Abstract

In this paper, we find some exact analytical solutions including bright soliton solution, dipole-mode soliton solution, double soliton solution and periodic solution when a slit laser beam propagates in Kerr-type nonlinear, nonlocal media with exponential response function. Furthermore, we address the energy flow is a monotonically growing function of d2 and the Hamiltonian decreases while the energy flow increases. And we also obtain an Airy-like soliton by numerical method.

© 2012 OSA

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  1. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, London, 2003).
  2. M. Segev and G. I. Stegeman, “Self-Trapping of Optical Beams: Spatial solitons,” Phys. Today 51(8), 42–45 (1998).
    [CrossRef]
  3. W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitonsand beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, 288–294 (2004).
    [CrossRef]
  4. M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of Multihump Multimode Solitons,” Phys. Rev. Lett. 80, 4657–4600 (1998).
    [CrossRef]
  5. 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, 1110–1113 (1997).
    [CrossRef]
  6. M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27, 1460–1462 (2002).
    [CrossRef]
  7. M. Peccianti, C. Conti, and G. Assanto, “Interplay between nonlocality and nonlinearity in nematic liquid crystals,” Opt. Lett. 30, 415–417 (2005).
    [CrossRef] [PubMed]
  8. 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).
  9. 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]
  10. 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] [PubMed]
  11. V. M. Perez-Garcia, V. V. Konotop, and J. J. García-Ripoll, “Dynamics of quasicollapse in nonlinear Schrödinger systems with nonlocal interactions,” Phys. Rev. E 62, 4300–4308 (2000).
    [CrossRef]
  12. W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612–016619 (2001).
    [CrossRef]
  13. M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602–025605 (2003).
    [CrossRef]
  14. S. K. Turitsyn, “Spatial dispersion of nonlinearity and stability of multidimensional solitons,” Theor. Math. Phys. 64, 797–801 (1985).
    [CrossRef]
  15. O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619–046623 (2002).
    [CrossRef]
  16. D. Neshev, G. McCarthy, and W. Królikowski, “Dipole-mode vector solitons in anisotropic nonlocal self-focusing media,” Opt. Lett. 26, 1185–1187 (2001).
    [CrossRef]
  17. W. Królikowski, O. Bang, and J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E 70, 036617–036621 (2004).
    [CrossRef]
  18. J. I. Yakimenko, Y.A. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71, 065603 (2005).
    [CrossRef]
  19. Y. Y. Lin, R. K. Lee, and B. A. Malomed, “Bragg solitons in nonlocal nonlinear media,” Phys. Rev. A 80, 013838–013844 (2009).
    [CrossRef]
  20. N. I. Nikolov, D. Neshev, W. Królikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Attraction of nonlocal dark optical solitons,” Opt. Lett. 29, 286–288 (2004).
    [CrossRef] [PubMed]
  21. A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Królikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. 96, 043901 (2006).
    [CrossRef] [PubMed]
  22. S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Królikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31, 1100–1102 (2006).
    [CrossRef] [PubMed]
  23. N. I. Nikolov, D. Neshev, O. Bang, and W.Z. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68, 036614–036618 (2003).
    [CrossRef]
  24. P. V. Larsen, M. P. Sørensen, O. Bang, W. Z. Królikowski, and S. Trillo, “Nonlocal description of X waves in quadratic nonlinear materials,” Phys. Rev. E 73, 036614–036623 (2006).
    [CrossRef]
  25. M. Bache, O. Bang, J. Moses, and F. W. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. 32, 2490–2492 (2007).
    [CrossRef] [PubMed]
  26. M. Bache, O. Bang, W. Królikowski, J. Moses, and F. W. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express 16, 3273–3287 (2008).
    [CrossRef] [PubMed]
  27. J. Wyller, W. Królikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615–066627 (2002).
    [CrossRef]
  28. Z. Y. Xu, Y. V. Kartashov, and L. Torner, “Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media,” Opt. Lett. 30, 3171–3173 (2005).
    [CrossRef] [PubMed]
  29. P. J. Olver, Applications of Lie Group to Differential Equations (Spinger,Berlin, 1986).
    [CrossRef]
  30. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
    [CrossRef]
  31. W. Królikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact solutions,” Phys. Rev. E 63, 016610–016615 (2000).
    [CrossRef]

