Abstract

We consider the temporal evolution of an incoherent optical wave that propagates in a noninstantaneous response nonlinear medium, such as single mode optical fibers. In contrast with the expected Raman-like spectral redshift due to a delayed nonlinear response, we show that a highly noninstantaneous response leads to a genuine modulational instability of the incoherent optical wave. We derive a Vlasov-like kinetic equation that provides a detailed description of this process of incoherent modulational instability in the temporal domain.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Mitchell, Z. Chen, M. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
    [CrossRef]
  2. See, e.g., M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
    [CrossRef]
  3. B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
    [CrossRef]
  4. A. Sauter, S. Pitois, G. Millot, and A. Picozzi, Opt. Lett. 30, 2143 (2005).
    [CrossRef]
  5. D. Dylov and J. Fleischer, in Localized States in Physics: Solitons and Patterns (Springer, 2011), Part 1, pp. 17–34.
  6. K. Hammani, B. Kibler, C. Finot, and A. Picozzi, Phys. Lett. A 374, 3585 (2010).
    [CrossRef]
  7. J. Garnier, J.-P. Ayanides, and O. Morice, J. Opt. Soc. Am. B 20, 1409 (2003).
    [CrossRef]
  8. U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori, J. Opt. Soc. Am. B 26, 2280 (2009).
    [CrossRef]
  9. Y. Silberberg, Y. Lahini, E. Small, and R. Morandotti, Phys. Rev. Lett. 102, 233904 (2009).
    [CrossRef]
  10. C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
    [CrossRef]
  11. R. Weill, B. Fischer, and O. Gat, Phys. Rev. Lett. 104, 173901 (2010).
    [CrossRef]
  12. S. Babin, D. Churkin, A. Ismagulov, S. Kablukov, and E. Podivilov, J. Opt. Soc. Am. B 24, 1729 (2007).
    [CrossRef]
  13. E. G. Turitsyna, G. Falkovich, V. K. Mezentsev, and S. K. Turitsyn, Phys. Rev. A 80, 031804 (2009).
    [CrossRef]
  14. V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, 1992).
  15. For a simple introduction in optics, see, e.g., A. Picozzi, Opt. Express 15, 9063 (2007).
    [CrossRef]
  16. P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, Phys. Rev. Lett. 104, 054101 (2010).
    [CrossRef]
  17. G. Düring, A. Picozzi, and S. Rica, Physica D 238, 1524 (2009).
    [CrossRef]
  18. C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. Fleischer, Nat. Phys., doi: 10.1038/NPHYS2278 (to be published).
  19. A. Picozzi and M. Haelterman, Phys. Rev. Lett. 92, 103901 (2004).
    [CrossRef]
  20. A. Picozzi, S. Pitois, and G. Millot, Phys. Rev. Lett. 101, 093901 (2008).
    [CrossRef]
  21. B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, Phys. Rev. E 84, 066605 (2011).
    [CrossRef]
  22. A. Picozzi and J. Garnier, Phys. Rev. Lett. 107, 233901 (2011).
    [CrossRef]
  23. C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, Phys. Rev. Lett. 105, 263902 (2010).
    [CrossRef]
  24. A. Hasegawa, Phys. Fluids 18, 77 (1975).
    [CrossRef]
  25. P. Béjot, B. Kibler, E. Hertz, B. Lavorel, and O. Faucher, Phys. Rev. A 83, 013830 (2011).
    [CrossRef]

2011 (3)

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, Phys. Rev. E 84, 066605 (2011).
[CrossRef]

A. Picozzi and J. Garnier, Phys. Rev. Lett. 107, 233901 (2011).
[CrossRef]

P. Béjot, B. Kibler, E. Hertz, B. Lavorel, and O. Faucher, Phys. Rev. A 83, 013830 (2011).
[CrossRef]

2010 (4)

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, Phys. Lett. A 374, 3585 (2010).
[CrossRef]

C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

R. Weill, B. Fischer, and O. Gat, Phys. Rev. Lett. 104, 173901 (2010).
[CrossRef]

P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, Phys. Rev. Lett. 104, 054101 (2010).
[CrossRef]

