Abstract

Using multivariate self-similarity transformation, we construct explicit spatial bright and dark solitary wave solutions of the generalized nonlinear Schrödinger equation with spatially Bessel-modulated nonlinearity and an external potential. Special kinds of explicit solutions, such as periodically breathing bright and dark solitary waves, are discussed in detail. The stability of these solutions is verified by means of direct numerical simulation.

© 2013 Optical Society of America

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  9. W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, and G. Chen, “Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. A 78, 023821 (2008).
    [CrossRef]
  10. M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (1+3)-dimensional nonlinear Schrödinger equation,” Phys. Rev. Lett. 101, 123904 (2008).
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  13. S. P. Burtsev, and I. R. Gabitov, “Alternative integrable equations of nonlinear optics,” Phys. Rev. A 49, 2065–2070 (1994).
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  16. V. V. Konotop, O. A. Chubykalo, and L. Vazquez, “Dynamics and interaction of solitons on an integrable inhomogeneous lattice,” Phys. Rev. E 48, 563–568 (1993).
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  19. A. T. Avelar, D. Bazeia, and W. B. Cardoso, “Solitons with cubic and quintic nonlinearities modulated in space and time,” Phys. Rev. E 79, 025602 (2009).
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  20. W. B. Cardoso, A. T. Avelar, and D. Bazeia, “Modulation of breathers in cigar- shaped Bose–Einstein condensates,” Phys. Lett. A 374, 2640–2645 (2010).
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  21. W. B. Cardoso, A. T. Avelar, D. Bazeia, and M. S. Hussein, “Solitons of two-component Bose–Einstein condensates modulated in space and time,” Phys. Lett. A 374, 2356–2360 (2010).
    [CrossRef]
  22. J. Belmonte-Beitia, and G. F. Calvo, “Exact solutions for the quintic nonlinear Schrödinger equation with time and space modulated nonlinearities and potentials,” Phys. Lett. A 373, 448–453 (2009).
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  23. J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98, 064102 (2007).
    [CrossRef]
  24. 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).
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    [CrossRef]
  27. G. Fibich, Y. Sivan, and M. I. Weinstein, “Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure,” Physica D 217, 31–57 (2006).
    [CrossRef]
  28. W. P. Zhong, M. Belić, G. Assanto, B. A. Malomed, and T. Huang, “Light bullets in the spatiotemporal nonlinear Schrödinger equation with a variable negative diffraction coefficient,” Phys. Rev. A 84, 043801 (2011).
    [CrossRef]
  29. W. P. Zhong and M. Belić, “Traveling wave and soliton solutions of coupled nonlinear Schrödinger equations with variable coefficients,” Phys. Rev. E 82, 047601 (2010).
    [CrossRef]
  30. W. P. Zhong, M. Belić, Y. Lu, and T. Huang, “Traveling and solitary wave solutions to the one-dimensional Gross-Pitaevskii equation,” Phys. Rev. E 81, 016605 (2010).
    [CrossRef]
  31. Z. Bouchal, “Nondifracting optical beams,” Czech. J. Phys. 53, 537–578 (2003).
    [CrossRef]
  32. J. Durnin, “Exact solutions for nondiffracting beams,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  33. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef]
  34. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
    [CrossRef]
  35. P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, “Nonlinear Bessel beams,” Opt. Commun. 222, 107–115(2003).
    [CrossRef]
  36. F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Light bullets in Bessel optical lattices with spatially modulated nonlinearity,” Opt. Express 17, 11328 (2009).
    [CrossRef]
  37. W.-P. Zhong, M. Belić, and T. Huang, “Three-dimensional Bessel light bullets in self-focusing Kerr media,” Phys. Rev. 82, 033834 (2010).
    [CrossRef]
  38. W. J. Tomlinson, R. J. Hawkins, A. M. Weiner, J. P. Heritage, and R. N. Thurston, “Dark optical solitons with finite-width background pulses,” J. Opt. Soc. Am. B 6, 329–334 (1989).
    [CrossRef]
  39. Y. S. Kivshar and X. Yang, “Dark solitons on background of finite extent,” Opt. Commun. 107, 93–98 (1994).
    [CrossRef]
  40. V. M. Perez-Garcia, P. J. Torres, and V. V. Konotop, “Similarity transformation for NLS equations with time-dependent coefficients,” Phys. D 221, 31–36 (2006).
    [CrossRef]
  41. B. Yang, W. P. Zhong, and M. Belić, “Self-similar HG spatial solitons in 2-dimensional nonlocal nonlinear media,” Commun. Theor. Phys. 53, 937–942 (2010).
    [CrossRef]
  42. W. P. Zhong, M. Belić, and G. Assanto, “Localized nonlinear wavepackets in with radial-azimuthal modulated nonlinearity and an external potential,” Phys. Scripta 84, 055001 (2011).
    [CrossRef]
  43. C. Q. Dai, R. P. Chen, and Y. Y. Wang, “Spatiotemporal self-similar solutions for the nonautonomous (3+1)D cubic-quintic GP equation,” Chin. Phys. B 21, 030508 (2012).
    [CrossRef]

