Abstract

Periodic and quasi-periodic breather multi-solitons solutions, the dipole-type breather soliton solution, the rogue wave solution, and the fission soliton solution of the general nonlocal Schrödinger equation are derived by using the similarity transformation and manipulating the external potential function. The stability of the exact solitary wave solutions with the white noise perturbation also is investigated numerically.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27, 1460–1462 (2002).
    [CrossRef]
  4. M. Peccianti, C. Conti, and G. Assanto, “Interplay between nonlocality and nonlinearity in nematic liquid crystals,” Opt. Lett. 30, 415–417 (2005).
    [CrossRef]
  5. A. G. Litvak, V. A. Mironov, G. M. Fraiman, and A. D. Yunakovskii, “Direct measurement of the attenuation length of extensive air showers,” Sov. J. Plasma Phys. 1, 31–33 (1975).
  6. J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
    [CrossRef]
  7. L. Santos, G. V. Shlyapnikov, P. Zoller, and M. Lewenstein, “Bose–Einstein condensation in trapped dipolar gases,” Phys. Rev. Lett. 85, 1791–1794 (2000).
    [CrossRef]
  8. V. M. Pérez-García, V. V. Konotop, and J. J. Garcia-Ripoll, “Dynamics of quasicollapse in nonlinear Schrödinger systems with nonlocal interactions,” Phys. Rev. E 62, 4300–4308 (2000).
    [CrossRef]
  9. W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612 (2001).
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  10. M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602 (2003).
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    [CrossRef]
  12. O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 (2002).
    [CrossRef]
  13. D. Neshev, G. McCarthy, and W. Krolikowski, “Dipole-mode vector solitons in anisotropic nonlocal self-focusing media,” Opt. Lett. 26, 1185–1187 (2001).
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  15. A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71, 065603(R) (2005).
    [CrossRef]
  16. N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Attraction of nonlocal dark optical solitons,” Opt. Lett. 29, 286–288 (2004).
    [CrossRef]
  17. S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31, 1100–1102 (2006).
    [CrossRef]
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  19. J. Jian and J. Lin, “Solitons in nonlocal nonlinear kerr media with exponential response,” Opt. Express 20, 7133–7479 (2012).
    [CrossRef]
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  23. V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, “Stable bright and vortex solitons in photonic crystal fibers with inhomogeneous defocusing nonlinearity,” Opt. Lett. 37, 1799–1801 (2012).
  24. Y. J. He and B. A. Malomed, “Solitary modes in nonlocal media with inhomogeneous self-repulsive nonlinearity,” Phys. Rev. A 87, 053812 (2013).
    [CrossRef]
  25. S. Abe and A. Ogura, “Solitary waves and their critical behavior in a nonlinear nonlocal medium with power-law response,” Phys. Rev. E 57, 6066–6070 (1998).
    [CrossRef]
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    [CrossRef]
  27. P. D. Rasmussen, F. H. Bennet, D. N. Dragomir, A. Andrey, C. R. Rosberg, W. Krolikowski, O. Bang, and Y. S. Kivshar, “Observation of two-dimensional nonlocal gap solitons,” Opt. Lett. 34, 295–297 (2009).
    [CrossRef]
  28. Z. W. Shi, G. Zheng, Q. Guo, and W. Hu, “Mixed-gap solitons supported by two different semi-infinite periodic lattices in nonlocal nonlinear media,” J. Opt. Soc. Am. B 26, 1526–1531 (2009).
    [CrossRef]
  29. V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous solitons in external potentials,” Phys. Rev. Lett. 98, 074102 (2007).
    [CrossRef]
  30. J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and V. V. Konotop, “Localized nonlinear waves in systems with time- and space-modulated nonlinearities,” Phys. Rev. Lett. 100, 164102 (2008).
  31. Z. X. Liang, Z. D. Zhang, and W. M. Liu, “Dynamics of a bright soliton in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential,” Phys. Rev. Lett. 94, 050402 (2005).
    [CrossRef]
  32. Z. Y. Yan, V. V. Konotop, A. V. Yulin, and W. M. Liu, “Two-dimensional superfluid flows in inhomogeneous Bose-Einstein condensates,” Phys. Rev. E 85, 016601 (2012).
    [CrossRef]
  33. X. Y. Tang and P. K. Shukla, “Solution of the one-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with an external potential,” Phys. Rev. A 76, 013612 (2007).
    [CrossRef]
  34. V. M. Pérez-García, P. Torres, and V. V. Konotop, “Similarity transformations for Schrödinger equations with time-dependent coefficients,” Physica D 221, 31–36 (2006).
    [CrossRef]
  35. N. I. Nikolov, D. Neshev, O. Bang, and W. Z. Krolikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68, 036614 (2003).
    [CrossRef]
  36. A. V. Buryak and Y. S. Kivshar, “Solitons due to second harmonic generation,” Phys. Lett. A 197, 407–412 (1995).
  37. G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111, 033601 (2013) and therein.
    [CrossRef]

