Abstract

We consider the statistics of light amplitude fluctuations for the propagation of a laser beam subjected to multiple filamentation in an amplified Kerr media, with both linear and nonlinear dissipation. Dissipation arrests the catastrophic collapse of filaments, causing their disintegration into almost linear waves. These waves form a nearly Gaussian random field that seeds new filaments. For small amplitudes the probability density function (PDF) of light amplitude is close to Gaussian, while for large amplitudes the PDF has a long powerlike tail that corresponds to strong non-Gaussian fluctuations, i.e., intermittency of strong optical turbulence. This tail is determined by the universal form of near singular filaments and the PDF for the maximum amplitudes of the filaments.

© 2010 Optical Society of America

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References

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  1. R. W. Boyd, Nonlinear Optics (Academic Press, 2003).
  2. C. Sulem and P. L. Sulem, Nonlinear Schroedinger Equations: Self-Focusing and Wave Collapse (Springer, 1999).
  3. J. D. Lindl, P. Amendt, R. L. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, Phys. Plasmas 11, 339 (2004).
    [CrossRef]
  4. P. M. Lushnikov and H. A. Rose, Phys. Rev. Lett. 92, 255003 (2004).
    [CrossRef] [PubMed]
  5. P. M. Lushnikov and H. A. Rose, Plasma Phys. Controlled Fusion 48, 1501 (2006).
    [CrossRef]
  6. M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, Phys. Rev. Lett. 83, 2938 (1999).
    [CrossRef]
  7. L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, Rep. Prog. Phys. 70, 1633 (2007).
    [CrossRef]
  8. S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, Physica D (Amsterdam) 57, 96 (1992).
    [CrossRef]
  9. U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge U. Press, 1995).
  10. A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, Phys. Rev. Lett. 103, 173901 (2009).
    [CrossRef] [PubMed]
  11. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, Nature 450, 1054 (2007).
    [CrossRef] [PubMed]
  12. H. Iwasaki and S. Toh, Progr. Theor. Phys. 87, 1127 (1992).
    [CrossRef]
  13. W. E, K. Khanin, A. Mazel, and Y. Sinai, Phys. Rev. Lett. 78, 1904 (1997).
    [CrossRef]
  14. M. Tlidi, M. Haelterman, and P. Mandel, Europhys. Lett. 42, 505 (1998).
    [CrossRef]
  15. P. M. Lushnikov and M. Saffman, Phys. Rev. E 62, 5793 (2000).
    [CrossRef]
  16. S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, Phys. Rev. Lett. 86, 5470 (2001).
    [CrossRef] [PubMed]
  17. L. Guyon, F. Courvoisier, V. Boutou, R. Nuter, A. Vinçotte, S. Champeaux, L. Bergé, P. Glorieux, J.-P. Wolf, Phys. Rev. A 73, 051802(R) (2006).
    [CrossRef]
  18. G. Fibich and G. Papanicolaou, SIAM J. Appl. Math. 60, 183 (1999).
    [CrossRef]
  19. Y. Chung, P. M. Lushnikov, and N. Vladimirova, AIP Conf. Proc. 1168, 1235 (2009).
    [CrossRef]

2009 (2)

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, Phys. Rev. Lett. 103, 173901 (2009).
[CrossRef] [PubMed]

Y. Chung, P. M. Lushnikov, and N. Vladimirova, AIP Conf. Proc. 1168, 1235 (2009).
[CrossRef]

2007 (2)

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, Nature 450, 1054 (2007).
[CrossRef] [PubMed]

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, Rep. Prog. Phys. 70, 1633 (2007).
[CrossRef]

2006 (2)

P. M. Lushnikov and H. A. Rose, Plasma Phys. Controlled Fusion 48, 1501 (2006).
[CrossRef]

L. Guyon, F. Courvoisier, V. Boutou, R. Nuter, A. Vinçotte, S. Champeaux, L. Bergé, P. Glorieux, J.-P. Wolf, Phys. Rev. A 73, 051802(R) (2006).
[CrossRef]

2004 (2)

J. D. Lindl, P. Amendt, R. L. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, Phys. Plasmas 11, 339 (2004).
[CrossRef]

P. M. Lushnikov and H. A. Rose, Phys. Rev. Lett. 92, 255003 (2004).
[CrossRef] [PubMed]

2001 (1)