2009 (4)

G. Düring, A. Picozzi, and S. Rica, Physica D 238, 1524 (2009).
[CrossRef]

Y. Silberberg, Y. Lahini, E. Small, and R. Morandotti, Phys. Rev. Lett. 102, 233904 (2009).
[CrossRef]

E. G. Turitsyna, G. Falkovich, V. K. Mezentsev, and S. K. Turitsyn, Phys. Rev. A 80, 031804 (2009).
[CrossRef]

U. Bortolozzo, J. Laurie, S. Nazarenko, and S. Residori, J. Opt. Soc. Am. B 26, 2280 (2009).
[CrossRef]

2008 (2)

A. Picozzi, S. Pitois, and G. Millot, Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef]

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

2007 (2)

2005 (1)

2004 (1)

A. Picozzi and M. Haelterman, Phys. Rev. Lett. 92, 103901 (2004).
[CrossRef]

2003 (1)

2002 (1)

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
[CrossRef]

2000 (1)

See, e.g., M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
[CrossRef]

1996 (1)

M. Mitchell, Z. Chen, M. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef]

1975 (1)

A. Hasegawa, Phys. Fluids 18, 77 (1975).
[CrossRef]

Anderson, D.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Angelani, L.

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

Ayanides, J.-P.

Babin, S.

Barsi, C.

C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. Fleischer, Nat. Phys., doi: 10.1038/NPHYS2278 (to be published).

Barviau, B.

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, Phys. Rev. E 84, 066605 (2011).
[CrossRef]

Béjot, P.

P. Béjot, B. Kibler, E. Hertz, B. Lavorel, and O. Faucher, Phys. Rev. A 83, 013830 (2011).
[CrossRef]

Biancalana, F.

C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

Bortolozzo, U.

Chen, Z.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef]

Christodoulides, D. N.

See, e.g., M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
[CrossRef]

Churkin, D.

Conti, C.

C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

Coskun, T.

See, e.g., M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
[CrossRef]

Düring, G.

G. Düring, A. Picozzi, and S. Rica, Physica D 238, 1524 (2009).
[CrossRef]

Dylov, D.

D. Dylov and J. Fleischer, in Localized States in Physics: Solitons and Patterns (Springer, 2011), Part 1, pp. 17–34.

Falkovich, G.

E. G. Turitsyna, G. Falkovich, V. K. Mezentsev, and S. K. Turitsyn, Phys. Rev. A 80, 031804 (2009).
[CrossRef]

V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, 1992).

Faucher, O.

P. Béjot, B. Kibler, E. Hertz, B. Lavorel, and O. Faucher, Phys. Rev. A 83, 013830 (2011).
[CrossRef]

Fedele, R.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Finot, C.

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, Phys. Lett. A 374, 3585 (2010).
[CrossRef]

Fischer, B.

R. Weill, B. Fischer, and O. Gat, Phys. Rev. Lett. 104, 173901 (2010).
[CrossRef]

Fleischer, J.

C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. Fleischer, Nat. Phys., doi: 10.1038/NPHYS2278 (to be published).

D. Dylov and J. Fleischer, in Localized States in Physics: Solitons and Patterns (Springer, 2011), Part 1, pp. 17–34.

Fratalocchi, A.

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

Garnier, J.

Gat, O.

R. Weill, B. Fischer, and O. Gat, Phys. Rev. Lett. 104, 173901 (2010).
[CrossRef]

Haelterman, M.

A. Picozzi and M. Haelterman, Phys. Rev. Lett. 92, 103901 (2004).
[CrossRef]

Hall, B.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Hammani, K.

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, Phys. Lett. A 374, 3585 (2010).
[CrossRef]

Hasegawa, A.

A. Hasegawa, Phys. Fluids 18, 77 (1975).
[CrossRef]

Hertz, E.

P. Béjot, B. Kibler, E. Hertz, B. Lavorel, and O. Faucher, Phys. Rev. A 83, 013830 (2011).
[CrossRef]

Ismagulov, A.

Jauslin, H. R.