2013 (2)

Y. J. He and B. A. Malomed, “Solitary modes in nonlocal media with inhomogeneous self-repulsive nonlinearity,” Phys. Rev. A 87, 053812 (2013).
[CrossRef]

G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111, 033601 (2013) and therein.
[CrossRef]

2012 (3)

V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, “Stable bright and vortex solitons in photonic crystal fibers with inhomogeneous defocusing nonlinearity,” Opt. Lett. 37, 1799–1801 (2012).

J. Jian and J. Lin, “Solitons in nonlocal nonlinear kerr media with exponential response,” Opt. Express 20, 7133–7479 (2012).
[CrossRef]

Z. Y. Yan, V. V. Konotop, A. V. Yulin, and W. M. Liu, “Two-dimensional superfluid flows in inhomogeneous Bose-Einstein condensates,” Phys. Rev. E 85, 016601 (2012).
[CrossRef]

2011 (3)

2009 (2)

2008 (1)

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and V. V. Konotop, “Localized nonlinear waves in systems with time- and space-modulated nonlinearities,” Phys. Rev. Lett. 100, 164102 (2008).

2007 (3)

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous solitons in external potentials,” Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef]

X. Y. Tang and P. K. Shukla, “Solution of the one-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with an external potential,” Phys. Rev. A 76, 013612 (2007).
[CrossRef]

I. Kaminer, C. Rotschild, O. Manela, and M. Segev, “Periodic solitons in nonlocal nonlinear media,” Opt. Lett. 32, 3209–3211 (2007).
[CrossRef]

2006 (3)

2005 (3)

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71, 065603(R) (2005).
[CrossRef]

M. Peccianti, C. Conti, and G. Assanto, “Interplay between nonlocality and nonlinearity in nematic liquid crystals,” Opt. Lett. 30, 415–417 (2005).
[CrossRef]

Z. X. Liang, Z. D. Zhang, and W. M. Liu, “Dynamics of a bright soliton in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential,” Phys. Rev. Lett. 94, 050402 (2005).
[CrossRef]

2004 (2)

2003 (2)

M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602 (2003).
[CrossRef]

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

2002 (2)

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 (2002).
[CrossRef]

M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27, 1460–1462 (2002).
[CrossRef]

2001 (2)

W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612 (2001).
[CrossRef]

D. Neshev, G. McCarthy, and W. Krolikowski, “Dipole-mode vector solitons in anisotropic nonlocal self-focusing media,” Opt. Lett. 26, 1185–1187 (2001).
[CrossRef]

2000 (2)

L. Santos, G. V. Shlyapnikov, P. Zoller, and M. Lewenstein, “Bose–Einstein condensation in trapped dipolar gases,” Phys. Rev. Lett. 85, 1791–1794 (2000).
[CrossRef]

V. M. Pérez-García, V. V. Konotop, and J. J. Garcia-Ripoll, “Dynamics of quasicollapse in nonlinear Schrödinger systems with nonlocal interactions,” Phys. Rev. E 62, 4300–4308 (2000).
[CrossRef]

1998 (2)