2012 (1)

C. Q. Dai, R. P. Chen, and Y. Y. Wang, “Spatiotemporal self-similar solutions for the nonautonomous (3+1)D cubic-quintic GP equation,” Chin. Phys. B 21, 030508 (2012).
[CrossRef]

2011 (2)

W. P. Zhong, M. Belić, and G. Assanto, “Localized nonlinear wavepackets in with radial-azimuthal modulated nonlinearity and an external potential,” Phys. Scripta 84, 055001 (2011).
[CrossRef]

W. P. Zhong, M. Belić, G. Assanto, B. A. Malomed, and T. Huang, “Light bullets in the spatiotemporal nonlinear Schrödinger equation with a variable negative diffraction coefficient,” Phys. Rev. A 84, 043801 (2011).
[CrossRef]

2010 (7)

W. P. Zhong and M. Belić, “Traveling wave and soliton solutions of coupled nonlinear Schrödinger equations with variable coefficients,” Phys. Rev. E 82, 047601 (2010).
[CrossRef]

W. P. Zhong, M. Belić, Y. Lu, and T. Huang, “Traveling and solitary wave solutions to the one-dimensional Gross-Pitaevskii equation,” Phys. Rev. E 81, 016605 (2010).
[CrossRef]

W.-P. Zhong, M. Belić, and T. Huang, “Three-dimensional Bessel light bullets in self-focusing Kerr media,” Phys. Rev. 82, 033834 (2010).
[CrossRef]

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous matter-wave solitons near the Feshbach resonance,” Phys. Rev. A 81, 023610 (2010).
[CrossRef]

W. B. Cardoso, A. T. Avelar, and D. Bazeia, “Modulation of breathers in cigar- shaped Bose–Einstein condensates,” Phys. Lett. A 374, 2640–2645 (2010).
[CrossRef]

W. B. Cardoso, A. T. Avelar, D. Bazeia, and M. S. Hussein, “Solitons of two-component Bose–Einstein condensates modulated in space and time,” Phys. Lett. A 374, 2356–2360 (2010).
[CrossRef]

B. Yang, W. P. Zhong, and M. Belić, “Self-similar HG spatial solitons in 2-dimensional nonlocal nonlinear media,” Commun. Theor. Phys. 53, 937–942 (2010).
[CrossRef]

2009 (4)

J. Belmonte-Beitia, and G. F. Calvo, “Exact solutions for the quintic nonlinear Schrödinger equation with time and space modulated nonlinearities and potentials,” Phys. Lett. A 373, 448–453 (2009).
[CrossRef]

A. T. Avelar, D. Bazeia, and W. B. Cardoso, “Solitons with cubic and quintic nonlinearities modulated in space and time,” Phys. Rev. E 79, 025602 (2009).
[CrossRef]

F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Light bullets in Bessel optical lattices with spatially modulated nonlinearity,” Opt. Express 17, 11328 (2009).
[CrossRef]

X. G. He, D. Zhao, L. Li, and H. G. Luo, “Engineering integrable nonautonomous nonlinear Schrödinger equations,” Phys. Rev. E 79, 056610 (2009).
[CrossRef]

2008 (3)

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).
[CrossRef]

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, and G. Chen, “Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. A 78, 023821 (2008).
[CrossRef]

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (1+3)-dimensional nonlinear Schrödinger equation,” Phys. Rev. Lett. 101, 123904 (2008).
[CrossRef]

2007 (2)

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef]

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

2006 (2)

G. Fibich, Y. Sivan, and M. I. Weinstein, “Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure,” Physica D 217, 31–57 (2006).
[CrossRef]

V. M. Perez-Garcia, P. J. Torres, and V. V. Konotop, “Similarity transformation for NLS equations with time-dependent coefficients,” Phys. D 221, 31–36 (2006).
[CrossRef]

2005 (2)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[CrossRef]

B. B. Baizakov, G. Filatrella, B. A. Malomed, and M. Salerno, “Double parametric resonance for matter-wave solitons in a time-modulated trap,” Phys. Rev. E 71, 036619 (2005).
[CrossRef]

2003 (3)

P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, “Nonlinear Bessel beams,” Opt. Commun. 222, 107–115(2003).
[CrossRef]

F. Kh. Abdullaev, A. M. Kamchatnov, V. V. Konotop, and V. A. Brazhnyi, “Adiabatic dynamics of periodic waves in Bose-Einstein condensates with time dependent atomic scattering length,” Phys. Rev. Lett. 90, 230402 (2003).
[CrossRef]

Z. Bouchal, “Nondifracting optical beams,” Czech. J. Phys. 53, 537–578 (2003).
[CrossRef]

2000 (1)

V. N. Serkin, and A. Hasegawa, “Novel soliton solutions of the nonlinear Schrödinger equation model,” Phys. Rev. Lett. 85, 4502–4505 (2000).
[CrossRef]

1999 (1)

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

1994 (2)

S. P. Burtsev, and I. R. Gabitov, “Alternative integrable equations of nonlinear optics,” Phys. Rev. A 49, 2065–2070 (1994).
[CrossRef]

