Abstract

We show that the intracavity field evolving in an externally driven passive Kerr resonator subject to weak normal dispersion undergoes wave-breaking, thus forming dispersive shock waves. At variance with the cavity-less propagation, such dispersive wave-breaking turns out to be strongly favored by cavity bistability and coexisting modulational instability.

© 2014 Optical Society of America

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  1. M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Lett. 17, 745 (1992).
    [CrossRef]
  2. M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
    [CrossRef]
  3. S. Coen and M. Haelterman, Phys. Rev. Lett. 91, 401 (1997).
  4. S. Coen, M. Tlidi, Ph. Emplit, and M. Haelterman, Phys. Rev. Lett. 83, 2328 (1999).
    [CrossRef]
  5. F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, Nature Photonics 4, 471 (2010).
    [CrossRef]
  6. M. Haelterman, S. Trillo, and S. Wabnitz, J. Opt. Soc. Am. B 11, 446 (1994).
    [CrossRef]
  7. F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
    [CrossRef]
  8. M. J. Schmidberger, D. Novoa, F. Biancalana, P. St. J. Russell, and N. Y. Joly, Opt. Express 22, 3045 (2014).
    [CrossRef]
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    [CrossRef]
  10. A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
    [CrossRef]
  11. I. S. Grudinin, N. Yu, and L. Maleki, Opt. Lett. 34, 878 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  15. A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2845 (2011).
    [CrossRef]
  16. S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, Opt. Lett. 38, 37 (2013).
    [CrossRef]
  17. Y. K. Chembo and C. R. Menyuk, Phys. Rev. A 87, 053852 (2013).
    [CrossRef]
  18. T. Hansson, D. Modotto, and S. Wabnitz, Phys. Rev. A 88, 023819 (2013).
    [CrossRef]
  19. S. Coen and M. Erkintalo, Opt. Lett. 38, 1790 (2013).
    [CrossRef]
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    [CrossRef]
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2014 (2)

2013 (9)

S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, Opt. Lett. 38, 37 (2013).
[CrossRef]

R. Wu, V. Torres-Company, D. E. Leaird, and A. M. Weiner, Opt. Express 21, 6045 (2013).
[CrossRef]

M. Conforti, F. Baronio, and S. Trillo, Opt. Lett. 38, 1648 (2013).
[CrossRef]

S. Coen and M. Erkintalo, Opt. Lett. 38, 1790 (2013).
[CrossRef]

M. Conforti and S. Trillo, Opt. Lett. 38, 3815 (2013).
[CrossRef]

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

Y. K. Chembo and C. R. Menyuk, Phys. Rev. A 87, 053852 (2013).
[CrossRef]

T. Hansson, D. Modotto, and S. Wabnitz, Phys. Rev. A 88, 023819 (2013).
[CrossRef]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[CrossRef]

2011 (2)

2010 (1)

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, Nature Photonics 4, 471 (2010).
[CrossRef]

2009 (2)

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

I. S. Grudinin, N. Yu, and L. Maleki, Opt. Lett. 34, 878 (2009).
[CrossRef]

2008 (1)

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

2007 (3)

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

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

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

2003 (1)

1999 (1)

S. Coen, M. Tlidi, Ph. Emplit, and M. Haelterman, Phys. Rev. Lett. 83, 2328 (1999).
[CrossRef]

1997 (1)

S. Coen and M. Haelterman, Phys. Rev. Lett. 91, 401 (1997).

1994 (1)

1992 (3)

1989 (1)

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

1987 (1)

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef]

1974 (1)

A. V. Gurevich and L. P. Pitaevskii, Sov. Phys. JETP 38, 291 (1974).

Anderson, D.

Arcizet, O.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

Baronio, F.

M. Conforti, F. Baronio, and S. Trillo, Phys. Rev. A 89, 013807 (2014).
[CrossRef]

M. Conforti, F. Baronio, and S. Trillo, Opt. Lett. 38, 1648 (2013).
[CrossRef]

Biancalana, F.

Brasch, V.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[CrossRef]

Buckley, J. R.

Chembo, Y. K.

Y. K. Chembo and C. R. Menyuk, Phys. Rev. A 87, 053852 (2013).
[CrossRef]

Clark, W. G.

Coen, S.