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, Phys. Rev. Lett. 86, 5470 (2001).
[CrossRef] [PubMed]

2000 (1)

P. M. Lushnikov and M. Saffman, Phys. Rev. E 62, 5793 (2000).
[CrossRef]

1999 (2)

G. Fibich and G. Papanicolaou, SIAM J. Appl. Math. 60, 183 (1999).
[CrossRef]

M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, Phys. Rev. Lett. 83, 2938 (1999).
[CrossRef]

1998 (1)

M. Tlidi, M. Haelterman, and P. Mandel, Europhys. Lett. 42, 505 (1998).
[CrossRef]

1997 (1)

W. E, K. Khanin, A. Mazel, and Y. Sinai, Phys. Rev. Lett. 78, 1904 (1997).
[CrossRef]

1992 (2)

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, Physica D (Amsterdam) 57, 96 (1992).
[CrossRef]

H. Iwasaki and S. Toh, Progr. Theor. Phys. 87, 1127 (1992).
[CrossRef]

Amendt, P.

J. D. Lindl, P. Amendt, R. L. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, Phys. Plasmas 11, 339 (2004).
[CrossRef]

Arecchi, F. T.

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, Phys. Rev. Lett. 103, 173901 (2009).
[CrossRef] [PubMed]

Bergé, L.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, Rep. Prog. Phys. 70, 1633 (2007).
[CrossRef]

L. Guyon, F. Courvoisier, V. Boutou, R. Nuter, A. Vinçotte, S. Champeaux, L. Bergé, P. Glorieux, J.-P. Wolf, Phys. Rev. A 73, 051802(R) (2006).
[CrossRef]

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, Phys. Rev. Lett. 86, 5470 (2001).
[CrossRef] [PubMed]

Berger, R. L.

J. D. Lindl, P. Amendt, R. L. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, Phys. Plasmas 11, 339 (2004).
[CrossRef]

Bortolozzo, U.

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, Phys. Rev. Lett. 103, 173901 (2009).
[CrossRef] [PubMed]

Boutou, V.

L. Guyon, F. Courvoisier, V. Boutou, R. Nuter, A. Vinçotte, S. Champeaux, L. Bergé, P. Glorieux, J.-P. Wolf, Phys. Rev. A 73, 051802(R) (2006).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic Press, 2003).

Champeaux, S.

L. Guyon, F. Courvoisier, V. Boutou, R. Nuter, A. Vinçotte, S. Champeaux, L. Bergé, P. Glorieux, J.-P. Wolf, Phys. Rev. A 73, 051802(R) (2006).
[CrossRef]

Chung, Y.

Y. Chung, P. M. Lushnikov, and N. Vladimirova, AIP Conf. Proc. 1168, 1235 (2009).
[CrossRef]

Couairon, A.

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, Phys. Rev. Lett. 86, 5470 (2001).
[CrossRef] [PubMed]

Courvoisier, F.

L. Guyon, F. Courvoisier, V. Boutou, R. Nuter, A. Vinçotte, S. Champeaux, L. Bergé, P. Glorieux, J.-P. Wolf, Phys. Rev. A 73, 051802(R) (2006).
[CrossRef]

Dyachenko, S.

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, Physica D (Amsterdam) 57, 96 (1992).
[CrossRef]

E, W.

W. E, K. Khanin, A. Mazel, and Y. Sinai, Phys. Rev. Lett. 78, 1904 (1997).
[CrossRef]

Fibich, G.

G. Fibich and G. Papanicolaou, SIAM J. Appl. Math. 60, 183 (1999).
[CrossRef]

Franco, M.

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, Phys. Rev. Lett. 86, 5470 (2001).
[CrossRef] [PubMed]

Frisch, U.

U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge U. Press, 1995).

Glendinning, S. G.

J. D. Lindl, P. Amendt, R. L. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, Phys. Plasmas 11, 339 (2004).
[CrossRef]

Glenzer, S. H.

J. D. Lindl, P. Amendt, R. L. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, Phys. Plasmas 11, 339 (2004).
[CrossRef]

Glorieux, P.