Y. S. Kivshar and X. Yang, “Dark solitons on background of finite extent,” Opt. Commun. 107, 93–98 (1994).
[CrossRef]

1993 (2)

V. V. Konotop, “Soliton on a disordered lattice,” Phys. Rev. E 47, 1423–1426 (1993).
[CrossRef]

V. V. Konotop, O. A. Chubykalo, and L. Vazquez, “Dynamics and interaction of solitons on an integrable inhomogeneous lattice,” Phys. Rev. E 48, 563–568 (1993).
[CrossRef]

1989 (1)

1987 (2)

J. Durnin, “Exact solutions for nondiffracting beams,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

1978 (1)

H. H. Chen, and C. S. Liu, “Nonlinear wave and soliton propagation in media with arbitrary inhomogeneities,” Phys. Fluids 21, 377–380 (1978).
[CrossRef]

1976 (2)

F. Calogero, and A. Degasperis, “Coupled nonlinear evolution equations solvable via the inverse spectral transform, and solitons that come back: the boomeron,” Lett. Nuovo Cimento 16, 425–433 (1976).
[CrossRef]

H. H. Chen, and C. S. Liu, “Solitons in nonuniform media,” Phys. Rev. Lett. 37, 693–697 (1976).
[CrossRef]

Abdullaev, F. Kh.

F. Kh. Abdullaev, A. M. Kamchatnov, V. V. Konotop, and V. A. Brazhnyi, “Adiabatic dynamics of periodic waves in Bose-Einstein condensates with time dependent atomic scattering length,” Phys. Rev. Lett. 90, 230402 (2003).
[CrossRef]

Agrawal, G. P.

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

Anderson, D.

P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, “Nonlinear Bessel beams,” Opt. Commun. 222, 107–115(2003).
[CrossRef]

Assanto, G.

W. P. Zhong, M. Belić, G. Assanto, B. A. Malomed, and T. Huang, “Light bullets in the spatiotemporal nonlinear Schrödinger equation with a variable negative diffraction coefficient,” Phys. Rev. A 84, 043801 (2011).
[CrossRef]

W. P. Zhong, M. Belić, and G. Assanto, “Localized nonlinear wavepackets in with radial-azimuthal modulated nonlinearity and an external potential,” Phys. Scripta 84, 055001 (2011).
[CrossRef]

Avelar, A. T.

W. B. Cardoso, A. T. Avelar, D. Bazeia, and M. S. Hussein, “Solitons of two-component Bose–Einstein condensates modulated in space and time,” Phys. Lett. A 374, 2356–2360 (2010).
[CrossRef]

W. B. Cardoso, A. T. Avelar, and D. Bazeia, “Modulation of breathers in cigar- shaped Bose–Einstein condensates,” Phys. Lett. A 374, 2640–2645 (2010).
[CrossRef]

A. T. Avelar, D. Bazeia, and W. B. Cardoso, “Solitons with cubic and quintic nonlinearities modulated in space and time,” Phys. Rev. E 79, 025602 (2009).
[CrossRef]

Baizakov, B. B.

B. B. Baizakov, G. Filatrella, B. A. Malomed, and M. Salerno, “Double parametric resonance for matter-wave solitons in a time-modulated trap,” Phys. Rev. E 71, 036619 (2005).
[CrossRef]

Bazeia, D.

W. B. Cardoso, A. T. Avelar, and D. Bazeia, “Modulation of breathers in cigar- shaped Bose–Einstein condensates,” Phys. Lett. A 374, 2640–2645 (2010).
[CrossRef]

W. B. Cardoso, A. T. Avelar, D. Bazeia, and M. S. Hussein, “Solitons of two-component Bose–Einstein condensates modulated in space and time,” Phys. Lett. A 374, 2356–2360 (2010).
[CrossRef]

A. T. Avelar, D. Bazeia, and W. B. Cardoso, “Solitons with cubic and quintic nonlinearities modulated in space and time,” Phys. Rev. E 79, 025602 (2009).
[CrossRef]

Belic, M.

W. P. Zhong, M. Belić, and G. Assanto, “Localized nonlinear wavepackets in with radial-azimuthal modulated nonlinearity and an external potential,” Phys. Scripta 84, 055001 (2011).
[CrossRef]

W. P. Zhong, M. Belić, G. Assanto, B. A. Malomed, and T. Huang, “Light bullets in the spatiotemporal nonlinear Schrödinger equation with a variable negative diffraction coefficient,” Phys. Rev. A 84, 043801 (2011).
[CrossRef]

W.-P. Zhong, M. Belić, and T. Huang, “Three-dimensional Bessel light bullets in self-focusing Kerr media,” Phys. Rev. 82, 033834 (2010).
[CrossRef]

W. P. Zhong and M. Belić, “Traveling wave and soliton solutions of coupled nonlinear Schrödinger equations with variable coefficients,” Phys. Rev. E 82, 047601 (2010).
[CrossRef]

W. P. Zhong, M. Belić, Y. Lu, and T. Huang, “Traveling and solitary wave solutions to the one-dimensional Gross-Pitaevskii equation,” Phys. Rev. E 81, 016605 (2010).
[CrossRef]

B. Yang, W. P. Zhong, and M. Belić, “Self-similar HG spatial solitons in 2-dimensional nonlocal nonlinear media,” Commun. Theor. Phys. 53, 937–942 (2010).
[CrossRef]

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, and G. Chen, “Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. A 78, 023821 (2008).
[CrossRef]

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (1+3)-dimensional nonlinear Schrödinger equation,” Phys. Rev. Lett. 101, 123904 (2008).
[CrossRef]

Belmonte-Beitia, J.