2009 (1)

Y. Y. Lin, R. K. Lee, and B. A. Malomed, “Bragg solitons in nonlocal nonlinear media,” Phys. Rev. A 80, 013838–013844 (2009).
[CrossRef]

2008 (1)

2007 (1)

2006 (3)

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

P. V. Larsen, M. P. Sørensen, O. Bang, W. Z. Królikowski, and S. Trillo, “Nonlocal description of X waves in quadratic nonlinear materials,” Phys. Rev. E 73, 036614–036623 (2006).
[CrossRef]

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Królikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

2005 (3)

2004 (3)

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

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitonsand beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, 288–294 (2004).
[CrossRef]

W. Królikowski, O. Bang, and J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E 70, 036617–036621 (2004).
[CrossRef]

2003 (2)

M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602–025605 (2003).
[CrossRef]

N. I. Nikolov, D. Neshev, O. Bang, and W.Z. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68, 036614–036618 (2003).
[CrossRef]

2002 (3)

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

O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619–046623 (2002).
[CrossRef]

J. Wyller, W. Królikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615–066627 (2002).
[CrossRef]

2001 (2)

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612–016619 (2001).
[CrossRef]

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

2000 (3)

W. Królikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact solutions,” Phys. Rev. E 63, 016610–016615 (2000).
[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] [PubMed]

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

1998 (2)

M. Segev and G. I. Stegeman, “Self-Trapping of Optical Beams: Spatial solitons,” Phys. Today 51(8), 42–45 (1998).
[CrossRef]

M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of Multihump Multimode Solitons,” Phys. Rev. Lett. 80, 4657–4600 (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, 1110–1113 (1997).
[CrossRef]

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).

1973 (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
[CrossRef]

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]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, London, 2003).

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, 1110–1113 (1997).
[CrossRef]

Assanto, G.

Bache, M.

Bang, O.

M. Bache, O. Bang, W. Królikowski, J. Moses, and F. W. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express 16, 3273–3287 (2008).
[CrossRef] [PubMed]

M. Bache, O. Bang, J. Moses, and F. W. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. 32, 2490–2492 (2007).
[CrossRef] [PubMed]

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Królikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

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

P. V. Larsen, M. P. Sørensen, O. Bang, W. Z. Królikowski, and S. Trillo, “Nonlocal description of X waves in quadratic nonlinear materials,” Phys. Rev. E 73, 036614–036623 (2006).
[CrossRef]

W. Królikowski, O. Bang, and J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E 70, 036617–036621 (2004).
[CrossRef]

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitonsand beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, 288–294 (2004).
[CrossRef]

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

N. I. Nikolov, D. Neshev, O. Bang, and W.Z. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68, 036614–036618 (2003).
[CrossRef]

J. Wyller, W. Królikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615–066627 (2002).
[CrossRef]

O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619–046623 (2002).
[CrossRef]

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612–016619 (2001).
[CrossRef]

W. Królikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact solutions,” Phys. Rev. E 63, 016610–016615 (2000).
[CrossRef]

Brzdakiewicz, K. A.

Christiansen, P. L.

Christodoulides, D. N.

M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of Multihump Multimode Solitons,” Phys. Rev. Lett. 80, 4657–4600 (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] [PubMed]

M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602–025605 (2003).
[CrossRef]

Desyatnikov, A. S.

Dreischuh, A.

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Królikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

Edmundson, D.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitonsand beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, 288–294 (2004).
[CrossRef]

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).

García-Ripoll, J. J.

V. M. Perez-Garcia, V. V. Konotop, and J. J. García-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]

Hasegawa, A.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
[CrossRef]

Kartashov, Y. V.

Kivshar, Y.

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

Kivshar, Y. S.

Konotop, V. V.

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

Królikowski, W.

M. Bache, O. Bang, W. Królikowski, J. Moses, and F. W. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express 16, 3273–3287 (2008).
[CrossRef] [PubMed]

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

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Królikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitonsand beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, 288–294 (2004).
[CrossRef]

W. Królikowski, O. Bang, and J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E 70, 036617–036621 (2004).
[CrossRef]

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

J. Wyller, W. Królikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615–066627 (2002).
[CrossRef]

O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619–046623 (2002).
[CrossRef]

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612–016619 (2001).
[CrossRef]

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

W. Królikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact solutions,” Phys. Rev. E 63, 016610–016615 (2000).
[CrossRef]

Królikowski, W. Z.

