Abstract

In the regime of normal dispersion and low-frequency detunings (or high powers), four-wave mixing is shown to undergo a hydrodynamic type of instability. Such instability involves the formation of shocks (steep fronts) from smooth initial data that are regularized through the appearance of trains of fast oscillations, which exhibit solitonlike behavior, colliding elastically.

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  1. J. R. Thompson and R. Roy, Phys. Rev. A 43, 4987 (1991).
    [CrossRef] [PubMed]
  2. C. S. West and T. A. B. Kennedy, Phys. Rev. A 47, 1252 (1993).
    [CrossRef] [PubMed]
  3. S. Trillo, S. Wabnitz, and T. A. B. Kennedy, Phys. Rev. A 50, 1732 (1994).
    [CrossRef] [PubMed]
  4. P. V. Mamyshev, P. G. Wigley, J. Wilson, C. Bosshard, and G. I. Stegeman, Appl. Phys. Lett. 64, 3374 (1994).
    [CrossRef]
  5. D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, Phys. Rev. E 57, 4757 (1998).
    [CrossRef]
  6. J. Fatome, S. Pitois, and G. Millot, IEEE J. Quantum Electron. 42, 1038 (2006).
    [CrossRef]
  7. E. Ciaramella, F. Curti, and S. Trillo, IEEE Photon. Technol. Lett. 13, 142 (2001).
    [CrossRef]
  8. P. V. Mamyshev, C. Bosshard, and G. I. Stegeman, J. Opt. Soc. Am. B 11, 1254 (1994).
    [CrossRef]
  9. S. Jia, W. Wan, and J. W. Fleischer, Opt. Lett. 32, 1668(2007).
    [CrossRef] [PubMed]
  10. A. V. Gurevich and L. P. Pitaevskii, Zh. Eksp. Teor. Fiz 65, 590 (1973) [Sov. Phys. JETP 38, 291 (1974)].
  11. Y. Kodama and S. Wabnitz, Opt. Lett. 20, 2291 (1995).
    [CrossRef] [PubMed]
  12. M. G. Forest and K. T. R. McLaughlin, J. Nonlinear Sci. 8, 43 (1998).
    [CrossRef]
  13. A. M. Kamchatnov, R. A. Kraenkel, and B. A. Umarov, Phys. Rev. E 66, 036609 (2002).
    [CrossRef]
  14. J. E. Rothenberg and D. Grischkowsky, Phys. Rev. Lett. 62, 531 (1989).
    [CrossRef] [PubMed]
  15. W. Wan, S. Jia, and J. W. Fleischer, Nature Phys. 3, 46 (2007).
    [CrossRef]
  16. S. Jia, W. Wan, and J. W. Fleischer, Phys. Rev. Lett. 99, 223901 (2007).
    [CrossRef]
  17. C. Barsi, W. Wan, C. Sun, and J. W. Fleischer, Opt. Lett. 32, 2930 (2007).
    [CrossRef] [PubMed]
  18. N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 99, 043903 (2007).
    [CrossRef] [PubMed]
  19. C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 102, 083902 (2009).
    [CrossRef] [PubMed]
  20. M. A. Hoefer and M. J. Ablowitz, Physica D (Amsterdam) 236, 44 (2007).
    [CrossRef]
  21. R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves (Cambridge, 1997).
    [CrossRef]
  22. X. D. Liu, S. Osher, and T. Chan, J. Comput. Phys. 115, 200 (1994).
    [CrossRef]
  23. A. Armaroli, S. Trillo, and A. Fratalocchi, Phys. Rev. A 80, 053803 (2009).
    [CrossRef]

2009 (2)

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

A. Armaroli, S. Trillo, and A. Fratalocchi, Phys. Rev. A 80, 053803 (2009).
[CrossRef]

2007 (6)

M. A. Hoefer and M. J. Ablowitz, Physica D (Amsterdam) 236, 44 (2007).
[CrossRef]

S. Jia, W. Wan, and J. W. Fleischer, Opt. Lett. 32, 1668(2007).
[CrossRef] [PubMed]

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

S. Jia, W. Wan, and J. W. Fleischer, Phys. Rev. Lett. 99, 223901 (2007).
[CrossRef]

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

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

2006 (1)

J. Fatome, S. Pitois, and G. Millot, IEEE J. Quantum Electron. 42, 1038 (2006).
[CrossRef]

2002 (1)