J. Belmonte-Beitia, and G. F. Calvo, “Exact solutions for the quintic nonlinear Schrödinger equation with time and space modulated nonlinearities and potentials,” Phys. Lett. A 373, 448–453 (2009).
[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).
[CrossRef]

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef]

Belyaeva, T. L.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous matter-wave solitons near the Feshbach resonance,” Phys. Rev. A 81, 023610 (2010).
[CrossRef]

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

Bouchal, Z.

Z. Bouchal, “Nondifracting optical beams,” Czech. J. Phys. 53, 537–578 (2003).
[CrossRef]

Brazhnyi, V. A.

F. Kh. Abdullaev, A. M. Kamchatnov, V. V. Konotop, and V. A. Brazhnyi, “Adiabatic dynamics of periodic waves in Bose-Einstein condensates with time dependent atomic scattering length,” Phys. Rev. Lett. 90, 230402 (2003).
[CrossRef]

Burtsev, S. P.

S. P. Burtsev, and I. R. Gabitov, “Alternative integrable equations of nonlinear optics,” Phys. Rev. A 49, 2065–2070 (1994).
[CrossRef]

Calogero, F.

F. Calogero, and A. Degasperis, “Coupled nonlinear evolution equations solvable via the inverse spectral transform, and solitons that come back: the boomeron,” Lett. Nuovo Cimento 16, 425–433 (1976).
[CrossRef]

Calvo, G. F.

J. Belmonte-Beitia, and G. F. Calvo, “Exact solutions for the quintic nonlinear Schrödinger equation with time and space modulated nonlinearities and potentials,” Phys. Lett. A 373, 448–453 (2009).
[CrossRef]

Cardoso, W. B.

W. B. Cardoso, A. T. Avelar, and D. Bazeia, “Modulation of breathers in cigar- shaped Bose–Einstein condensates,” Phys. Lett. A 374, 2640–2645 (2010).
[CrossRef]

W. B. Cardoso, A. T. Avelar, D. Bazeia, and M. S. Hussein, “Solitons of two-component Bose–Einstein condensates modulated in space and time,” Phys. Lett. A 374, 2356–2360 (2010).
[CrossRef]

A. T. Avelar, D. Bazeia, and W. B. Cardoso, “Solitons with cubic and quintic nonlinearities modulated in space and time,” Phys. Rev. E 79, 025602 (2009).
[CrossRef]

Chen, G.

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (1+3)-dimensional nonlinear Schrödinger equation,” Phys. Rev. Lett. 101, 123904 (2008).
[CrossRef]

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, and G. Chen, “Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. A 78, 023821 (2008).
[CrossRef]

Chen, H. H.

H. H. Chen, and C. S. Liu, “Nonlinear wave and soliton propagation in media with arbitrary inhomogeneities,” Phys. Fluids 21, 377–380 (1978).
[CrossRef]

H. H. Chen, and C. S. Liu, “Solitons in nonuniform media,” Phys. Rev. Lett. 37, 693–697 (1976).
[CrossRef]

Chen, R. P.

C. Q. Dai, R. P. Chen, and Y. Y. Wang, “Spatiotemporal self-similar solutions for the nonautonomous (3+1)D cubic-quintic GP equation,” Chin. Phys. B 21, 030508 (2012).
[CrossRef]

Chubykalo, O. A.

V. V. Konotop, O. A. Chubykalo, and L. Vazquez, “Dynamics and interaction of solitons on an integrable inhomogeneous lattice,” Phys. Rev. E 48, 563–568 (1993).
[CrossRef]

Dai, C. Q.

C. Q. Dai, R. P. Chen, and Y. Y. Wang, “Spatiotemporal self-similar solutions for the nonautonomous (3+1)D cubic-quintic GP equation,” Chin. Phys. B 21, 030508 (2012).
[CrossRef]

Dalfovo, F.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

Degasperis, A.

F. Calogero, and A. Degasperis, “Coupled nonlinear evolution equations solvable via the inverse spectral transform, and solitons that come back: the boomeron,” Lett. Nuovo Cimento 16, 425–433 (1976).
[CrossRef]

Dholakia, K.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[CrossRef]

Durnin, J.

J. Durnin, “Exact solutions for nondiffracting beams,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Fibich, G.

G. Fibich, Y. Sivan, and M. I. Weinstein, “Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure,” Physica D 217, 31–57 (2006).
[CrossRef]

Filatrella, G.