L. Guyon, F. Courvoisier, V. Boutou, R. Nuter, A. Vinçotte, S. Champeaux, L. Bergé, P. Glorieux, J.-P. Wolf, Phys. Rev. A 73, 051802(R) (2006).
[CrossRef]

Guyon, L.

L. Guyon, F. Courvoisier, V. Boutou, R. Nuter, A. Vinçotte, S. Champeaux, L. Bergé, P. Glorieux, J.-P. Wolf, Phys. Rev. A 73, 051802(R) (2006).
[CrossRef]

Haan, S. W.

J. D. Lindl, P. Amendt, R. L. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, Phys. Plasmas 11, 339 (2004).
[CrossRef]

Haelterman, M.

M. Tlidi, M. Haelterman, and P. Mandel, Europhys. Lett. 42, 505 (1998).
[CrossRef]

Iwasaki, H.

H. Iwasaki and S. Toh, Progr. Theor. Phys. 87, 1127 (1992).
[CrossRef]

Jalali, B.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, Nature 450, 1054 (2007).
[CrossRef] [PubMed]

Kasparian, J.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, Rep. Prog. Phys. 70, 1633 (2007).
[CrossRef]

Kauffman, R. L.

J. D. Lindl, P. Amendt, R. L. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, Phys. Plasmas 11, 339 (2004).
[CrossRef]

Khanin, K.

W. E, K. Khanin, A. Mazel, and Y. Sinai, Phys. Rev. Lett. 78, 1904 (1997).
[CrossRef]

Kolesik, M.

M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, Phys. Rev. Lett. 83, 2938 (1999).
[CrossRef]

Koonath, P.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, Nature 450, 1054 (2007).
[CrossRef] [PubMed]

Landen, O. L.

J. D. Lindl, P. Amendt, R. L. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, Phys. Plasmas 11, 339 (2004).
[CrossRef]

Lindl, J. D.

J. D. Lindl, P. Amendt, R. L. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, Phys. Plasmas 11, 339 (2004).
[CrossRef]

Lushnikov, P. M.

Y. Chung, P. M. Lushnikov, and N. Vladimirova, AIP Conf. Proc. 1168, 1235 (2009).
[CrossRef]

P. M. Lushnikov and H. A. Rose, Plasma Phys. Controlled Fusion 48, 1501 (2006).
[CrossRef]

P. M. Lushnikov and H. A. Rose, Phys. Rev. Lett. 92, 255003 (2004).
[CrossRef] [PubMed]

P. M. Lushnikov and M. Saffman, Phys. Rev. E 62, 5793 (2000).
[CrossRef]

Mandel, P.

M. Tlidi, M. Haelterman, and P. Mandel, Europhys. Lett. 42, 505 (1998).
[CrossRef]

Mazel, A.

W. E, K. Khanin, A. Mazel, and Y. Sinai, Phys. Rev. Lett. 78, 1904 (1997).
[CrossRef]

Mlejnek, M.

M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, Phys. Rev. Lett. 83, 2938 (1999).
[CrossRef]

Moloney, J. V.

M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, Phys. Rev. Lett. 83, 2938 (1999).
[CrossRef]

Montina, A.

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, Phys. Rev. Lett. 103, 173901 (2009).
[CrossRef] [PubMed]

Mysyrowicz, A.

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, Phys. Rev. Lett. 86, 5470 (2001).
[CrossRef] [PubMed]

Newell, A. C.

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, Physica D (Amsterdam) 57, 96 (1992).
[CrossRef]

Nuter, R.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, Rep. Prog. Phys. 70, 1633 (2007).
[CrossRef]

L. Guyon, F. Courvoisier, V. Boutou, R. Nuter, A. Vinçotte, S. Champeaux, L. Bergé, P. Glorieux, J.-P. Wolf, Phys. Rev. A 73, 051802(R) (2006).
[CrossRef]

Papanicolaou, G.

G. Fibich and G. Papanicolaou, SIAM J. Appl. Math. 60, 183 (1999).
[CrossRef]

Prade, B.

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, Phys. Rev. Lett. 86, 5470 (2001).
[CrossRef] [PubMed]

Pushkarev, A.

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, Physica D (Amsterdam) 57, 96 (1992).
[CrossRef]

Residori, S.

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, Phys. Rev. Lett. 103, 173901 (2009).
[CrossRef] [PubMed]

Ropers, C.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, Nature 450, 1054 (2007).
[CrossRef] [PubMed]

Rose, H. A.