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

2001 (1)

E. Ciaramella, F. Curti, and S. Trillo, IEEE Photon. Technol. Lett. 13, 142 (2001).
[CrossRef]

1998 (2)

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, Phys. Rev. E 57, 4757 (1998).
[CrossRef]

M. G. Forest and K. T. R. McLaughlin, J. Nonlinear Sci. 8, 43 (1998).
[CrossRef]

1995 (1)

1994 (4)

P. V. Mamyshev, C. Bosshard, and G. I. Stegeman, J. Opt. Soc. Am. B 11, 1254 (1994).
[CrossRef]

S. Trillo, S. Wabnitz, and T. A. B. Kennedy, Phys. Rev. A 50, 1732 (1994).
[CrossRef] [PubMed]

P. V. Mamyshev, P. G. Wigley, J. Wilson, C. Bosshard, and G. I. Stegeman, Appl. Phys. Lett. 64, 3374 (1994).
[CrossRef]

X. D. Liu, S. Osher, and T. Chan, J. Comput. Phys. 115, 200 (1994).
[CrossRef]

1993 (1)

C. S. West and T. A. B. Kennedy, Phys. Rev. A 47, 1252 (1993).
[CrossRef] [PubMed]

1991 (1)

J. R. Thompson and R. Roy, Phys. Rev. A 43, 4987 (1991).
[CrossRef] [PubMed]

1989 (1)

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

1973 (1)

A. V. Gurevich and L. P. Pitaevskii, Zh. Eksp. Teor. Fiz 65, 590 (1973) [Sov. Phys. JETP 38, 291 (1974)].

Ablowitz, M. J.

M. A. Hoefer and M. J. Ablowitz, Physica D (Amsterdam) 236, 44 (2007).
[CrossRef]

Armaroli, A.

A. Armaroli, S. Trillo, and A. Fratalocchi, Phys. Rev. A 80, 053803 (2009).
[CrossRef]

Barsi, C.

Beletic, J. W.

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, Phys. Rev. E 57, 4757 (1998).
[CrossRef]

Bosshard, C.

P. V. Mamyshev, P. G. Wigley, J. Wilson, C. Bosshard, and G. I. Stegeman, Appl. Phys. Lett. 64, 3374 (1994).
[CrossRef]

P. V. Mamyshev, C. Bosshard, and G. I. Stegeman, J. Opt. Soc. Am. B 11, 1254 (1994).
[CrossRef]

Chan, T.

X. D. Liu, S. Osher, and T. Chan, J. Comput. Phys. 115, 200 (1994).
[CrossRef]

Ciaramella, E.

E. Ciaramella, F. Curti, and S. Trillo, IEEE Photon. Technol. Lett. 13, 142 (2001).
[CrossRef]

Conti, C.

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

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

Curti, F.

E. Ciaramella, F. Curti, and S. Trillo, IEEE Photon. Technol. Lett. 13, 142 (2001).
[CrossRef]

Fatome, J.

J. Fatome, S. Pitois, and G. Millot, IEEE J. Quantum Electron. 42, 1038 (2006).
[CrossRef]

Fleischer, J. W.

S. Jia, W. Wan, and J. W. Fleischer, Opt. Lett. 32, 1668(2007).
[CrossRef] [PubMed]

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

S. Jia, W. Wan, and J. W. Fleischer, Phys. Rev. Lett. 99, 223901 (2007).
[CrossRef]

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

Forest, M. G.

M. G. Forest and K. T. R. McLaughlin, J. Nonlinear Sci. 8, 43 (1998).
[CrossRef]

Fratalocchi, A.

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

A. Armaroli, S. Trillo, and A. Fratalocchi, Phys. Rev. A 80, 053803 (2009).
[CrossRef]

Ghofraniha, N.

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

Grischkowsky, D.

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

Gurevich, A. V.

A. V. Gurevich and L. P. Pitaevskii, Zh. Eksp. Teor. Fiz 65, 590 (1973) [Sov. Phys. JETP 38, 291 (1974)].

Hart, D. L.

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, Phys. Rev. E 57, 4757 (1998).
[CrossRef]

Hoefer, M. A.

M. A. Hoefer and M. J. Ablowitz, Physica D (Amsterdam) 236, 44 (2007).
[CrossRef]

Jia, S.