M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of multihump multimode solitons,” Phys. Rev. Lett. 80, 4657–4660 (1998).
[CrossRef]

S. Abe and A. Ogura, “Solitary waves and their critical behavior in a nonlinear nonlocal medium with power-law response,” Phys. Rev. E 57, 6066–6070 (1998).
[CrossRef]

1997 (1)

A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, “Bound dipole solitary solutions in anisotropic nonlocal self-focusing media,” Phys. Rev. A 56, R1110–R1113 (1997).
[CrossRef]

1995 (1)

A. V. Buryak and Y. S. Kivshar, “Solitons due to second harmonic generation,” Phys. Lett. A 197, 407–412 (1995).

1985 (1)

S. K. Turitsyn, “Spatial dispersion of nonlinearity and stability of multidimensional solitons,” Theor. Math. Phys. 64, 797–801 (1985).
[CrossRef]

1975 (1)

A. G. Litvak, V. A. Mironov, G. M. Fraiman, and A. D. Yunakovskii, “Direct measurement of the attenuation length of extensive air showers,” Sov. J. Plasma Phys. 1, 31–33 (1975).

1965 (1)

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Abe, S.

S. Abe and A. Ogura, “Solitary waves and their critical behavior in a nonlinear nonlocal medium with power-law response,” Phys. Rev. E 57, 6066–6070 (1998).
[CrossRef]

Anderson, D. Z.

A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, “Bound dipole solitary solutions in anisotropic nonlocal self-focusing media,” Phys. Rev. A 56, R1110–R1113 (1997).
[CrossRef]

Andrey, A.

Assanto, G.

Bang, O.

P. D. Rasmussen, F. H. Bennet, D. N. Dragomir, A. Andrey, C. R. Rosberg, W. Krolikowski, O. Bang, and Y. S. Kivshar, “Observation of two-dimensional nonlocal gap solitons,” Opt. Lett. 34, 295–297 (2009).
[CrossRef]

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31, 1100–1102 (2006).
[CrossRef]

N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Attraction of nonlocal dark optical solitons,” Opt. Lett. 29, 286–288 (2004).
[CrossRef]

W. Krolikowski, O. Bang, and J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E 70, 036617 (2004).
[CrossRef]

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

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 (2002).
[CrossRef]

W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612 (2001).
[CrossRef]

Belmonte-Beitia, J.

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and V. V. Konotop, “Localized nonlinear waves in systems with time- and space-modulated nonlinearities,” Phys. Rev. Lett. 100, 164102 (2008).

Belyaeva, T. L.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous solitons in external potentials,” Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef]

Bennet, F. H.

Borovkova, O. V.

Brzdakiewicz, K. A.

Buryak, A. V.

A. V. Buryak and Y. S. Kivshar, “Solitons due to second harmonic generation,” Phys. Lett. A 197, 407–412 (1995).

Christiansen, P. L.

Christodoulides, D. N.

M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of multihump multimode solitons,” Phys. Rev. Lett. 80, 4657–4660 (1998).
[CrossRef]

Conti, C.

M. Peccianti, C. Conti, and G. Assanto, “Interplay between nonlocality and nonlinearity in nematic liquid crystals,” Opt. Lett. 30, 415–417 (2005).
[CrossRef]

M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602 (2003).
[CrossRef]

Desyatnikov, A. S.

Dragomir, D. N.

Fraiman, G. M.

A. G. Litvak, V. A. Mironov, G. M. Fraiman, and A. D. Yunakovskii, “Direct measurement of the attenuation length of extensive air showers,” Sov. J. Plasma Phys. 1, 31–33 (1975).

Garcia-Ripoll, J. J.

V. M. Pérez-García, V. V. Konotop, and J. J. Garcia-Ripoll, “Dynamics of quasicollapse in nonlinear Schrödinger systems with nonlocal interactions,” Phys. Rev. E 62, 4300–4308 (2000).
[CrossRef]

Gordon, J. P.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Guo, Q.

Halfmann, T.

G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111, 033601 (2013) and therein.
[CrossRef]

Hasegawa, A.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous solitons in external potentials,” Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef]

He, Y. J.

