Abstract

We investigate properties of dark solitons in nonlocal materials with an arbitrary degree of nonlocality. We employ the variational technique and describe dark solitons, for the first time to our knowledge, in the whole range of degree of nonlocality.

© 2010 Optical Society of America

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  1. G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
    [CrossRef] [PubMed]
  2. Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
    [CrossRef]
  3. W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B: Quantum Semiclass. Opt. 6, S288 (2004).
    [CrossRef]
  4. F. W. Dabby and J. B. Whinnery, Appl. Phys. Lett. 13, 284(1968).
    [CrossRef]
  5. S. Skupin, M. Saffman, and W. Krolikowski, Phys. Rev. Lett. 98, 263902 (2007).
    [CrossRef] [PubMed]
  6. D. Suter and T. Blasberg, Phys. Rev. A 48, 4583 (1993).
    [CrossRef] [PubMed]
  7. E. A. Ultanir, D. Michaelis, F. Lederer, and G. I. Stegeman, Opt. Lett. 28, 251 (2003).
    [CrossRef] [PubMed]
  8. S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewenstein, Phys. Rev. Lett. 835198 (1999).
    [CrossRef]
  9. E. Braun, L. P. Faucheux, and A. Libchaber, Phys. Rev. A 48, 611 (1993).
    [CrossRef] [PubMed]
  10. O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
    [CrossRef]
  11. Y. Y. Lin, R. K. Lee, and Y. S. Kivshar, J. Opt. Soc. Am. B 25, 576 (2008).
    [CrossRef]
  12. A. Armaroli, S. Trillo, and A. Fratalocchi, Phys. Rev. A 80, 053803 (2009).
    [CrossRef]
  13. S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, Phys. Rev. E 73, 066603 (2006).
    [CrossRef]
  14. A. W. Snyder and D. J. Mitchell, Science 276, 1538(1997).
    [CrossRef]
  15. M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, Opt. Lett. 27, 1460 (2002).
    [CrossRef]
  16. P. D. Rasmussen, O. Bang, and W. Krolikowski, Phys. Rev. E 72, 066611 (2005).
    [CrossRef]
  17. A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, Phys. Rev. Lett. 96, 043901 (2006).
    [CrossRef] [PubMed]
  18. Y. V. Kartashov and L. Torner, Opt. Lett. 32, 946 (2007).
    [CrossRef] [PubMed]
  19. S. Ouyang and Q. Guo, Opt. Express 17, 5170 (2009).
    [CrossRef] [PubMed]
  20. W. Krolikowski and O. Bang, Phys. Rev. E 63, 016610(2000).
    [CrossRef]
  21. N. I. Nikolov, D. Neshev, O. Bang, and W. Z. Krolikowski, Phys. Rev. E 68, 036614 (2003).
    [CrossRef]
  22. Y. S. Kivshar and W. Krolikowski, Opt. Commun. 114, 353 (1995).
    [CrossRef]
  23. L. J. Ge, Q. Wang, M. Shen, J. L. Shi, Q. Kong, and P. Hou, J. Opt. A Pure: Appl. Opt. 11, 065207 (2009).
    [CrossRef]

2009 (3)

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

S. Ouyang and Q. Guo, Opt. Express 17, 5170 (2009).
[CrossRef] [PubMed]

L. J. Ge, Q. Wang, M. Shen, J. L. Shi, Q. Kong, and P. Hou, J. Opt. A Pure: Appl. Opt. 11, 065207 (2009).
[CrossRef]

2008 (1)

2007 (2)

Y. V. Kartashov and L. Torner, Opt. Lett. 32, 946 (2007).
[CrossRef] [PubMed]

S. Skupin, M. Saffman, and W. Krolikowski, Phys. Rev. Lett. 98, 263902 (2007).
[CrossRef] [PubMed]

2006 (2)

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, Phys. Rev. E 73, 066603 (2006).
[CrossRef]

2005 (1)

P. D. Rasmussen, O. Bang, and W. Krolikowski, Phys. Rev. E 72, 066611 (2005).
[CrossRef]

2004 (1)

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B: Quantum Semiclass. Opt. 6, S288 (2004).
[CrossRef]

2003 (2)

