. Ilan, arXiv: 1105.3702 [nlin.PS] (2011).

M. J. Connelly, Semiconductor Optical Amplifiers (Springer, 2002).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Maximal and minimal amplitudes of shock wave and (b) renormalized energy flow versus pr at α=0.6. (c) Critical defect depth for shock-wave existence versus α. (d) Profiles of stationary shock waves at pr=0.10 (curve 1) and pr=0.27 (curve 2) corresponding to α=0.6 and circles in (a) and (b). Left wings decay exponentially according to Eq. (2), where κ does not depend on pr; its theoretical value κth=0.0938 agrees rather well with numerical result κnum=0.0974.

Fig. 2.
Fig. 2.

Profiles of stationary shock waves corresponding to (a) α=1 for pr=0.2 (curve 1) and pr=0.6 (curve 2) and to (b) α=2 for pr=0.4 (curve 1) and pr=0.7 (curve 2).

Fig. 3.
Fig. 3.

(a) Maximal and minimal amplitudes of shock wave and (b) renormalized energy flow versus pr at α=1.4. (c) Threshold defect depth for shock-wave stability versus α. (d) Profiles of shock waves at pr=0.5 (curve 1) and pr=1.5 (curve 2) corresponding to α=1.4 and circles in (a) and (b). Asymptotic theoretical period of oscillations (3) on left wing dth=3.206 agrees well with numerical result dnum=3.226. Theoretical κth=0.0729 and numerically calculated κnum=0.0738 decay rates of envelopes are also close.

Fig. 4.
Fig. 4.

Dynamics of propagation of perturbed shock waves at α=1.4, pr=0.9 (top left panel), α=1.4, pr=1.1 (top right panel), and α=0.6, pr=0.25 (bottom panel).

Equations (3)

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iqξ=122qη2+q|q|2iσiq|q|2+ipiq+prR(η)q.
κ=2σiχ2αχ2α2
d=π(α2χ2)1/2,κ=σiχ2αα2χ2.

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