P. V. Larsen, M. P. Sørensen, O. Bang, W. Z. Królikowski, and S. Trillo, “Nonlocal description of X waves in quadratic nonlinear materials,” Phys. Rev. E 73, 036614–036623 (2006).
[CrossRef]

Królikowski, W.Z.

N. I. Nikolov, D. Neshev, O. Bang, and W.Z. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68, 036614–036618 (2003).
[CrossRef]

Larsen, P. V.

P. V. Larsen, M. P. Sørensen, O. Bang, W. Z. Królikowski, and S. Trillo, “Nonlocal description of X waves in quadratic nonlinear materials,” Phys. Rev. E 73, 036614–036623 (2006).
[CrossRef]

Lee, R. K.

Y. Y. Lin, R. K. Lee, and B. A. Malomed, “Bragg solitons in nonlocal nonlinear media,” Phys. Rev. A 80, 013838–013844 (2009).
[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] [PubMed]

Lin, Y. Y.

Y. Y. Lin, R. K. Lee, and B. A. Malomed, “Bragg solitons in nonlocal nonlinear media,” Phys. Rev. A 80, 013838–013844 (2009).
[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).

Lopez-Aguayo, S.

Malomed, B. A.

Y. Y. Lin, R. K. Lee, and B. A. Malomed, “Bragg solitons in nonlocal nonlinear media,” Phys. Rev. A 80, 013838–013844 (2009).
[CrossRef]

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, 1110–1113 (1997).
[CrossRef]

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, 1110–1113 (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, 4657–4600 (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]

Moses, J.

Neshev, D.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitonsand beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, 288–294 (2004).
[CrossRef]

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

N. I. Nikolov, D. Neshev, O. Bang, and W.Z. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68, 036614–036618 (2003).
[CrossRef]

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

Neshev, D. N.

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Królikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

Nikolov, N. I.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitonsand beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, 288–294 (2004).
[CrossRef]

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

N. I. Nikolov, D. Neshev, O. Bang, and W.Z. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68, 036614–036618 (2003).
[CrossRef]

Olver, P. J.

P. J. Olver, Applications of Lie Group to Differential Equations (Spinger,Berlin, 1986).
[CrossRef]

Peccianti, M.

Perez-Garcia, V. M.

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

Petersen, D. E.

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Królikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

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.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitonsand beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, 288–294 (2004).
[CrossRef]

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

J. Wyller, W. Królikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615–066627 (2002).
[CrossRef]

O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619–046623 (2002).
[CrossRef]

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612–016619 (2001).
[CrossRef]

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, 1110–1113 (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] [PubMed]

Segev, M.

M. Segev and G. I. Stegeman, “Self-Trapping of Optical Beams: Spatial solitons,” Phys. Today 51(8), 42–45 (1998).
[CrossRef]

M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of Multihump Multimode Solitons,” Phys. Rev. Lett. 80, 4657–4600 (1998).
[CrossRef]

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] [PubMed]

Skupin, S.

Sørensen, M. P.

P. V. Larsen, M. P. Sørensen, O. Bang, W. Z. Królikowski, and S. Trillo, “Nonlocal description of X waves in quadratic nonlinear materials,” Phys. Rev. E 73, 036614–036623 (2006).
[CrossRef]

Stegeman, G. I.

M. Segev and G. I. Stegeman, “Self-Trapping of Optical Beams: Spatial solitons,” Phys. Today 51(8), 42–45 (1998).
[CrossRef]

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
[CrossRef]

Torner, L.

Trillo, S.

P. V. Larsen, M. P. Sørensen, O. Bang, W. Z. Królikowski, and S. Trillo, “Nonlocal description of X waves in quadratic nonlinear materials,” Phys. Rev. E 73, 036614–036623 (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]

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]

Wise, F. W.