Y. J. He and B. A. Malomed, “Solitary modes in nonlocal media with inhomogeneous self-repulsive nonlinearity,” Phys. Rev. A 87, 053812 (2013).
[CrossRef]

Heinze, G.

G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111, 033601 (2013) and therein.
[CrossRef]

Hu, W.

Hubrich, C.

G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111, 033601 (2013) and therein.
[CrossRef]

Jian, J.

J. Jian and J. Lin, “Solitons in nonlocal nonlinear kerr media with exponential response,” Opt. Express 20, 7133–7479 (2012).
[CrossRef]

Kaminer, I.

Kartashov, Y. V.

Kivshar, Y.

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71, 065603(R) (2005).
[CrossRef]

Kivshar, Y. S.

Konotop, V. V.

Z. Y. Yan, V. V. Konotop, A. V. Yulin, and W. M. Liu, “Two-dimensional superfluid flows in inhomogeneous Bose-Einstein condensates,” Phys. Rev. E 85, 016601 (2012).
[CrossRef]

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and V. V. Konotop, “Localized nonlinear waves in systems with time- and space-modulated nonlinearities,” Phys. Rev. Lett. 100, 164102 (2008).

V. M. Pérez-García, P. Torres, and V. V. Konotop, “Similarity transformations for Schrödinger equations with time-dependent coefficients,” Physica D 221, 31–36 (2006).
[CrossRef]

V. M. Pérez-García, V. V. Konotop, and J. J. Garcia-Ripoll, “Dynamics of quasicollapse in nonlinear Schrödinger systems with nonlocal interactions,” Phys. Rev. E 62, 4300–4308 (2000).
[CrossRef]

Krolikowski, W.

Krolikowski, W. Z.

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

Leite, R. C. C.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Lewenstein, M.

L. Santos, G. V. Shlyapnikov, P. Zoller, and M. Lewenstein, “Bose–Einstein condensation in trapped dipolar gases,” Phys. Rev. Lett. 85, 1791–1794 (2000).
[CrossRef]

Liang, Z. X.

Z. X. Liang, Z. D. Zhang, and W. M. Liu, “Dynamics of a bright soliton in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential,” Phys. Rev. Lett. 94, 050402 (2005).
[CrossRef]

Lin, J.

J. Jian and J. Lin, “Solitons in nonlocal nonlinear kerr media with exponential response,” Opt. Express 20, 7133–7479 (2012).
[CrossRef]

Litvak, A. G.

A. G. Litvak, V. A. Mironov, G. M. Fraiman, and A. D. Yunakovskii, “Direct measurement of the attenuation length of extensive air showers,” Sov. J. Plasma Phys. 1, 31–33 (1975).

Liu, W. M.

Z. Y. Yan, V. V. Konotop, A. V. Yulin, and W. M. Liu, “Two-dimensional superfluid flows in inhomogeneous Bose-Einstein condensates,” Phys. Rev. E 85, 016601 (2012).
[CrossRef]

Z. X. Liang, Z. D. Zhang, and W. M. Liu, “Dynamics of a bright soliton in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential,” Phys. Rev. Lett. 94, 050402 (2005).
[CrossRef]

Lobanov, V. E.

Lopez-Aguayo, S.

Malomed, B. A.

Mamaev, A. V.

A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, “Bound dipole solitary solutions in anisotropic nonlocal self-focusing media,” Phys. Rev. A 56, R1110–R1113 (1997).
[CrossRef]

Manela, O.

McCarthy, G.

Mezentsev, V. K.

A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, “Bound dipole solitary solutions in anisotropic nonlocal self-focusing media,” Phys. Rev. A 56, R1110–R1113 (1997).
[CrossRef]

Mironov, V. A.

A. G. Litvak, V. A. Mironov, G. M. Fraiman, and A. D. Yunakovskii, “Direct measurement of the attenuation length of extensive air showers,” Sov. J. Plasma Phys. 1, 31–33 (1975).

Mitchell, M.