S. Coen and M. Erkintalo, Opt. Lett. 38, 1790 (2013).
[CrossRef]

S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, Opt. Lett. 38, 37 (2013).
[CrossRef]

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, Nature Photonics 4, 471 (2010).
[CrossRef]

S. Coen, M. Tlidi, Ph. Emplit, and M. Haelterman, Phys. Rev. Lett. 83, 2328 (1999).
[CrossRef]

S. Coen and M. Haelterman, Phys. Rev. Lett. 91, 401 (1997).

Conforti, M.

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]

Del’Haye, P.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

Desaix, M.

Emplit, P.

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, Nature Photonics 4, 471 (2010).
[CrossRef]

Emplit, Ph.

S. Coen, M. Tlidi, Ph. Emplit, and M. Haelterman, Phys. Rev. Lett. 83, 2328 (1999).
[CrossRef]

Erkintalo, M.

Fleischer, J. W.

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

Foster, M. A.

Fratalocchi, A.

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

Gaeta, A. L.

Ghofraniha, N.

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

Gorodetsky, M. L.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[CrossRef]

Gorza, S.-P.

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, Nature Photonics 4, 471 (2010).
[CrossRef]

Grischkowsky, D.

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

Grudinin, I. S.

Gurevich, A. V.

A. V. Gurevich and L. P. Pitaevskii, Sov. Phys. JETP 38, 291 (1974).

Haelterman, M.

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, Nature Photonics 4, 471 (2010).
[CrossRef]

S. Coen, M. Tlidi, Ph. Emplit, and M. Haelterman, Phys. Rev. Lett. 83, 2328 (1999).
[CrossRef]

S. Coen and M. Haelterman, Phys. Rev. Lett. 91, 401 (1997).

M. Haelterman, S. Trillo, and S. Wabnitz, J. Opt. Soc. Am. B 11, 446 (1994).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Lett. 17, 745 (1992).
[CrossRef]

Hansson, T.

T. Hansson, D. Modotto, and S. Wabnitz, Phys. Rev. A 88, 023819 (2013).
[CrossRef]

Herr, T.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[CrossRef]

Holzwarth, R.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

Ilchenko, V. S.

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2845 (2011).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

Ilday, F. Ö.

Jia, S.

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

Joly, N. Y.

Jost, J. D.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[CrossRef]

Kippenberg, T. J.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[CrossRef]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

Kockaert, P.

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, Nature Photonics 4, 471 (2010).
[CrossRef]

Kondratiev, N. M.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[CrossRef]

Kuzucu, O.

Leaird, D. E.

Lefever, R.

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef]

Leo, F.

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, Nature Photonics 4, 471 (2010).
[CrossRef]

Levy, J. S.

Liang, W.

Lim, H.

Lipson, M.

Lisak, M.

Lugiato, L. A.

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef]

Maleki, L.

Matsko, A. B.

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2845 (2011).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

Menyuk, C. R.

Y. K. Chembo and C. R. Menyuk, Phys. Rev. A 87, 053852 (2013).
[CrossRef]

Modotto, D.

T. Hansson, D. Modotto, and S. Wabnitz, Phys. Rev. A 88, 023819 (2013).
[CrossRef]

Mussot, A.

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

Novoa, D.

Peccianti, M.

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

Pitaevskii, L. P.

A. V. Gurevich and L. P. Pitaevskii, Sov. Phys. JETP 38, 291 (1974).

Quiroiga-Teixeiro, M. L.

Randle, H. G.

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]

Russell, P. St. J.

Saha, K.

Savchenkov, A. A.

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2845 (2011).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

Schliesser, A.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

Schmidberger, M. J.

Seidel, D.

A. B. Matsko, A. A. Savchenkov, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, Opt. Lett. 36, 2845 (2011).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

Solomatine, I.

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

Sylvestre, T.

Taki, M.

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

Tlidi, M.

S. Coen, M. Tlidi, Ph. Emplit, and M. Haelterman, Phys. Rev. Lett. 83, 2328 (1999).
[CrossRef]

Torres-Company, V.

Trillo, S.

M. Conforti, F. Baronio, and S. Trillo, Phys. Rev. A 89, 013807 (2014).
[CrossRef]

M. Conforti, F. Baronio, and S. Trillo, Opt. Lett. 38, 1648 (2013).
[CrossRef]

M. Conforti and S. Trillo, Opt. Lett. 38, 3815 (2013).
[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]

M. Haelterman, S. Trillo, and S. Wabnitz, J. Opt. Soc. Am. B 11, 446 (1994).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Lett. 17, 745 (1992).
[CrossRef]

Wabnitz, S.