E. A. Ultanir, D. Michaelis, F. Lederer, and G. I. Stegeman, Opt. Lett. 28, 251 (2003).
[CrossRef] [PubMed]

N. I. Nikolov, D. Neshev, O. Bang, and W. Z. Krolikowski, Phys. Rev. E 68, 036614 (2003).
[CrossRef]

2002 (2)

M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, Opt. Lett. 27, 1460 (2002).
[CrossRef]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
[CrossRef]

2000 (1)

W. Krolikowski and O. Bang, Phys. Rev. E 63, 016610(2000).
[CrossRef]

1999 (1)

S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewenstein, Phys. Rev. Lett. 835198 (1999).
[CrossRef]

1998 (1)

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

1997 (1)

A. W. Snyder and D. J. Mitchell, Science 276, 1538(1997).
[CrossRef]

1995 (1)

Y. S. Kivshar and W. Krolikowski, Opt. Commun. 114, 353 (1995).
[CrossRef]

1993 (2)

E. Braun, L. P. Faucheux, and A. Libchaber, Phys. Rev. A 48, 611 (1993).
[CrossRef] [PubMed]

D. Suter and T. Blasberg, Phys. Rev. A 48, 4583 (1993).
[CrossRef] [PubMed]

1991 (1)

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

1968 (1)

F. W. Dabby and J. B. Whinnery, Appl. Phys. Lett. 13, 284(1968).
[CrossRef]

Andersen, D. R.

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Armaroli, A.

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

Assanto, G.

Bang, O.

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, Phys. Rev. E 73, 066603 (2006).
[CrossRef]

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

P. D. Rasmussen, O. Bang, and W. Krolikowski, Phys. Rev. E 72, 066611 (2005).
[CrossRef]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B: Quantum Semiclass. Opt. 6, S288 (2004).
[CrossRef]

N. I. Nikolov, D. Neshev, O. Bang, and W. Z. Krolikowski, Phys. Rev. E 68, 036614 (2003).
[CrossRef]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
[CrossRef]

W. Krolikowski and O. Bang, Phys. Rev. E 63, 016610(2000).
[CrossRef]

Blasberg, T.

D. Suter and T. Blasberg, Phys. Rev. A 48, 4583 (1993).
[CrossRef] [PubMed]

Bongs, K.

S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewenstein, Phys. Rev. Lett. 835198 (1999).
[CrossRef]

Braun, E.

E. Braun, L. P. Faucheux, and A. Libchaber, Phys. Rev. A 48, 611 (1993).
[CrossRef] [PubMed]

Brzdakiewicz, K. A.

Burger, S.

S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewenstein, Phys. Rev. Lett. 835198 (1999).
[CrossRef]

Dabby, F. W.

F. W. Dabby and J. B. Whinnery, Appl. Phys. Lett. 13, 284(1968).
[CrossRef]

Dettmer, S.

S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewenstein, Phys. Rev. Lett. 835198 (1999).
[CrossRef]

Dreischuh, A.

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

Edmundson, D.

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, Phys. Rev. E 73, 066603 (2006).
[CrossRef]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B: Quantum Semiclass. Opt. 6, S288 (2004).
[CrossRef]

Ertmer, W.

S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewenstein, Phys. Rev. Lett. 835198 (1999).
[CrossRef]

Faucheux, L. P.

E. Braun, L. P. Faucheux, and A. Libchaber, Phys. Rev. A 48, 611 (1993).
[CrossRef] [PubMed]

Fratalocchi, A.

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

Ge, L. J.

L. J. Ge, Q. Wang, M. Shen, J. L. Shi, Q. Kong, and P. Hou, J. Opt. A Pure: Appl. Opt. 11, 065207 (2009).
[CrossRef]

Guo, Q.

Hou, P.

L. J. Ge, Q. Wang, M. Shen, J. L. Shi, Q. Kong, and P. Hou, J. Opt. A Pure: Appl. Opt. 11, 065207 (2009).
[CrossRef]

Kaplan, A. E.

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Kartashov, Y. V.

Kivshar, Y. S.

Y. Y. Lin, R. K. Lee, and Y. S. Kivshar, J. Opt. Soc. Am. B 25, 576 (2008).
[CrossRef]

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

Y. S. Kivshar and W. Krolikowski, Opt. Commun. 114, 353 (1995).
[CrossRef]

Kong, Q.