M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of multihump multimode solitons,” Phys. Rev. Lett. 80, 4657–4660 (1998).
[CrossRef]

Moore, R. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Neshev, D.

Nikolov, N. I.

N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Attraction of nonlocal dark optical solitons,” Opt. Lett. 29, 286–288 (2004).
[CrossRef]

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

Ogura, A.

S. Abe and A. Ogura, “Solitary waves and their critical behavior in a nonlinear nonlocal medium with power-law response,” Phys. Rev. E 57, 6066–6070 (1998).
[CrossRef]

Peccianti, M.

Pérez-García, V. M.

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and V. V. Konotop, “Localized nonlinear waves in systems with time- and space-modulated nonlinearities,” Phys. Rev. Lett. 100, 164102 (2008).

V. M. Pérez-García, P. Torres, and V. V. Konotop, “Similarity transformations for Schrödinger equations with time-dependent coefficients,” Physica D 221, 31–36 (2006).
[CrossRef]

V. M. Pérez-García, V. V. Konotop, and J. J. Garcia-Ripoll, “Dynamics of quasicollapse in nonlinear Schrödinger systems with nonlocal interactions,” Phys. Rev. E 62, 4300–4308 (2000).
[CrossRef]

Porto, S. P. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Rasmussen, J. J.

N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Attraction of nonlocal dark optical solitons,” Opt. Lett. 29, 286–288 (2004).
[CrossRef]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 (2002).
[CrossRef]

W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612 (2001).
[CrossRef]

Rasmussen, P. D.

Rosberg, C. R.

Rotschild, C.

Saffman, M.

A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, “Bound dipole solitary solutions in anisotropic nonlocal self-focusing media,” Phys. Rev. A 56, R1110–R1113 (1997).
[CrossRef]

Santos, L.

L. Santos, G. V. Shlyapnikov, P. Zoller, and M. Lewenstein, “Bose–Einstein condensation in trapped dipolar gases,” Phys. Rev. Lett. 85, 1791–1794 (2000).
[CrossRef]

Segev, M.

I. Kaminer, C. Rotschild, O. Manela, and M. Segev, “Periodic solitons in nonlocal nonlinear media,” Opt. Lett. 32, 3209–3211 (2007).
[CrossRef]

M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of multihump multimode solitons,” Phys. Rev. Lett. 80, 4657–4660 (1998).
[CrossRef]

Serkin, V. N.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous solitons in external potentials,” Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef]

Shi, Z. W.

Shlyapnikov, G. V.

L. Santos, G. V. Shlyapnikov, P. Zoller, and M. Lewenstein, “Bose–Einstein condensation in trapped dipolar gases,” Phys. Rev. Lett. 85, 1791–1794 (2000).
[CrossRef]

Shukla, P. K.

X. Y. Tang and P. K. Shukla, “Solution of the one-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with an external potential,” Phys. Rev. A 76, 013612 (2007).
[CrossRef]

Skupin, S.

Tang, X. Y.

X. Y. Tang and P. K. Shukla, “Solution of the one-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with an external potential,” Phys. Rev. A 76, 013612 (2007).
[CrossRef]

Torner, L.

Torres, P.

V. M. Pérez-García, P. Torres, and V. V. Konotop, “Similarity transformations for Schrödinger equations with time-dependent coefficients,” Physica D 221, 31–36 (2006).
[CrossRef]

Turitsyn, S. K.

S. K. Turitsyn, “Spatial dispersion of nonlinearity and stability of multidimensional solitons,” Theor. Math. Phys. 64, 797–801 (1985).
[CrossRef]

Vekslerchik, V.

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and V. V. Konotop, “Localized nonlinear waves in systems with time- and space-modulated nonlinearities,” Phys. Rev. Lett. 100, 164102 (2008).

Vysloukh, V. A.

Whinnery, J. R.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Wyller, J.

W. Krolikowski, O. Bang, and J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E 70, 036617 (2004).
[CrossRef]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 (2002).
[CrossRef]

W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612 (2001).
[CrossRef]

Yakimenko, A. I.