Wyller, J.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitonsand beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, 288–294 (2004).
[CrossRef]

W. Królikowski, O. Bang, and J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E 70, 036617–036621 (2004).
[CrossRef]

O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619–046623 (2002).
[CrossRef]

J. Wyller, W. Królikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615–066627 (2002).
[CrossRef]

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612–016619 (2001).
[CrossRef]

Xu, Z. Y.

Yakimenko, J. I.

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

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

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] [PubMed]

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, 1110–1113 (1997).
[CrossRef]

Appl. Phys. Lett. (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
[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. B. (1)

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitonsand beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, 288–294 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (7)

Phys. Rev. A (2)

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, 1110–1113 (1997).
[CrossRef]

Y. Y. Lin, R. K. Lee, and B. A. Malomed, “Bragg solitons in nonlocal nonlinear media,” Phys. Rev. A 80, 013838–013844 (2009).
[CrossRef]

Phys. Rev. E (10)

N. I. Nikolov, D. Neshev, O. Bang, and W.Z. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68, 036614–036618 (2003).
[CrossRef]

P. V. Larsen, M. P. Sørensen, O. Bang, W. Z. Królikowski, and S. Trillo, “Nonlocal description of X waves in quadratic nonlinear materials,” Phys. Rev. E 73, 036614–036623 (2006).
[CrossRef]

O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619–046623 (2002).
[CrossRef]

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

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612–016619 (2001).
[CrossRef]

M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602–025605 (2003).
[CrossRef]

W. Królikowski, O. Bang, and J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E 70, 036617–036621 (2004).
[CrossRef]

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

J. Wyller, W. Królikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615–066627 (2002).
[CrossRef]

W. Królikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact solutions,” Phys. Rev. E 63, 016610–016615 (2000).
[CrossRef]

Phys. Rev. Lett. (3)

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] [PubMed]

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Królikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of Multihump Multimode Solitons,” Phys. Rev. Lett. 80, 4657–4600 (1998).
[CrossRef]

Phys. Today (1)

M. Segev and G. I. Stegeman, “Self-Trapping of Optical Beams: Spatial solitons,” Phys. Today 51(8), 42–45 (1998).
[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]

Other (2)

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, London, 2003).

P. J. Olver, Applications of Lie Group to Differential Equations (Spinger,Berlin, 1986).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

(a) Evolution of bright soliton u and (b) evolution of refractive index n in Eq. (16) for d2 = 3, c0 = 3, c1 = 3. (c) relationship between U and d2. (d) relationship between H and U.

Fig. 2
Fig. 2

(a) Profiles of dipole-mode soliton u when ξ = 0(black line), 2(red line), 4(blue line) with d2 = 2, c0 = 2, c1 = 3. (b) refractive index n when ξ = 0(black line), 2(red line), 4(blue line) from left to right with d2 = 2, c0 = 2, c1 = 3

Fig. 3
Fig. 3

(a), (b) and (c) Profiles of the double soliton u when ξ = 0, 4, 8 with d2 = 1, c0 = 2, c1 = 3. (d) refractive index when ξ = 0(black line), 2(red line), 4(blue line) from left to right with d2 = 1, c0 = 2, c1 = 3

Fig. 4
Fig. 4

(a) Profiles of period solution u where d2 = 1(black line), 2(red line) and 3(blue line) with c0 = 3, c1 = 3 and m = 0.5. (b) refractive index n with d2 = 2, c0 = 3, c1 = 3 and m = 0.5.

Fig. 5
Fig. 5

(a) and (b) are profiles of a Airy-like solution u and refractive index n at ρ = −8, κ = −0.01, λ = 3.1 and d = 0.1.

Equations (32)