L. J. Ge, Q. Wang, M. Shen, J. L. Shi, Q. Kong, and P. Hou, J. Opt. A Pure: Appl. Opt. 11, 065207 (2009).
[CrossRef]

Krolikowski, W.

S. Skupin, M. Saffman, and W. Krolikowski, Phys. Rev. Lett. 98, 263902 (2007).
[CrossRef] [PubMed]

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, Phys. Rev. E 73, 066603 (2006).
[CrossRef]

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

P. D. Rasmussen, O. Bang, and W. Krolikowski, Phys. Rev. E 72, 066611 (2005).
[CrossRef]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B: Quantum Semiclass. Opt. 6, S288 (2004).
[CrossRef]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
[CrossRef]

W. Krolikowski and O. Bang, Phys. Rev. E 63, 016610(2000).
[CrossRef]

Y. S. Kivshar and W. Krolikowski, Opt. Commun. 114, 353 (1995).
[CrossRef]

Krolikowski, W. Z.

N. I. Nikolov, D. Neshev, O. Bang, and W. Z. Krolikowski, Phys. Rev. E 68, 036614 (2003).
[CrossRef]

Lederer, F.

Lee, R. K.

Lewenstein, M.

S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewenstein, Phys. Rev. Lett. 835198 (1999).
[CrossRef]

Libchaber, A.

E. Braun, L. P. Faucheux, and A. Libchaber, Phys. Rev. A 48, 611 (1993).
[CrossRef] [PubMed]

Lin, Y. Y.

Luther-Davies, B.

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

Michaelis, D.

Mitchell, D. J.

A. W. Snyder and D. J. Mitchell, Science 276, 1538(1997).
[CrossRef]

Neshev, D.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B: Quantum Semiclass. Opt. 6, S288 (2004).
[CrossRef]

N. I. Nikolov, D. Neshev, O. Bang, and W. Z. Krolikowski, Phys. Rev. E 68, 036614 (2003).
[CrossRef]

Neshev, D. N.

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

Nikolov, N. I.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B: Quantum Semiclass. Opt. 6, S288 (2004).
[CrossRef]

N. I. Nikolov, D. Neshev, O. Bang, and W. Z. Krolikowski, Phys. Rev. E 68, 036614 (2003).
[CrossRef]

Ouyang, S.

Peccianti, M.

Petersen, D. E.

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

Rasmussen, J. J.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B: Quantum Semiclass. Opt. 6, S288 (2004).
[CrossRef]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
[CrossRef]

Rasmussen, P. D.

P. D. Rasmussen, O. Bang, and W. Krolikowski, Phys. Rev. E 72, 066611 (2005).
[CrossRef]

Regan, J. J.

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Saffman, M.

S. Skupin, M. Saffman, and W. Krolikowski, Phys. Rev. Lett. 98, 263902 (2007).
[CrossRef] [PubMed]

Sanpera, A.

S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewenstein, Phys. Rev. Lett. 835198 (1999).
[CrossRef]

Sengstock, K.

S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewenstein, Phys. Rev. Lett. 835198 (1999).
[CrossRef]

Shen, M.

L. J. Ge, Q. Wang, M. Shen, J. L. Shi, Q. Kong, and P. Hou, J. Opt. A Pure: Appl. Opt. 11, 065207 (2009).
[CrossRef]

Shi, J. L.

L. J. Ge, Q. Wang, M. Shen, J. L. Shi, Q. Kong, and P. Hou, J. Opt. A Pure: Appl. Opt. 11, 065207 (2009).
[CrossRef]

Shlyapnikov, G. V.

S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewenstein, Phys. Rev. Lett. 835198 (1999).
[CrossRef]

Skupin, S.

S. Skupin, M. Saffman, and W. Krolikowski, Phys. Rev. Lett. 98, 263902 (2007).
[CrossRef] [PubMed]

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, Phys. Rev. E 73, 066603 (2006).
[CrossRef]

Snyder, A. W.

A. W. Snyder and D. J. Mitchell, Science 276, 1538(1997).
[CrossRef]

Stegeman, G. I.

Suter, D.

D. Suter and T. Blasberg, Phys. Rev. A 48, 4583 (1993).
[CrossRef] [PubMed]

Swartzlander, G. A.

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Torner, L.

Trillo, S.

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

Ultanir, E. A.

Wang, Q.