P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, Phys. Rev. Lett. 104, 054101 (2010).
[CrossRef]

Jia, S.

C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. Fleischer, Nat. Phys., doi: 10.1038/NPHYS2278 (to be published).

Kablukov, S.

Kibler, B.

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, Phys. Rev. E 84, 066605 (2011).
[CrossRef]

P. Béjot, B. Kibler, E. Hertz, B. Lavorel, and O. Faucher, Phys. Rev. A 83, 013830 (2011).
[CrossRef]

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, Phys. Lett. A 374, 3585 (2010).
[CrossRef]

Kudlinski, A.

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, Phys. Rev. E 84, 066605 (2011).
[CrossRef]

L’vov, V. S.

V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, 1992).

Lahini, Y.

Y. Silberberg, Y. Lahini, E. Small, and R. Morandotti, Phys. Rev. Lett. 102, 233904 (2009).
[CrossRef]

Laurie, J.

Lavorel, B.

P. Béjot, B. Kibler, E. Hertz, B. Lavorel, and O. Faucher, Phys. Rev. A 83, 013830 (2011).
[CrossRef]

Leonetti, M.

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

Lisak, M.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Mezentsev, V. K.

E. G. Turitsyna, G. Falkovich, V. K. Mezentsev, and S. K. Turitsyn, Phys. Rev. A 80, 031804 (2009).
[CrossRef]

Michel, C.

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, Phys. Rev. E 84, 066605 (2011).
[CrossRef]

Millot, G.

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, Phys. Rev. E 84, 066605 (2011).
[CrossRef]

A. Picozzi, S. Pitois, and G. Millot, Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef]

A. Sauter, S. Pitois, G. Millot, and A. Picozzi, Opt. Lett. 30, 2143 (2005).
[CrossRef]

Mitchell, M.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef]

Morandotti, R.

Y. Silberberg, Y. Lahini, E. Small, and R. Morandotti, Phys. Rev. Lett. 102, 233904 (2009).
[CrossRef]

Morice, O.

Nazarenko, S.

Picozzi, A.

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, Phys. Rev. E 84, 066605 (2011).
[CrossRef]

A. Picozzi and J. Garnier, Phys. Rev. Lett. 107, 233901 (2011).
[CrossRef]

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, Phys. Lett. A 374, 3585 (2010).
[CrossRef]

P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, Phys. Rev. Lett. 104, 054101 (2010).
[CrossRef]

G. Düring, A. Picozzi, and S. Rica, Physica D 238, 1524 (2009).
[CrossRef]

A. Picozzi, S. Pitois, and G. Millot, Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef]

For a simple introduction in optics, see, e.g., A. Picozzi, Opt. Express 15, 9063 (2007).
[CrossRef]

A. Sauter, S. Pitois, G. Millot, and A. Picozzi, Opt. Lett. 30, 2143 (2005).
[CrossRef]

A. Picozzi and M. Haelterman, Phys. Rev. Lett. 92, 103901 (2004).
[CrossRef]

C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. Fleischer, Nat. Phys., doi: 10.1038/NPHYS2278 (to be published).

Pitois, S.

A. Picozzi, S. Pitois, and G. Millot, Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef]

A. Sauter, S. Pitois, G. Millot, and A. Picozzi, Opt. Lett. 30, 2143 (2005).
[CrossRef]

Podivilov, E.

Randoux, S.

P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, Phys. Rev. Lett. 104, 054101 (2010).
[CrossRef]

Residori, S.

Rica, S.

G. Düring, A. Picozzi, and S. Rica, Physica D 238, 1524 (2009).
[CrossRef]

C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. Fleischer, Nat. Phys., doi: 10.1038/NPHYS2278 (to be published).

Ruocco, G.

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

Russell, P. St. J.

C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

Sauter, A.

Schmidt, M. A.

C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

Segev, M.

See, e.g., M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
[CrossRef]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef]

Semenov, V. E.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Shih, M.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef]

Silberberg, Y.

Y. Silberberg, Y. Lahini, E. Small, and R. Morandotti, Phys. Rev. Lett. 102, 233904 (2009).
[CrossRef]

Small, E.