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A = i q ξ + 1 2 2 q η 2 + q n = 0 , B = n d 2 n η 2 | q | 2 = 0 ,
{ η , ξ , q , q * , n } { η , ξ , q , q * , n } + ε { χ , τ , ζ , ζ * , δ }
V = χ x + τ z + ζ u + ζ * u * + δ n .
Pr ( 2 ) V A = 0 , Pr ( 2 ) V B = 0.
Pr ( 2 ) V = V + ζ ξ q ξ + ζ η η q η η + δ η η q η η ,
ζ ξ = D ξ ( ζ χ q η τ q ξ ) + χ q η ξ + τ q ξ ξ , ζ η η = D η η ( ζ χ q η τ q ξ ) + χ q η η η + τ q η η ξ , δ η η = D η η ( δ χ n η τ n ξ ) + χ n η η η + τ n η η ξ .
χ = b ξ + c 0 , τ = c 1 , ζ = i b q η + i a q , ζ * = i b q * η + i a q * , δ = 0 ,
d η χ = d ξ τ = d q ζ = d q * ζ * = d n δ .
χ = c 0 , τ = c 1 , ζ = i a q , ζ * = i a q * , δ = 0.
ψ = c 1 η c 0 ξ , q = u ( ψ ) exp ( i a ξ c 1 ) exp ( i ( c 2 u ( ψ ) 2 + c 0 c 1 2 ) d ψ ) , n = Q ,
2 a c 1 u + c 0 2 c 1 2 u c 2 2 c 1 2 u 3 + u ψ ψ c 1 2 + 2 u Q = 0 ,
Q d Q ψ ψ c 1 2 u 2 = 0 ,
ψ = c 1 η b ξ 2 2 c 0 ξ , q = u ( ψ ) exp ( i b c 1 2 ( c 1 η ξ b ξ 3 3 c 0 ξ 2 2 + c 1 a ξ b ) ) exp ( i ( c 3 u ( ψ ) 2 + c 0 c 1 2 ) d ψ ) , n = Q .
u ψ ψ c 1 2 ( 2 b ψ c 1 2 + 2 a c 1 c 0 2 c 1 2 2 Q ) u c 3 2 c 1 2 u 3 = 0 ,
Q d Q ψ ψ c 1 2 u 2 = 0.
q = 3 d 2 2 sech ( 3 d 2 ( c 1 η c 0 ξ ) 3 c 1 ) 2 exp ( i ξ ( 3 c 0 2 + 4 c 1 2 d 2 ) 6 c 1 2 ) exp ( i c 0 c 1 2 d ( c 1 η c 0 ξ ) ) , n = d 2 sech ( 3 d 2 ( c 1 η c 0 ξ ) 3 c 1 ) 2 ,
U = | q | 2 d η = 2 3 d 2 ,
H = 1 2 ( | q η | 2 n | q | 2 ) d η = 8 ( d 2 ) 3 2 5 3 = U 3 45 .
q = 3 d 2 2 sech ( 3 d 2 η 3 ) 2 exp ( i ξ 2 d 2 3 ) , n = d 2 sech ( 3 d 2 η 3 ) 2 .
q = 3 d 2 sech ( 3 d 2 ( c 1 η c 0 ξ ) 3 c 1 ) tanh ( 3 d 2 ( c 1 η c 0 ξ ) 3 c 1 ) exp ( i ξ ( 3 c 0 2 + c 1 2 d 2 ) 6 c 1 2 ) exp ( i c 0 c 1 2 d ( c 1 η c 0 ξ ) ) , n = d 2 sech ( 3 d 2 ( c 1 η c 0 ξ ) 3 c 1 ) 2 .
q = 3 d 2 tanh ( 3 d 2 ( c 1 η c 0 ξ ) 3 c 1 ) 2 3 d 2 tanh ( 3 d 2 ( c 1 η c 0 ξ ) 3 c 1 ) 4 exp ( i ξ ( 3 c 0 2 + 4 c 1 2 d 2 ) 6 c 1 2 ) exp ( i c 0 c 1 2 d ( c 1 η c 0 ξ ) ) , n = d 2 sech ( 3 d 2 ( c 1 η c 0 ξ ) 3 c 1 ) 2 .