L. J. Ge, Q. Wang, M. Shen, J. L. Shi, Q. Kong, and P. Hou, J. Opt. A Pure: Appl. Opt. 11, 065207 (2009).
[CrossRef]

Whinnery, J. B.

F. W. Dabby and J. B. Whinnery, Appl. Phys. Lett. 13, 284(1968).
[CrossRef]

Wyller, J.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B: Quantum Semiclass. Opt. 6, S288 (2004).
[CrossRef]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
[CrossRef]

Yin, H.

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

F. W. Dabby and J. B. Whinnery, Appl. Phys. Lett. 13, 284(1968).
[CrossRef]

J. Opt. A Pure: Appl. Opt. (1)

L. J. Ge, Q. Wang, M. Shen, J. L. Shi, Q. Kong, and P. Hou, J. Opt. A Pure: Appl. Opt. 11, 065207 (2009).
[CrossRef]

J. Opt. B: Quantum Semiclass. Opt. (1)

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B: Quantum Semiclass. Opt. 6, S288 (2004).
[CrossRef]

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

Opt. Commun. (1)

Y. S. Kivshar and W. Krolikowski, Opt. Commun. 114, 353 (1995).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rep. (1)

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

Phys. Rev. A (3)

D. Suter and T. Blasberg, Phys. Rev. A 48, 4583 (1993).
[CrossRef] [PubMed]

E. Braun, L. P. Faucheux, and A. Libchaber, Phys. Rev. A 48, 611 (1993).
[CrossRef] [PubMed]

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

Phys. Rev. E (5)

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, Phys. Rev. E 73, 066603 (2006).
[CrossRef]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
[CrossRef]

P. D. Rasmussen, O. Bang, and W. Krolikowski, Phys. Rev. E 72, 066611 (2005).
[CrossRef]

W. Krolikowski and O. Bang, Phys. Rev. E 63, 016610(2000).
[CrossRef]

N. I. Nikolov, D. Neshev, O. Bang, and W. Z. Krolikowski, Phys. Rev. E 68, 036614 (2003).
[CrossRef]

Phys. Rev. Lett. (4)

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov, and M. Lewenstein, Phys. Rev. Lett. 835198 (1999).
[CrossRef]

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

S. Skupin, M. Saffman, and W. Krolikowski, Phys. Rev. Lett. 98, 263902 (2007).
[CrossRef] [PubMed]

Science (1)

A. W. Snyder and D. J. Mitchell, Science 276, 1538(1997).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Width of the dark nonlocal soliton (normalized to its value in the local limit) as a function of the degree of nonlocality σ: solid blue curve, variational results with rectangular response function; dashed red curve, exact solution in a weakly nonlocal limit [20]. The inset depicts the magnified region of weak nonlocality. Here, B 1 and A 0 .

Fig. 2
Fig. 2

Dynamics of formation of dark spatial solitons in (a) almost local, σ = 0.2 , and (b) nonlocal, σ = 3 media with rectangular response function. The insets depict the soliton profile at the beginning (bottom) and end (top) of the propagation distance.

Equations (7)

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

i u z + 1 2 2 u x 2 - u + R ( x - ξ ) | u ( ξ , z ) | 2 d ξ = 0.
L = i 2 ( u * u z u u * z ) ( 1 1 | u | 2 ) - 1 2 | u x | 2 1 2 ( | u | 2 1 ) - R ( x - ξ ) ( | u ( ξ , z ) | 2 1 ) d ξ .
R ( x ) = { 1 2 σ σ x σ , 0 otherwise ,
u ( x , z ) = B tanh [ D ( x - x 0 ) ] + i A , A 2 + B 2 = 1 ,
L = L ( u ) d x = 2 d x 0 d z [ A B + tan - 1 ( B A ) ] - 2 3 B 2 D + B 4 D [ csch 2 ( D σ ) coth ( D σ ) D σ ] .
coth ( D σ ) D σ [ 1 D 2 σ 2 csch 2 ( D σ ) ] = 1 3 B 2 , d x 0 d z = A [ D 3 B B D ( csch 2 ( D σ ) coth ( D σ ) D σ ) ] .
- + R ( x ξ ) | u ( ξ , z ) | 2 d ξ | u ( x ) | 2 + γ 2 | u ( x ) | 2 x 2 ,

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