P. M. Lushnikov and H. A. Rose, Plasma Phys. Controlled Fusion 48, 1501 (2006).
[CrossRef]

P. M. Lushnikov and H. A. Rose, Phys. Rev. Lett. 92, 255003 (2004).
[CrossRef] [PubMed]

Saffman, M.

P. M. Lushnikov and M. Saffman, Phys. Rev. E 62, 5793 (2000).
[CrossRef]

Sinai, Y.

W. E, K. Khanin, A. Mazel, and Y. Sinai, Phys. Rev. Lett. 78, 1904 (1997).
[CrossRef]

Skupin, S.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, Rep. Prog. Phys. 70, 1633 (2007).
[CrossRef]

Solli, D. R.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, Nature 450, 1054 (2007).
[CrossRef] [PubMed]

Sulem, C.

C. Sulem and P. L. Sulem, Nonlinear Schroedinger Equations: Self-Focusing and Wave Collapse (Springer, 1999).

Sulem, P. L.

C. Sulem and P. L. Sulem, Nonlinear Schroedinger Equations: Self-Focusing and Wave Collapse (Springer, 1999).

Suter, L. J.

J. D. Lindl, P. Amendt, R. L. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, Phys. Plasmas 11, 339 (2004).
[CrossRef]

Tlidi, M.

M. Tlidi, M. Haelterman, and P. Mandel, Europhys. Lett. 42, 505 (1998).
[CrossRef]

Toh, S.

H. Iwasaki and S. Toh, Progr. Theor. Phys. 87, 1127 (1992).
[CrossRef]

Tzortzakis, S.

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, Phys. Rev. Lett. 86, 5470 (2001).
[CrossRef] [PubMed]

Vinçotte, A.

L. Guyon, F. Courvoisier, V. Boutou, R. Nuter, A. Vinçotte, S. Champeaux, L. Bergé, P. Glorieux, J.-P. Wolf, Phys. Rev. A 73, 051802(R) (2006).
[CrossRef]

Vladimirova, N.

Y. Chung, P. M. Lushnikov, and N. Vladimirova, AIP Conf. Proc. 1168, 1235 (2009).
[CrossRef]

Wolf, J.-P.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, Rep. Prog. Phys. 70, 1633 (2007).
[CrossRef]

L. Guyon, F. Courvoisier, V. Boutou, R. Nuter, A. Vinçotte, S. Champeaux, L. Bergé, P. Glorieux, J.-P. Wolf, Phys. Rev. A 73, 051802(R) (2006).
[CrossRef]

Wright, E. M.

M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, Phys. Rev. Lett. 83, 2938 (1999).
[CrossRef]

Zakharov, V. E.

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, Physica D (Amsterdam) 57, 96 (1992).
[CrossRef]

AIP Conf. Proc. (1)

Y. Chung, P. M. Lushnikov, and N. Vladimirova, AIP Conf. Proc. 1168, 1235 (2009).
[CrossRef]

Europhys. Lett. (1)

M. Tlidi, M. Haelterman, and P. Mandel, Europhys. Lett. 42, 505 (1998).
[CrossRef]

Nature (1)

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, Nature 450, 1054 (2007).
[CrossRef] [PubMed]

Phys. Plasmas (1)

J. D. Lindl, P. Amendt, R. L. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman, O. L. Landen, and L. J. Suter, Phys. Plasmas 11, 339 (2004).
[CrossRef]

Phys. Rev. A (1)

L. Guyon, F. Courvoisier, V. Boutou, R. Nuter, A. Vinçotte, S. Champeaux, L. Bergé, P. Glorieux, J.-P. Wolf, Phys. Rev. A 73, 051802(R) (2006).
[CrossRef]

Phys. Rev. E (1)

P. M. Lushnikov and M. Saffman, Phys. Rev. E 62, 5793 (2000).
[CrossRef]

Phys. Rev. Lett. (5)

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, Phys. Rev. Lett. 86, 5470 (2001).
[CrossRef] [PubMed]

