Abstract

We investigate dispersive shock waves developing via a gradient catastrophe during propagation of a dark beam in Kerr defocusing media, showing that a good degree of control, and even shock suppression, is possible by introducing a suitable phase chirp. Insight into the process is obtained by means of a suitable reduction of the hydrodynamic limit of the governing nonlinear Schrödinger equation.

© 2010 Optical Society of America

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  1. G. B. Whitham, Linear and Nonlinear Waves (Wiley, 1974).
  2. A. V. Gurevich and L. P. Pitaevskii, Zh. Eksp. Teor. Fiz 65, 590 (1973) [Sov. Phys. JETP 38, 291 (1974) (in Russian)].
  3. W. J. Tomlinson, R. H. Stolen, and A. M. Johnson, Opt. Lett. 10, 457 (1985).
    [Crossref] [PubMed]
  4. J. E. Rothenberg and D. Grischkowsky, Phys. Rev. Lett. 62, 531 (1989).
    [Crossref] [PubMed]
  5. M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, Phys. Rev. A 74, 023623(2006).
    [Crossref]
  6. W. Wan, S. Jia, and J. W. Fleischer, Nat. Phys. 3, 46 (2007).
    [Crossref]
  7. N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 99, 043903 (2007).
    [Crossref] [PubMed]
  8. A. M. Kamchatnov, R. A. Kraenkel, and B. A. Umarov, Phys. Rev. E 66, 036609 (2002).
    [Crossref]
  9. Z. Dutton, M. Budde, C. Slowe, and L. V. Hau, Science 293, 663 (2001).
    [Crossref] [PubMed]
  10. C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, Phys. Rev. Lett. 102, 083902 (2009).
    [Crossref] [PubMed]
  11. Y. Kodama and S. Wabnitz, Opt. Lett. 20, 2291 (1995).
    [Crossref] [PubMed]
  12. M. G. Forest, J. N. Kutz, and K. T. R. McLaughlin, J. Opt. Soc. Am. B 16, 1856 (1999).
    [Crossref]
  13. D. Anderson, M. Desaix, M. Karlsson, M. Lisak, and M. L. Quiroiga-Teixeiro, J. Opt. Soc. Am. B 10, 1185 (1993).
    [Crossref]
  14. J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, Nat. Phys. 3, 597 (2007).
    [Crossref]
  15. L. D. Landau and E. M. Lifshitz, Fluid Mechanics(Butterworth-Heinemann, 1987).
  16. O. C. Wright, M. G. Forest, and K. T. R. McLaughlin, Phys. Lett. A 257, 170 (1999).
    [Crossref]
  17. P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (Society for Industrial and Applied Mathematics, 1973).
    [Crossref]

2009 (1)

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

2007 (3)

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, Nat. Phys. 3, 597 (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] [PubMed]

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]

2002 (1)

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

2001 (1)

Z. Dutton, M. Budde, C. Slowe, and L. V. Hau, Science 293, 663 (2001).
[Crossref] [PubMed]

1999 (2)

O. C. Wright, M. G. Forest, and K. T. R. McLaughlin, Phys. Lett. A 257, 170 (1999).
[Crossref]

M. G. Forest, J. N. Kutz, and K. T. R. McLaughlin, J. Opt. Soc. Am. B 16, 1856 (1999).
[Crossref]

1995 (1)

1993 (1)

1989 (1)

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

1985 (1)

1973 (1)

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

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]

Anderson, D.

Budde, M.

Z. Dutton, M. Budde, C. Slowe, and L. V. Hau, Science 293, 663 (2001).
[Crossref] [PubMed]

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]

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]

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.

Dudley, J. M.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, Nat. Phys. 3, 597 (2007).
[Crossref]

Dutton, Z.

Z. Dutton, M. Budde, C. Slowe, and L. V. Hau, Science 293, 663 (2001).
[Crossref] [PubMed]

Engels, P.

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

Finot, C.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, Nat. Phys. 3, 597 (2007).
[Crossref]

Fleischer, J. W.

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

Forest, M. G.

O. C. Wright, M. G. Forest, and K. T. R. McLaughlin, Phys. Lett. A 257, 170 (1999).
[Crossref]

M. G. Forest, J. N. Kutz, and K. T. R. McLaughlin, J. Opt. Soc. Am. B 16, 1856 (1999).
[Crossref]

Fratalocchi, A.

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

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) (in Russian)].

Hau, L. V.

Z. Dutton, M. Budde, C. Slowe, and L. V. Hau, Science 293, 663 (2001).
[Crossref] [PubMed]

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]

Jia, S.