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71, 065603(R) (2005).
[CrossRef]

Yan, Z. Y.

Z. Y. Yan, V. V. Konotop, A. V. Yulin, and W. M. Liu, “Two-dimensional superfluid flows in inhomogeneous Bose-Einstein condensates,” Phys. Rev. E 85, 016601 (2012).
[CrossRef]

Yulin, A. V.

Z. Y. Yan, V. V. Konotop, A. V. Yulin, and W. M. Liu, “Two-dimensional superfluid flows in inhomogeneous Bose-Einstein condensates,” Phys. Rev. E 85, 016601 (2012).
[CrossRef]

Yunakovskii, A. D.

A. G. Litvak, V. A. Mironov, G. M. Fraiman, and A. D. Yunakovskii, “Direct measurement of the attenuation length of extensive air showers,” Sov. J. Plasma Phys. 1, 31–33 (1975).

Zaliznyak, Y. A.

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71, 065603(R) (2005).
[CrossRef]

Zhang, Z. D.

Z. X. Liang, Z. D. Zhang, and W. M. Liu, “Dynamics of a bright soliton in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential,” Phys. Rev. Lett. 94, 050402 (2005).
[CrossRef]

Zheng, G.

Zoller, P.

L. Santos, G. V. Shlyapnikov, P. Zoller, and M. Lewenstein, “Bose–Einstein condensation in trapped dipolar gases,” Phys. Rev. Lett. 85, 1791–1794 (2000).
[CrossRef]

Zozulya, A. A.

A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, “Bound dipole solitary solutions in anisotropic nonlocal self-focusing media,” Phys. Rev. A 56, R1110–R1113 (1997).
[CrossRef]

J. Appl. Phys. (1)

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

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

Opt. Express (1)

J. Jian and J. Lin, “Solitons in nonlocal nonlinear kerr media with exponential response,” Opt. Express 20, 7133–7479 (2012).
[CrossRef]

Opt. Lett. (11)

O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, “Algebraic bright and vortex solitons in defocusing media,” Opt. Lett. 36, 3088–3099 (2011).
[CrossRef]

N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Attraction of nonlocal dark optical solitons,” Opt. Lett. 29, 286–288 (2004).
[CrossRef]

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31, 1100–1102 (2006).
[CrossRef]

I. Kaminer, C. Rotschild, O. Manela, and M. Segev, “Periodic solitons in nonlocal nonlinear media,” Opt. Lett. 32, 3209–3211 (2007).
[CrossRef]

D. Neshev, G. McCarthy, and W. Krolikowski, “Dipole-mode vector solitons in anisotropic nonlocal self-focusing media,” Opt. Lett. 26, 1185–1187 (2001).
[CrossRef]

M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27, 1460–1462 (2002).
[CrossRef]

M. Peccianti, C. Conti, and G. Assanto, “Interplay between nonlocality and nonlinearity in nematic liquid crystals,” Opt. Lett. 30, 415–417 (2005).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, L. Torner, and B. A. Malomed, “Self-trapping and splitting of bright vector solitons under inhomogeneous defocusing nonlinearities,” Opt. Lett. 36, 4587–4589 (2011).
[CrossRef]

V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, “Stable bright and vortex solitons in photonic crystal fibers with inhomogeneous defocusing nonlinearity,” Opt. Lett. 37, 1799–1801 (2012).

Y. V. Kartashov, L. Torner, and V. A. Vysloukh, “Lattice-supported surface solitons in nonlocal nonlinear media,” Opt. Lett. 31, 2595–2597 (2006).
[CrossRef]

P. D. Rasmussen, F. H. Bennet, D. N. Dragomir, A. Andrey, C. R. Rosberg, W. Krolikowski, O. Bang, and Y. S. Kivshar, “Observation of two-dimensional nonlocal gap solitons,” Opt. Lett. 34, 295–297 (2009).
[CrossRef]

Phys. Lett. A (1)

A. V. Buryak and Y. S. Kivshar, “Solitons due to second harmonic generation,” Phys. Lett. A 197, 407–412 (1995).

Phys. Rev. A (3)