q = 3 d 2 2 csch ( 3 d 2 ( c 1 η c 0 ξ ) 3 c 1 ) 2 exp [ i ξ ( 3 c 0 2 + 4 c 1 2 d 2 ) 6 c 1 2 ] exp ( i c 0 c 1 2 d ( c 1 η c 0 ξ ) ) , n = d 2 csch ( 3 d 2 ( c 1 η c 0 ξ ) 3 c 1 ) 2 .
u = d 2 ( m 2 + 1 + γ 3 m 2 s n ( 3 d 2 ψ 3 m c 1 , m ) 2 ) m 6 d 2 γ , Q = d 2 ( m 2 + 1 + γ 3 m 2 s n ( 3 d 2 ψ 3 m c 1 , m ) 2 ) 3 m 2 ,
d 2 = 3 m 2 ( ( m 2 + 1 ) γ + ( m 4 + m 2 + 1 ) ) 2 d ( ( 2 + 3 m 6 ) + α γ ) , a = 3 m 2 c 0 2 ( α + 2 γ ( m 2 + 1 ) ) + 4 c 1 2 d 2 ( 2 ( 1 + m 6 ) + γ α ) 6 c 1 m 2 ( α + 2 γ + 2 m 2 γ ) .
q = d 2 ( m 2 1 + γ 3 m 2 s n ( 3 d 2 ( c 1 η c 0 ξ ) 3 m c 1 , m ) 2 ) m 6 d 2 γ exp ( i c 0 c 1 2 d ( c 1 η c 0 ξ ) ) exp ( i ξ ( 3 m 2 c 0 2 ( α + 2 γ ( m 2 + 1 ) ) + 4 c 1 2 d 2 ( 2 ( 1 + m 6 ) + γ α ) ) 6 c 1 m 2 ( α + 2 γ + 2 m 2 γ ) ) , n = d 2 ( m 2 + 1 γ 3 m 2 s n ( 3 d 2 ( c 1 η c 0 ξ ) 3 m c 1 , m ) 2 ) 3 m 2 .
q = ( d 2 ( m 2 + 1 + γ ) m 6 d 2 γ + 6 d 2 γ 2 γ s n ( k ψ , m ) 2 ) exp ( i a ξ c 1 ) exp ( i c 0 c 1 2 d ( c 1 η c 0 ξ ) ) , n = d 2 ( m 2 + 1 + γ ) 3 m 2 d 2 1 s n ( k ψ , m ) 2 ,
d 2 = 3 ( m 4 + m 2 + 1 ) + 3 ( m 2 + 1 ) γ 2 d ( α γ 2 ( m 6 + 1 ) ) , a = 4 d 2 2 c 1 2 ( 2 ( 1 + m 2 ) + α γ ) ) 12 c 1 d 2 ( ( 2 m 2 + 1 ) γ + ( 1 + m 2 + m 4 ) ) + 3 d 0 c 0 2 ( α + 2 ( 1 + m 2 ) γ ) 12 c 1 d 2 ( ( 2 m 2 + 1 ) γ + ( 1 + m 2 + m 4 ) ) ,
q = d 2 ( 3 m 2 + 3 ) c n ( 3 d 2 ( c 1 η c 0 ξ ) 3 m c 1 , m ) d n ( 3 d 2 ( c 1 η c 0 ξ ) 3 m c 1 , m ) 1 + m 2 exp ( i a ξ c 1 ) exp ( i c 0 c 1 2 d ( c 1 η c 0 ξ ) ) , n = d 2 ( 2 ( m 2 + 1 ) s n ( 3 d 2 ( c 1 η c 0 ξ ) 3 m c 1 , m ) 2 ) 1 + m 2 ,
d 2 = 3 m 2 2 d ( 1 + m 2 ) , a = c 1 2 d 1 ( 1 10 m 2 + m 4 ) + 3 m 2 c 0 2 ( 1 + m 2 ) 6 m 2 c 1 ( 1 + m 2 ) .
q = m 3 d 2 m 2 2 s n ( 3 d 2 ( c 1 η c 0 ξ ) 3 m c 1 ) c n ( 3 d 2 ( c 1 η c 0 ξ ) 3 m c 1 ) exp ( i a ξ c 1 ) exp ( i c 0 c 1 2 d ( c 1 η c 0 ξ ) ) , n = d 2 ( 1 ( m 2 2 ) s n ( 3 d 2 ( c 1 η c 0 ξ ) 3 m c 1 , m ) 2 ) m 2 2 .
a = c 1 2 d 1 ( m 2 1 ) ( 8 8 m 2 m 4 ) + 3 m 2 c 0 2 ( 4 m 2 4 m 4 ) 6 m 4 c 1 ( 4 + m 2 4 m 2 ) , d 2 = 3 m 2 2 d ( m 2 2 ) .
u ψ ψ ρ ψ u + κ u + λ u Q = 0.

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