B. B. Baizakov, G. Filatrella, B. A. Malomed, and M. Salerno, “Double parametric resonance for matter-wave solitons in a time-modulated trap,” Phys. Rev. E 71, 036619 (2005).
[CrossRef]

Gabitov, I. R.

S. P. Burtsev, and I. R. Gabitov, “Alternative integrable equations of nonlinear optics,” Phys. Rev. A 49, 2065–2070 (1994).
[CrossRef]

Giorgini, S.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

Hasegawa, A.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous matter-wave solitons near the Feshbach resonance,” Phys. Rev. A 81, 023610 (2010).
[CrossRef]

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

V. N. Serkin, and A. Hasegawa, “Novel soliton solutions of the nonlinear Schrödinger equation model,” Phys. Rev. Lett. 85, 4502–4505 (2000).
[CrossRef]

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford University, 1995).

Hawkins, R. J.

He, X. G.

X. G. He, D. Zhao, L. Li, and H. G. Luo, “Engineering integrable nonautonomous nonlinear Schrödinger equations,” Phys. Rev. E 79, 056610 (2009).
[CrossRef]

Heritage, J. P.

Hu, B.

Huang, T.

W. P. Zhong, M. Belić, G. Assanto, B. A. Malomed, and T. Huang, “Light bullets in the spatiotemporal nonlinear Schrödinger equation with a variable negative diffraction coefficient,” Phys. Rev. A 84, 043801 (2011).
[CrossRef]

W.-P. Zhong, M. Belić, and T. Huang, “Three-dimensional Bessel light bullets in self-focusing Kerr media,” Phys. Rev. 82, 033834 (2010).
[CrossRef]

W. P. Zhong, M. Belić, Y. Lu, and T. Huang, “Traveling and solitary wave solutions to the one-dimensional Gross-Pitaevskii equation,” Phys. Rev. E 81, 016605 (2010).
[CrossRef]

Hussein, M. S.

W. B. Cardoso, A. T. Avelar, D. Bazeia, and M. S. Hussein, “Solitons of two-component Bose–Einstein condensates modulated in space and time,” Phys. Lett. A 374, 2356–2360 (2010).
[CrossRef]

Johannisson, P.

P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, “Nonlinear Bessel beams,” Opt. Commun. 222, 107–115(2003).
[CrossRef]

Kamchatnov, A. M.

F. Kh. Abdullaev, A. M. Kamchatnov, V. V. Konotop, and V. A. Brazhnyi, “Adiabatic dynamics of periodic waves in Bose-Einstein condensates with time dependent atomic scattering length,” Phys. Rev. Lett. 90, 230402 (2003).
[CrossRef]

Kartashov, Y. V.

Kivshar, Y. S.

Y. S. Kivshar and X. Yang, “Dark solitons on background of finite extent,” Opt. Commun. 107, 93–98 (1994).
[CrossRef]

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

Kodama, Y.

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford University, 1995).

Konotop, V. 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).
[CrossRef]

V. M. Perez-Garcia, P. J. Torres, and V. V. Konotop, “Similarity transformation for NLS equations with time-dependent coefficients,” Phys. D 221, 31–36 (2006).
[CrossRef]

F. Kh. Abdullaev, A. M. Kamchatnov, V. V. Konotop, and V. A. Brazhnyi, “Adiabatic dynamics of periodic waves in Bose-Einstein condensates with time dependent atomic scattering length,” Phys. Rev. Lett. 90, 230402 (2003).
[CrossRef]

V. V. Konotop, “Soliton on a disordered lattice,” Phys. Rev. E 47, 1423–1426 (1993).
[CrossRef]

V. V. Konotop, O. A. Chubykalo, and L. Vazquez, “Dynamics and interaction of solitons on an integrable inhomogeneous lattice,” Phys. Rev. E 48, 563–568 (1993).
[CrossRef]

Li, L.

X. G. He, D. Zhao, L. Li, and H. G. Luo, “Engineering integrable nonautonomous nonlinear Schrödinger equations,” Phys. Rev. E 79, 056610 (2009).
[CrossRef]

Lisak, M.

P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, “Nonlinear Bessel beams,” Opt. Commun. 222, 107–115(2003).
[CrossRef]

Liu, C. S.

H. H. Chen, and C. S. Liu, “Nonlinear wave and soliton propagation in media with arbitrary inhomogeneities,” Phys. Fluids 21, 377–380 (1978).
[CrossRef]

H. H. Chen, and C. S. Liu, “Solitons in nonuniform media,” Phys. Rev. Lett. 37, 693–697 (1976).
[CrossRef]

Lu, Y.

W. P. Zhong, M. Belić, Y. Lu, and T. Huang, “Traveling and solitary wave solutions to the one-dimensional Gross-Pitaevskii equation,” Phys. Rev. E 81, 016605 (2010).
[CrossRef]

Luo, H. G.

X. G. He, D. Zhao, L. Li, and H. G. Luo, “Engineering integrable nonautonomous nonlinear Schrödinger equations,” Phys. Rev. E 79, 056610 (2009).
[CrossRef]

Malomed, B. A.