X. Y. Tang and P. K. Shukla, “Solution of the one-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrödinger equation with an external potential,” Phys. Rev. A 76, 013612 (2007).
[CrossRef]

Y. J. He and B. A. Malomed, “Solitary modes in nonlocal media with inhomogeneous self-repulsive nonlinearity,” Phys. Rev. A 87, 053812 (2013).
[CrossRef]

A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, “Bound dipole solitary solutions in anisotropic nonlocal self-focusing media,” Phys. Rev. A 56, R1110–R1113 (1997).
[CrossRef]

Phys. Rev. E (10)

V. M. Pérez-García, V. V. Konotop, and J. J. Garcia-Ripoll, “Dynamics of quasicollapse in nonlinear Schrödinger systems with nonlocal interactions,” Phys. Rev. E 62, 4300–4308 (2000).
[CrossRef]

W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612 (2001).
[CrossRef]

M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602 (2003).
[CrossRef]

W. Krolikowski, O. Bang, and J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E 70, 036617 (2004).
[CrossRef]

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71, 065603(R) (2005).
[CrossRef]

O. V. Borovkova, Y. V. Kartashov, L. Torner, and B. A. Malomed, “Bright solitons from defocusing nonlinearities,” Phys. Rev. E 84, 035602(R) (2011).
[CrossRef]

S. Abe and A. Ogura, “Solitary waves and their critical behavior in a nonlinear nonlocal medium with power-law response,” Phys. Rev. E 57, 6066–6070 (1998).
[CrossRef]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619 (2002).
[CrossRef]

Z. Y. Yan, V. V. Konotop, A. V. Yulin, and W. M. Liu, “Two-dimensional superfluid flows in inhomogeneous Bose-Einstein condensates,” Phys. Rev. E 85, 016601 (2012).
[CrossRef]

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

Phys. Rev. Lett. (6)

G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111, 033601 (2013) and therein.
[CrossRef]

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous solitons in external potentials,” Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef]

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and V. V. Konotop, “Localized nonlinear waves in systems with time- and space-modulated nonlinearities,” Phys. Rev. Lett. 100, 164102 (2008).

Z. X. Liang, Z. D. Zhang, and W. M. Liu, “Dynamics of a bright soliton in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential,” Phys. Rev. Lett. 94, 050402 (2005).
[CrossRef]

L. Santos, G. V. Shlyapnikov, P. Zoller, and M. Lewenstein, “Bose–Einstein condensation in trapped dipolar gases,” Phys. Rev. Lett. 85, 1791–1794 (2000).
[CrossRef]

M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of multihump multimode solitons,” Phys. Rev. Lett. 80, 4657–4660 (1998).
[CrossRef]

Physica D (1)

V. M. Pérez-García, P. Torres, and V. V. Konotop, “Similarity transformations for Schrödinger equations with time-dependent coefficients,” Physica D 221, 31–36 (2006).
[CrossRef]

Sov. J. Plasma Phys. (1)

A. G. Litvak, V. A. Mironov, G. M. Fraiman, and A. D. Yunakovskii, “Direct measurement of the attenuation length of extensive air showers,” Sov. J. Plasma Phys. 1, 31–33 (1975).

Theor. Math. Phys. (1)

S. K. Turitsyn, “Spatial dispersion of nonlinearity and stability of multidimensional solitons,” Theor. Math. Phys. 64, 797–801 (1985).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Propagating behavior of Eq. (3.13) |ψ|2 and n with G=3, D=2, c1=2, and c2=3: (a) |ψ|2, (b) n, (c) V(x,z).

Fig. 2.
Fig. 2.

Propagating behavior of Eq. (3.13) |ψ|2 and n with G=3, D=0.2, c1=2, and c2=3: (a) |ψ|2, (b) n, (c) A(x,z), and the density distributions of (d) n and (e) |ψ|2.

Fig. 3.
Fig. 3.

Intensity |ψ|2 and n with G=3, D=2, c1=2, c2=3: (a) |ψ|2, (b) n, (c) V(x,z), (d). The evolutions of amplitudes of |ψ|2 at x=0.8 and n at x=0 along the propagating direction.