T. Hansson, D. Modotto, and S. Wabnitz, Phys. Rev. A 88, 023819 (2013).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, J. Opt. Soc. Am. B 11, 446 (1994).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Lett. 17, 745 (1992).
[CrossRef]

Wan, W.

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

Wang, C. Y.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[CrossRef]

Weiner, A. M.

Wilken, T.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

Wise, F. W.

Wu, R.

Yu, N.

J. Opt. Soc. Am. B (2)

Nat. Photonics (1)

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat. Photonics 8, 145 (2013).
[CrossRef]

Nat. Phys. (1)

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

Nature (1)

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef]

Nature Photonics (1)

F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman, Nature Photonics 4, 471 (2010).
[CrossRef]

Opt. Commun. (1)

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992).
[CrossRef]

Opt. Express (3)

Opt. Lett. (8)

Phys. Rev. A (3)

Y. K. Chembo and C. R. Menyuk, Phys. Rev. A 87, 053852 (2013).
[CrossRef]

T. Hansson, D. Modotto, and S. Wabnitz, Phys. Rev. A 88, 023819 (2013).
[CrossRef]

M. Conforti, F. Baronio, and S. Trillo, Phys. Rev. A 89, 013807 (2014).
[CrossRef]

Phys. Rev. Lett. (8)

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

L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef]

F. Leo, A. Mussot, P. Kockaert, P. Emplit, M. Haelterman, and M. Taki, Phys. Rev. Lett. 110, 104103 (2013).
[CrossRef]

S. Coen and M. Haelterman, Phys. Rev. Lett. 91, 401 (1997).

S. Coen, M. Tlidi, Ph. Emplit, and M. Haelterman, Phys. Rev. Lett. 83, 2328 (1999).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett. 101, 093902 (2008).
[CrossRef]

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

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

Sov. Phys. JETP (1)

A. V. Gurevich and L. P. Pitaevskii, Sov. Phys. JETP 38, 291 (1974).

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Figures (5)

Fig. 1.
Fig. 1.

DSWs ruled by Eq. (3). (a) Steady response and superimposed addressing power profile P(t) (red curve) sitting on CW bias [blue bullet at (P,Pu)=(0.0041,0.064)]; dashed lines gives bistable knees. (b) Snapshots of intracavity power |u(t)|2. (c) Level plot of |u(t,z)|2. The average power of the DSW upper state is highlighted by red bullets in (a) and (b) and dashed line in (b). Here ε=0.1, δ=π/10, α=0.03.

Fig. 2.
Fig. 2.

As in Fig. 1, showing MI-assisted DWSs occurring under CW bias (P,Pu)=(0.0034,0.048) [blue bullet in (a)] and peak power of addressing pulse (red curve) well below the lower knee of the bistable response.

Fig. 3.
Fig. 3.

(a) MI gain curves g(Ω) as a function of intracavity power Pu [from Eq. (5)], with black dashed lines highlighting knee values Pu±. The region in between such lines where g(Ω=0)0 corresponds to the negative slope branch of steady response. Curves for PuPu correspond to the lower branch (upper branch Pu>Pu+ is always stable with g=0). (b) Gain curves g(Ω) corresponding to the dynamics displayed in Fig. 2 (blue curve, lower gain; Pu=0.048) and Fig. 4 (red curve, high gain; Pu=0.064), respectively.

Fig. 4.
Fig. 4.

As in Fig. 1, MI-assisted DSWs occurring with the same bias as in Fig. 1 (blue bullet) but weaker addressing pulse below the bistable knee.

Fig. 5.
Fig. 5.

As in Fig. 1 for higher losses α=0.14, showing the transition from DSWs toward smooth shocks dominated by dissipation. The bias (blue bullet) is (P,Pu)=(0.0062,0.086) [Pin=110mW and Ppeak=216mW]. The red and green bullets in (a) and relative dashed lines in (b) stand for the maximal Pu and the power Pu of the upper state of the front, respectively.

Equations (5)

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un+1(0,t)=ρexp(iδ)un(z=ϵ,t)+S,
iϵunzϵ222unt2+|un|2un=0,
iϵuzϵ222ut2+|u|2u=[δiα]u+iS,
P=Pu[α2+(δPu)2],
g(Ω)=1ϵ4PuδΩδΩ23Pu2αϵ,

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