W. P. Zhong, M. Belić, G. Assanto, B. A. Malomed, and T. Huang, “Light bullets in the spatiotemporal nonlinear Schrödinger equation with a variable negative diffraction coefficient,” Phys. Rev. A 84, 043801 (2011).
[CrossRef]

B. B. Baizakov, G. Filatrella, B. A. Malomed, and M. Salerno, “Double parametric resonance for matter-wave solitons in a time-modulated trap,” Phys. Rev. E 71, 036619 (2005).
[CrossRef]

B. A. Malomed, Soliton Manegement in Periodic Systems (Springer, 2006).

Marklund, M.

P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, “Nonlinear Bessel beams,” Opt. Commun. 222, 107–115(2003).
[CrossRef]

McGloin, D.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

Perez-Garcia, V. M.

V. M. Perez-Garcia, P. J. Torres, and V. V. Konotop, “Similarity transformation for NLS equations with time-dependent coefficients,” Phys. D 221, 31–36 (2006).
[CrossRef]

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).
[CrossRef]

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef]

Petrovic, N.

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, and G. Chen, “Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. A 78, 023821 (2008).
[CrossRef]

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (1+3)-dimensional nonlinear Schrödinger equation,” Phys. Rev. Lett. 101, 123904 (2008).
[CrossRef]

Pitaevski, L. P.

L. P. Pitaevski and S. Stringari, Bose-Einstein Condensation (Oxford University, 2003).

Pitaevskii, L. P.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

Salerno, M.

B. B. Baizakov, G. Filatrella, B. A. Malomed, and M. Salerno, “Double parametric resonance for matter-wave solitons in a time-modulated trap,” Phys. Rev. E 71, 036619 (2005).
[CrossRef]

Serkin, V. N.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous matter-wave solitons near the Feshbach resonance,” Phys. Rev. A 81, 023610 (2010).
[CrossRef]

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

V. N. Serkin, and A. Hasegawa, “Novel soliton solutions of the nonlinear Schrödinger equation model,” Phys. Rev. Lett. 85, 4502–4505 (2000).
[CrossRef]

Sivan, Y.

G. Fibich, Y. Sivan, and M. I. Weinstein, “Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure,” Physica D 217, 31–57 (2006).
[CrossRef]

Stringari, S.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

L. P. Pitaevski and S. Stringari, Bose-Einstein Condensation (Oxford University, 2003).

Sulem, C.

C. Sulem, and P. L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (Springer-Verlag, 1999).

Sulem, P. L.

C. Sulem, and P. L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (Springer-Verlag, 1999).

Thurston, R. N.

Tomlinson, W. J.

Torner, L.

Torres, P. J.

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef]

V. M. Perez-Garcia, P. J. Torres, and V. V. Konotop, “Similarity transformation for NLS equations with time-dependent coefficients,” Phys. D 221, 31–36 (2006).
[CrossRef]

Vazquez, L.

V. V. Konotop, O. A. Chubykalo, and L. Vazquez, “Dynamics and interaction of solitons on an integrable inhomogeneous lattice,” Phys. Rev. E 48, 563–568 (1993).
[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).
[CrossRef]

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef]

Wang, Y. Y.

C. Q. Dai, R. P. Chen, and Y. Y. Wang, “Spatiotemporal self-similar solutions for the nonautonomous (3+1)D cubic-quintic GP equation,” Chin. Phys. B 21, 030508 (2012).
[CrossRef]

Weiner, A. M.

Weinstein, M. I.

G. Fibich, Y. Sivan, and M. I. Weinstein, “Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure,” Physica D 217, 31–57 (2006).
[CrossRef]

Xie, R. H.

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (1+3)-dimensional nonlinear Schrödinger equation,” Phys. Rev. Lett. 101, 123904 (2008).
[CrossRef]

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, and G. Chen, “Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. A 78, 023821 (2008).
[CrossRef]

Yang, B.

B. Yang, W. P. Zhong, and M. Belić, “Self-similar HG spatial solitons in 2-dimensional nonlocal nonlinear media,” Commun. Theor. Phys. 53, 937–942 (2010).
[CrossRef]

Yang, X.

Y. S. Kivshar and X. Yang, “Dark solitons on background of finite extent,” Opt. Commun. 107, 93–98 (1994).
[CrossRef]

Ye, F.

Zhao, D.

X. G. He, D. Zhao, L. Li, and H. G. Luo, “Engineering integrable nonautonomous nonlinear Schrödinger equations,” Phys. Rev. E 79, 056610 (2009).
[CrossRef]

Zhong, W. P.