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

Johnson, A. M.

Kamchatnov, A. M.

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

Karlsson, M.

Kodama, Y.

Kraenkel, R. A.

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

Kutz, J. N.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Fluid Mechanics(Butterworth-Heinemann, 1987).

Lax, P. D.

P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (Society for Industrial and Applied Mathematics, 1973).
[Crossref]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Fluid Mechanics(Butterworth-Heinemann, 1987).

Lisak, M.

McLaughlin, K. T. R.

M. G. Forest, J. N. Kutz, and K. T. R. McLaughlin, J. Opt. Soc. Am. B 16, 1856 (1999).
[Crossref]

O. C. Wright, M. G. Forest, and K. T. R. McLaughlin, Phys. Lett. A 257, 170 (1999).
[Crossref]

Millot, G.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, Nat. Phys. 3, 597 (2007).
[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) (in Russian)].

Quiroiga-Teixeiro, M. L.

Richardson, D. J.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, Nat. Phys. 3, 597 (2007).
[Crossref]

Rothenberg, J. E.

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

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]

Slowe, C.

Z. Dutton, M. Budde, C. Slowe, and L. V. Hau, Science 293, 663 (2001).
[Crossref] [PubMed]

Stolen, R. H.

Tomlinson, W. J.

Trillo, S.

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]

Umarov, B. A.

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

Wabnitz, S.

Wan, W.

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

Whitham, G. B.

G. B. Whitham, Linear and Nonlinear Waves (Wiley, 1974).

Wright, O. C.

O. C. Wright, M. G. Forest, and K. T. R. McLaughlin, Phys. Lett. A 257, 170 (1999).
[Crossref]

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

Nat. Phys. (2)

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, Nat. Phys. 3, 597 (2007).
[Crossref]

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

Opt. Lett. (2)

Phys. Lett. A (1)

O. C. Wright, M. G. Forest, and K. T. R. McLaughlin, Phys. Lett. A 257, 170 (1999).
[Crossref]

Phys. Rev. A (1)

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

Phys. Rev. E (1)

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

Phys. Rev. Lett. (3)

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

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

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

Science (1)

Z. Dutton, M. Budde, C. Slowe, and L. V. Hau, Science 293, 663 (2001).
[Crossref] [PubMed]

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) (in Russian)].

Other (3)

G. B. Whitham, Linear and Nonlinear Waves (Wiley, 1974).

P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (Society for Industrial and Applied Mathematics, 1973).
[Crossref]

L. D. Landau and E. M. Lifshitz, Fluid Mechanics(Butterworth-Heinemann, 1987).

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Figures (3)

Fig. 1
Fig. 1

(a) Profiles ruled by Eq. (4) with ρ 0 = exp ( x 2 ) (the thick solid curve corresponds to shock formation at z = z c = 0.55 ). (b) As in (a) for a dark beam ρ 0 = tanh 2 ( x ) , where z c = 0.33 (input phase profile θ 0 ( x ) in the inset). (c) Characteristic lines associated with the dynamics shown in (b). (d) Formation of a cusp singularity ruled by Eq. (2) with v 0 ( x ) = 0 .

Fig. 2
Fig. 2

Full NLSE solution for ε = 0.05 and the same chirped input that leads to the dynamics shown in Fig. 1b. (a) Level plot of the intensity | ψ ( x , z ) | 2 . (b) Snapshots showing the asymmetric front right after-shock formation ( z = 0.4 ) and the undular bore in its early stage ( z = 1 ) or full stage of formation ( z = 3 ).

Fig. 3
Fig. 3

Intensity level plots obtained with input phase (a)  θ 0 = 2 log ( cosh x ) (shock suppression) and (b) θ 0 = 2 log ( cosh x ) (shock enhancement). (c) Snapshots comparing smooth profiles obtained at z = 1 with input phase profile (see inset) θ 0 = x 2 / 2 (solid curve) or θ 0 = 2 log ( cosh x ) (dashed–dotted curve). (d) Intensity profile beyond breaking obtained with convex parabolic phase θ 0 = x 2 (see inset). All results are obtained numerically from Eq. (1) with ε = 0.05 .

Equations (4)

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i ε ψ z + ε 2 2 ψ x x | ψ | 2 ψ = 0.
{ ρ z + ( ρ v ) x = 0 , v z + v v x + ρ x = 0 ,
ρ z ± 3 ρ ρ x = 0 ,
ρ = ρ 0 ( x 3 ρ z ) , v = ± 2 ρ 0 ( x 3 ρ z ) .

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