Y. Silberberg, Y. Lahini, E. Small, and R. Morandotti, Phys. Rev. Lett. 102, 233904 (2009).
[CrossRef]

Soljacic, M.

See, e.g., M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
[CrossRef]

Sun, C.

C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. Fleischer, Nat. Phys., doi: 10.1038/NPHYS2278 (to be published).

Suret, P.

P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, Phys. Rev. Lett. 104, 054101 (2010).
[CrossRef]

Turitsyn, S. K.

E. G. Turitsyna, G. Falkovich, V. K. Mezentsev, and S. K. Turitsyn, Phys. Rev. A 80, 031804 (2009).
[CrossRef]

Turitsyna, E. G.

E. G. Turitsyna, G. Falkovich, V. K. Mezentsev, and S. K. Turitsyn, Phys. Rev. A 80, 031804 (2009).
[CrossRef]

Vishwanath, A.

See, e.g., M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
[CrossRef]

Weill, R.

R. Weill, B. Fischer, and O. Gat, Phys. Rev. Lett. 104, 173901 (2010).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, 1992).

J. Opt. Soc. Am. B (3)

Opt. Express (1)

Opt. Lett. (1)

Phys. Fluids (1)

A. Hasegawa, Phys. Fluids 18, 77 (1975).
[CrossRef]

Phys. Lett. A (1)

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, Phys. Lett. A 374, 3585 (2010).
[CrossRef]

Phys. Rev. A (2)

E. G. Turitsyna, G. Falkovich, V. K. Mezentsev, and S. K. Turitsyn, Phys. Rev. A 80, 031804 (2009).
[CrossRef]

P. Béjot, B. Kibler, E. Hertz, B. Lavorel, and O. Faucher, Phys. Rev. A 83, 013830 (2011).
[CrossRef]

Phys. Rev. E (2)

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, Phys. Rev. E 84, 066605 (2011).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Phys. Rev. Lett. (10)

P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, Phys. Rev. Lett. 104, 054101 (2010).
[CrossRef]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef]

See, e.g., M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
[CrossRef]

Y. Silberberg, Y. Lahini, E. Small, and R. Morandotti, Phys. Rev. Lett. 102, 233904 (2009).
[CrossRef]

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

R. Weill, B. Fischer, and O. Gat, Phys. Rev. Lett. 104, 173901 (2010).
[CrossRef]

A. Picozzi and J. Garnier, Phys. Rev. Lett. 107, 233901 (2011).
[CrossRef]

C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

A. Picozzi and M. Haelterman, Phys. Rev. Lett. 92, 103901 (2004).
[CrossRef]

A. Picozzi, S. Pitois, and G. Millot, Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef]

Physica D (1)

G. Düring, A. Picozzi, and S. Rica, Physica D 238, 1524 (2009).
[CrossRef]

Other (3)

C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. Fleischer, Nat. Phys., doi: 10.1038/NPHYS2278 (to be published).

V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, 1992).

D. Dylov and J. Fleischer, in Localized States in Physics: Solitons and Patterns (Springer, 2011), Part 1, pp. 17–34.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1.
Fig. 1.

Numerical simulation of the NLSE [Eq. (1)] showing the evolution of the spectrograms of the wave at different propagation lengths (left column), and corresponding simulation of the Vlasov equation [Eq. (2)] with the same parameters (τR=100 [τ0], Δω/2π=0.05 [τ01]), except that the incoherent MI is seeded (right column). We underline the agreement between the NLSE and Vlasov simulations.

Fig. 2.
Fig. 2.

Plots of MI gain profiles [from Eq. (4)] for (a) different Δω (in units of τ01) and (b) different τR (in units of τ0).

Equations (4)

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

izA=βttA+γA+R(tt)|A(z,t)|2dt,
znω(t,z)=2βωtnω(t,z)+tV(t,z)ωnω(t,z),
1=2γβΩ2R˜(Ω)+nω0(2βΩω+ip)2dω.
p(Ω)=|Ω|(2βγPR˜(Ω)2βΔω),

Metrics