S. Jia, W. Wan, and J. W. Fleischer, Phys. Rev. Lett. 99, 223901 (2007).
[CrossRef]

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

S. Jia, W. Wan, and J. W. Fleischer, Opt. Lett. 32, 1668(2007).
[CrossRef] [PubMed]

Johnson, R. S.

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves (Cambridge, 1997).
[CrossRef]

Judy, A. F.

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, Phys. Rev. E 57, 4757 (1998).
[CrossRef]

Kamchatnov, A. M.

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

Kennedy, T. A. B.

S. Trillo, S. Wabnitz, and T. A. B. Kennedy, Phys. Rev. A 50, 1732 (1994).
[CrossRef] [PubMed]

C. S. West and T. A. B. Kennedy, Phys. Rev. A 47, 1252 (1993).
[CrossRef] [PubMed]

Kodama, Y.

Kraenkel, R. A.

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

Liu, X. D.

X. D. Liu, S. Osher, and T. Chan, J. Comput. Phys. 115, 200 (1994).
[CrossRef]

Mamyshev, P. V.

P. V. Mamyshev, C. Bosshard, and G. I. Stegeman, J. Opt. Soc. Am. B 11, 1254 (1994).
[CrossRef]

P. V. Mamyshev, P. G. Wigley, J. Wilson, C. Bosshard, and G. I. Stegeman, Appl. Phys. Lett. 64, 3374 (1994).
[CrossRef]

McLaughlin, K. T. R.

M. G. Forest and K. T. R. McLaughlin, J. Nonlinear Sci. 8, 43 (1998).
[CrossRef]

Millot, G.

J. Fatome, S. Pitois, and G. Millot, IEEE J. Quantum Electron. 42, 1038 (2006).
[CrossRef]

Osher, S.

X. D. Liu, S. Osher, and T. Chan, J. Comput. Phys. 115, 200 (1994).
[CrossRef]

Peccianti, M.

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

Pitaevskii, L. P.

A. V. Gurevich and L. P. Pitaevskii, Zh. Eksp. Teor. Fiz 65, 590 (1973) [Sov. Phys. JETP 38, 291 (1974)].

Pitois, S.

J. Fatome, S. Pitois, and G. Millot, IEEE J. Quantum Electron. 42, 1038 (2006).
[CrossRef]

Rothenberg, J. E.

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

Roy, R.

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, Phys. Rev. E 57, 4757 (1998).
[CrossRef]

J. R. Thompson and R. Roy, Phys. Rev. A 43, 4987 (1991).
[CrossRef] [PubMed]

Ruocco, G.

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

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

Stegeman, G. I.

P. V. Mamyshev, P. G. Wigley, J. Wilson, C. Bosshard, and G. I. Stegeman, Appl. Phys. Lett. 64, 3374 (1994).
[CrossRef]

P. V. Mamyshev, C. Bosshard, and G. I. Stegeman, J. Opt. Soc. Am. B 11, 1254 (1994).
[CrossRef]

Sun, C.

Thompson, J. R.

J. R. Thompson and R. Roy, Phys. Rev. A 43, 4987 (1991).
[CrossRef] [PubMed]

Trillo, S.

A. Armaroli, S. Trillo, and A. Fratalocchi, Phys. Rev. A 80, 053803 (2009).
[CrossRef]

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

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

E. Ciaramella, F. Curti, and S. Trillo, IEEE Photon. Technol. Lett. 13, 142 (2001).
[CrossRef]

S. Trillo, S. Wabnitz, and T. A. B. Kennedy, Phys. Rev. A 50, 1732 (1994).
[CrossRef] [PubMed]

Umarov, B. A.

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

Wabnitz, S.

Y. Kodama and S. Wabnitz, Opt. Lett. 20, 2291 (1995).
[CrossRef] [PubMed]

S. Trillo, S. Wabnitz, and T. A. B. Kennedy, Phys. Rev. A 50, 1732 (1994).
[CrossRef] [PubMed]

Wan, W.

S. Jia, W. Wan, and J. W. Fleischer, Opt. Lett. 32, 1668(2007).
[CrossRef] [PubMed]

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

S. Jia, W. Wan, and J. W. Fleischer, Phys. Rev. Lett. 99, 223901 (2007).
[CrossRef]

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

West, C. S.