W. P. Zhong, M. Belić, G. Assanto, B. A. Malomed, and T. Huang, “Light bullets in the spatiotemporal nonlinear Schrödinger equation with a variable negative diffraction coefficient,” Phys. Rev. A 84, 043801 (2011).
[CrossRef]

W. P. Zhong, M. Belić, and G. Assanto, “Localized nonlinear wavepackets in with radial-azimuthal modulated nonlinearity and an external potential,” Phys. Scripta 84, 055001 (2011).
[CrossRef]

B. Yang, W. P. Zhong, and M. Belić, “Self-similar HG spatial solitons in 2-dimensional nonlocal nonlinear media,” Commun. Theor. Phys. 53, 937–942 (2010).
[CrossRef]

W. P. Zhong and M. Belić, “Traveling wave and soliton solutions of coupled nonlinear Schrödinger equations with variable coefficients,” Phys. Rev. E 82, 047601 (2010).
[CrossRef]

W. P. Zhong, M. Belić, Y. Lu, and T. Huang, “Traveling and solitary wave solutions to the one-dimensional Gross-Pitaevskii equation,” Phys. Rev. E 81, 016605 (2010).
[CrossRef]

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (1+3)-dimensional nonlinear Schrödinger equation,” Phys. Rev. Lett. 101, 123904 (2008).
[CrossRef]

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, and G. Chen, “Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. A 78, 023821 (2008).
[CrossRef]

Zhong, W.-P.

W.-P. Zhong, M. Belić, and T. Huang, “Three-dimensional Bessel light bullets in self-focusing Kerr media,” Phys. Rev. 82, 033834 (2010).
[CrossRef]

Chin. Phys. B (1)

C. Q. Dai, R. P. Chen, and Y. Y. Wang, “Spatiotemporal self-similar solutions for the nonautonomous (3+1)D cubic-quintic GP equation,” Chin. Phys. B 21, 030508 (2012).
[CrossRef]

Commun. Theor. Phys. (1)

B. Yang, W. P. Zhong, and M. Belić, “Self-similar HG spatial solitons in 2-dimensional nonlocal nonlinear media,” Commun. Theor. Phys. 53, 937–942 (2010).
[CrossRef]

Contemp. Phys. (1)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[CrossRef]

Czech. J. Phys. (1)

Z. Bouchal, “Nondifracting optical beams,” Czech. J. Phys. 53, 537–578 (2003).
[CrossRef]

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

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

Lett. Nuovo Cimento (1)

F. Calogero, and A. Degasperis, “Coupled nonlinear evolution equations solvable via the inverse spectral transform, and solitons that come back: the boomeron,” Lett. Nuovo Cimento 16, 425–433 (1976).
[CrossRef]

Opt. Commun. (2)

P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, “Nonlinear Bessel beams,” Opt. Commun. 222, 107–115(2003).
[CrossRef]

Y. S. Kivshar and X. Yang, “Dark solitons on background of finite extent,” Opt. Commun. 107, 93–98 (1994).
[CrossRef]

Opt. Express (1)

Phys. D (1)

V. M. Perez-Garcia, P. J. Torres, and V. V. Konotop, “Similarity transformation for NLS equations with time-dependent coefficients,” Phys. D 221, 31–36 (2006).
[CrossRef]

Phys. Fluids (1)

H. H. Chen, and C. S. Liu, “Nonlinear wave and soliton propagation in media with arbitrary inhomogeneities,” Phys. Fluids 21, 377–380 (1978).
[CrossRef]

Phys. Lett. A (3)

W. B. Cardoso, A. T. Avelar, and D. Bazeia, “Modulation of breathers in cigar- shaped Bose–Einstein condensates,” Phys. Lett. A 374, 2640–2645 (2010).
[CrossRef]

W. B. Cardoso, A. T. Avelar, D. Bazeia, and M. S. Hussein, “Solitons of two-component Bose–Einstein condensates modulated in space and time,” Phys. Lett. A 374, 2356–2360 (2010).
[CrossRef]

J. Belmonte-Beitia, and G. F. Calvo, “Exact solutions for the quintic nonlinear Schrödinger equation with time and space modulated nonlinearities and potentials,” Phys. Lett. A 373, 448–453 (2009).
[CrossRef]

Phys. Rev. (1)

W.-P. Zhong, M. Belić, and T. Huang, “Three-dimensional Bessel light bullets in self-focusing Kerr media,” Phys. Rev. 82, 033834 (2010).
[CrossRef]

Phys. Rev. A (4)

W. P. Zhong, M. Belić, G. Assanto, B. A. Malomed, and T. Huang, “Light bullets in the spatiotemporal nonlinear Schrödinger equation with a variable negative diffraction coefficient,” Phys. Rev. A 84, 043801 (2011).
[CrossRef]

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous matter-wave solitons near the Feshbach resonance,” Phys. Rev. A 81, 023610 (2010).
[CrossRef]

S. P. Burtsev, and I. R. Gabitov, “Alternative integrable equations of nonlinear optics,” Phys. Rev. A 49, 2065–2070 (1994).
[CrossRef]

W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, and G. Chen, “Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. A 78, 023821 (2008).
[CrossRef]

Phys. Rev. E (7)

V. V. Konotop, “Soliton on a disordered lattice,” Phys. Rev. E 47, 1423–1426 (1993).
[CrossRef]

V. V. Konotop, O. A. Chubykalo, and L. Vazquez, “Dynamics and interaction of solitons on an integrable inhomogeneous lattice,” Phys. Rev. E 48, 563–568 (1993).
[CrossRef]