Fig. 4.
Fig. 4.

Propagating behaviors of solutions |ψ|2 and n with Eq. (3.15) and G=3, D=2: (a) |ψ|2, (b) n and the numerical evolution of the exact solution (c) |ψ|2 with the white noise.

Fig. 5.
Fig. 5.

Dynamics of solution |ψ|2 and n with Eq. (3.19) and G=2, D=1 and the external potential V(x,z): (a) |ψ|2, (b) n, (c) V(x,z).

Fig. 6.
Fig. 6.

Propagating behavior of solution |ψ| and n with Eq. (3.19) and G=13, D=0.1: (a) |ψ|2, (b) n.

Equations (36)

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

iψz+12ψxxV(x,z)ψg(x,z)nψ=0,nA(x,z)nxx=B(x,z)|ψ|2,
ψ(x,z)=ρ(x,z)eiτ(x,z)Φ(ξ(x,z)),n(x,z)=ρ(x,z)χ(ξ(x,z)).
μΦ+Φξξ+GΦχ=0,χDχξξΦ2=0,
2ρρz+(ρ2τx)x=0,
(ρ2ξx)x=0,2gρ=Gξx2
V+τz+(τx2+μξx2ρ1ρxx)/2=0,
ξz+τxξx=0,
ρAρxxBρ2=0,
Aξx2DBρ=0.
ρ(x,z)=γ(z)F(η),
τ(x,z)=γz2γx2δzγx+α(z)
g(x,z)=G2γ4ρ5,
V(x,z)=ρxx2ρτzτx22μγ42ρ4
A(x,z)=ρ4Dγ4+ρ3ρxxD,
B(x,z)=γ4ργ4+ρ4ρxxD,
g(x,z)=G2γ32e5η22.
V(x,z)=ω2(z)x2+f(z)x+h(z)μ2γ2e2η2,
A(x,z)=γ2e2η2D1+(η21)e2η2D,
ω2(z)=γ6+γγzz2γz22γ2,
f(z)=γ5δ+γδzz2γzδzγ2,
h(z)=γ4δ2γ4δz22γ2αz2γ2,
V(x,z)=ω2(z)x2.
φzz+2ω2(z)φ=1φ3.
ω2(z)=(1+cos(ω0z))/2,
γ=34c1c22c12(C(19,118,3z)S(19,118,3z)C(19,118,3z)S(19,118,3z))2c1C(19,118,3z)S(19,118,3z)+2c2S(19,118,3z)2+2c1c2C(19,118,3z)2,
ξ(x,z)=0γ(z)xeη2dη=i2πerf(iη).
Φ(ξ)=3m(m2+1)GDcn(kξ,m)dn(kξ,m),χ(ξ)=12GD3m2(1+m2)GDsn(kξ,m)2,
|ψ|2=9m2γeη2(1+m2)2GDcn(kξ,m)2dn(kξ,m)2,n=γ12eη22(12GD3m3(1+m2)GDsn(kξ,m)2).
φ=1γ=1+cos(2z)22.
Φ(ξ)=3GDsech(ξ2D)tanh(ξ2D),
χ(ξ)=3GDsech(ξ2D)2.
γ(z)=4c1c22c12(C(2,12,z)S(2,12,z)C(2,12,z)S(2,12,z))2c1C(2,12,z)S(2,12,z)+2c2S(2,12,z)2+2c1c2C(2,12,z)2.
ψ=62e(0.5+0.15i)x2sech(ξ2)tanh(ξ2)[1+0.1random(x)].
Φ(ξ)=32GD(tanh(ξ2D)1),χ(ξ)=32GD(1tanh(ξ2D)2).
V(x,z)=((1+0.2z2)4+0.4(10.6z2)(1+0.2z2)2)x2+0.5(1+0.2z2)2e2(1+0.2z2)2x2),
V(x,z)=0.5(sech(0.1z)40.01)x2+sech(0.1z)2e2sech(0.1z)2x2),

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