C. S. West and T. A. B. Kennedy, Phys. Rev. A 47, 1252 (1993).
[CrossRef] [PubMed]

Wigley, P. G.

P. V. Mamyshev, P. G. Wigley, J. Wilson, C. Bosshard, and G. I. Stegeman, Appl. Phys. Lett. 64, 3374 (1994).
[CrossRef]

Wilson, J.

P. V. Mamyshev, P. G. Wigley, J. Wilson, C. Bosshard, and G. I. Stegeman, Appl. Phys. Lett. 64, 3374 (1994).
[CrossRef]

Appl. Phys. Lett. (1)

P. V. Mamyshev, P. G. Wigley, J. Wilson, C. Bosshard, and G. I. Stegeman, Appl. Phys. Lett. 64, 3374 (1994).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Fatome, S. Pitois, and G. Millot, IEEE J. Quantum Electron. 42, 1038 (2006).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

E. Ciaramella, F. Curti, and S. Trillo, IEEE Photon. Technol. Lett. 13, 142 (2001).
[CrossRef]

J. Comput. Phys. (1)

X. D. Liu, S. Osher, and T. Chan, J. Comput. Phys. 115, 200 (1994).
[CrossRef]

J. Nonlinear Sci. (1)

M. G. Forest and K. T. R. McLaughlin, J. Nonlinear Sci. 8, 43 (1998).
[CrossRef]

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

Nature Phys. (1)

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

Opt. Lett. (3)

Phys. Rev. A (4)

A. Armaroli, S. Trillo, and A. Fratalocchi, Phys. Rev. A 80, 053803 (2009).
[CrossRef]

J. R. Thompson and R. Roy, Phys. Rev. A 43, 4987 (1991).
[CrossRef] [PubMed]

C. S. West and T. A. B. Kennedy, Phys. Rev. A 47, 1252 (1993).
[CrossRef] [PubMed]

S. Trillo, S. Wabnitz, and T. A. B. Kennedy, Phys. Rev. A 50, 1732 (1994).
[CrossRef] [PubMed]

Phys. Rev. E (2)

D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, Phys. Rev. E 57, 4757 (1998).
[CrossRef]

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

Phys. Rev. Lett. (4)

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

S. Jia, W. Wan, and J. W. Fleischer, Phys. Rev. Lett. 99, 223901 (2007).
[CrossRef]

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

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

Physica D (Amsterdam) (1)

M. A. Hoefer and M. J. Ablowitz, Physica D (Amsterdam) 236, 44 (2007).
[CrossRef]

Zh. Eksp. Teor. Fiz (1)

A. V. Gurevich and L. P. Pitaevskii, Zh. Eksp. Teor. Fiz 65, 590 (1973) [Sov. Phys. JETP 38, 291 (1974)].

Other (1)

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves (Cambridge, 1997).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Recursive FWM dynamics from Eq. (1) with ε = 0.4 : (a) color level plot of power | u ( z , t ) | 2 ; (b) snapshots, comparing the output (red dots, z = 3.7 ) and the stage of maximum compression ( z = 1.85 , black solid) with the input (blue dashed).

Fig. 2
Fig. 2

(a) Same as in Fig. 1 for ε = 0.04 . Snapshots at (b) point of singularity z = 0.34 and (c) point of collision between adjacent DSWs, z = 0.95 .

Fig. 3
Fig. 3

Colliding DSWs from an imbalanced input ( η = 0.9 , ε = 0.02 ): (a) NLS dynamics (color level plot of power). Snapshots of (b) power ρ and (c) chirp v comparing outcomes of NLS (solid lines) and hydrodynamic limit [Eqs. (4), red dots] at z = 0.66 (where the left front forms, the right one exhibiting already the onset of oscillations), with the input (dashed).

Fig. 4
Fig. 4

Comparison of FWM spectra characterizing the recursive case [(a) ε = 0.4 ] and DSW generation [(b) ε = 0.04 ]. Here η = 0.5 .

Equations (4)

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

i ε u z ε 2 2 2 u t 2 + | u | 2 u = 0.
u 0 ( t ) = η exp ( i ω t / 2 ) + 1 η exp ( i ω t / 2 ) ,
ε L n l L d = Δ f 4 k γ P t .
{ ρ z + ( ρ v ) t = 0 v z + v v t + ρ t = 0 ,

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