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

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]

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

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.

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]

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.

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]

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)

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

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

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

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]

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)

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

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

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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