A. Montina, U. Bortolozzo, S. Residori, and F. T. Arecchi, Phys. Rev. Lett. 103, 173901 (2009).
[CrossRef] [PubMed]

W. E, K. Khanin, A. Mazel, and Y. Sinai, Phys. Rev. Lett. 78, 1904 (1997).
[CrossRef]

P. M. Lushnikov and H. A. Rose, Phys. Rev. Lett. 92, 255003 (2004).
[CrossRef] [PubMed]

M. Mlejnek, M. Kolesik, J. V. Moloney, and E. M. Wright, Phys. Rev. Lett. 83, 2938 (1999).
[CrossRef]

Physica D (Amsterdam) (1)

S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, Physica D (Amsterdam) 57, 96 (1992).
[CrossRef]

Plasma Phys. Controlled Fusion (1)

P. M. Lushnikov and H. A. Rose, Plasma Phys. Controlled Fusion 48, 1501 (2006).
[CrossRef]

Progr. Theor. Phys. (1)

H. Iwasaki and S. Toh, Progr. Theor. Phys. 87, 1127 (1992).
[CrossRef]

Rep. Prog. Phys. (1)

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, Rep. Prog. Phys. 70, 1633 (2007).
[CrossRef]

SIAM J. Appl. Math. (1)

G. Fibich and G. Papanicolaou, SIAM J. Appl. Math. 60, 183 (1999).
[CrossRef]

Other (3)

U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge U. Press, 1995).

R. W. Boyd, Nonlinear Optics (Academic Press, 2003).

C. Sulem and P. L. Sulem, Nonlinear Schroedinger Equations: Self-Focusing and Wave Collapse (Springer, 1999).

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Figures (4)

Fig. 1
Fig. 1

(a) N ( z ) from simulation of RNLS (1) with b = 20 , a = c = 1 , ϵ = 0.01 . (b) Zoom of N ( z ) in a smaller interval in z (solid curve, scale on the right) superimposed the | ψ | max ( z ) (dashed curve, scale on the left). All simulations used the fourth-order pseudospectral split-step algorithm on 12.8 x , y 12.8 with periodic boundary conditions at resolution 4096 × 4096 grid points. Initial conditions were a superposition of 100 randomly placed real-valued Gaussians with amplitudes and radii on [ - 2 , 2 ] and [1,2].

Fig. 2
Fig. 2

Snapshot of | ψ | (vertical axis) versus spatial coordinates ( x , y ) for simulation of Fig. 1.

Fig. 3
Fig. 3

Dependence of | ψ | max ( z ) for multiple individual collapsing filaments in (a) the nonrescaled units and (b) the rescaled units (see the text for the description of rescaling). Individual filaments are extracted from simulation of RNLS (1) with parameters of Fig. 1.

Fig. 4
Fig. 4

(a) P ( h ) for | ψ | = h (solid curve) for the same simulation as in Fig. 1. The dashed line shows fit to the Gaussian distribution, and the dotted line shows h 6 power law. Circles correspond to the solution of Eq. (7). (b) H max ( h max ) for | ψ | max max . 2231 collapse events with | ψ | max max > 10 are included in simulations in log–log scale (red dots). The dotted line shows h 1 power law. Scattering of data points for h 110 is due to lack of statistical ensemble for large collapses and is reduced for larger simulation times.

Equations (7)

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

i z ψ + ( 1 i a ϵ ) 2 ψ + ( 1 + i c ϵ ) | ψ | 2 ψ = i ϵ b ψ ,
i z ψ + 2 ψ + | ψ | 2 ψ = 0 .
| ψ ( r , z ) | = 1 L ( z ) R 0 ( ρ ) , ρ = r L ( z ) , r | r | ,
γ L d L d z ,
P ( h ) = δ ( | ψ ( r , z ) | h ) d r d z d r d z .
P ( h | h max ) z max d z d r δ ( h 1 L ( z ) R 0 ( r L ( z ) ) ) d ρ ρ L ( z max ) d L L 3 γ δ ( h 1 L ( z ) R 0 ( ρ ) ) d ρ ρ γ h 5 [ R 0 ( ρ ) ] 4 Θ ( R 0 ( 0 ) L ( z max ) h ) = C o n s t h 5 Θ ( h max h ) ,
P ( h ) = d h max P ( h | h max ) P max ( h max ) Const h 5 d h max Θ ( h max h ) P max ( h max ) = Const h 5 H max ( h ) ,

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