B. B. Baizakov, G. Filatrella, B. A. Malomed, and M. Salerno, “Double parametric resonance for matter-wave solitons in a time-modulated trap,” Phys. Rev. E 71, 036619 (2005).
[CrossRef]

A. T. Avelar, D. Bazeia, and W. B. Cardoso, “Solitons with cubic and quintic nonlinearities modulated in space and time,” Phys. Rev. E 79, 025602 (2009).
[CrossRef]

X. G. He, D. Zhao, L. Li, and H. G. Luo, “Engineering integrable nonautonomous nonlinear Schrödinger equations,” Phys. Rev. E 79, 056610 (2009).
[CrossRef]

W. P. Zhong and M. Belić, “Traveling wave and soliton solutions of coupled nonlinear Schrödinger equations with variable coefficients,” Phys. Rev. E 82, 047601 (2010).
[CrossRef]

W. P. Zhong, M. Belić, Y. Lu, and T. Huang, “Traveling and solitary wave solutions to the one-dimensional Gross-Pitaevskii equation,” Phys. Rev. E 81, 016605 (2010).
[CrossRef]

Phys. Rev. Lett. (8)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef]

F. Kh. Abdullaev, A. M. Kamchatnov, V. V. Konotop, and V. A. Brazhnyi, “Adiabatic dynamics of periodic waves in Bose-Einstein condensates with time dependent atomic scattering length,” Phys. Rev. Lett. 90, 230402 (2003).
[CrossRef]

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98, 064102 (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).
[CrossRef]

V. N. Serkin, and A. Hasegawa, “Novel soliton solutions of the nonlinear Schrödinger equation model,” Phys. Rev. Lett. 85, 4502–4505 (2000).
[CrossRef]

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

M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (1+3)-dimensional nonlinear Schrödinger equation,” Phys. Rev. Lett. 101, 123904 (2008).
[CrossRef]

H. H. Chen, and C. S. Liu, “Solitons in nonuniform media,” Phys. Rev. Lett. 37, 693–697 (1976).
[CrossRef]

Phys. Scripta (1)

W. P. Zhong, M. Belić, and G. Assanto, “Localized nonlinear wavepackets in with radial-azimuthal modulated nonlinearity and an external potential,” Phys. Scripta 84, 055001 (2011).
[CrossRef]

Physica D (1)

G. Fibich, Y. Sivan, and M. I. Weinstein, “Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure,” Physica D 217, 31–57 (2006).
[CrossRef]

Rev. Mod. Phys. (1)

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

Other (5)

L. P. Pitaevski and S. Stringari, Bose-Einstein Condensation (Oxford University, 2003).

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford University, 1995).

C. Sulem, and P. L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (Springer-Verlag, 1999).

B. A. Malomed, Soliton Manegement in Periodic Systems (Springer, 2006).

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

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Figures (6)

Fig. 1.
Fig. 1.

Comparison of the nonlinearity coefficient χ(x) and the diffraction coefficient β(x), for different n. (a) Nonlinearity coefficient with the parameters b0=0.3, λ=10. (b) Diffraction coefficient with the parameter b0=0.1.

Fig. 2.
Fig. 2.

Intensity distributions of the solitary wave (left column) and the profiles of the external potential (right column) when n=1/2; the parameters are as in the text. (a) Bright soliton and (b) Dark soliton.

Fig. 3.
Fig. 3.

Breather solitons (left column) and the external potentials (right column) for n=1/2, for the parameters from Eq. (12B). The figure setup is as in Fig. 2.

Fig. 4.
Fig. 4.

Intensity of the solitary waves (11)—bright soliton (top row) and dark soliton (bottom row)—for different values of n; other parameters b0=0.3, λ0=1, a=1, θ=0, ω0=2. The left column is the intensity distribution, the right column the external potential.

Fig. 5.
Fig. 5.

Distribution plots of the bright and dark breathers; the parameters are as in the text. The setup is as in Fig. 3.

Fig. 6.
Fig. 6.

Comparing the analytical solution with the numerical simulation at z=80. The left column represents the analytical solution given by Eqs. (11), while the right column depicts the numerical simulation of Eq. (1) with w=10. (a) Bright soliton and (b) Dark soliton.

Equations (20)

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

iuz+12β(z,x)2ux2+χ(z,x)|u|2u+R(z,x)u=0,
u(z,x)=A(z,x)V(z,X)eiB(z,x),
iVz+β2(Xx)22VX2+χA2|V|2V=0.
β(Xx)2=1,
χA2=σ,
iVz+122VX2+σ|V|2V=0,
V(z,X)=sech(X)eiz2
V(z,X)=tanh(X)eiz
x(A2Xx)=0,
Bz+β2A2Ax2β2(Bx)2+R=0,
Bx=1βXzXx,
zA2+βx(A2Bx)=0,
A2=λ2(z)Xx,
λzλXxXxzXx2+βx2βXzXx+XzXxxXx2=0,
X(z,x)=λ(z)F(x)+θ(z),
R(z,x)=12λλzzF23Fxx22FxFxxx8λ2Fx4+φz.
uB(z,x)=λ|Fx|sech(λF+θ)ei(12λλzF2λωzF+z2+φ),
uD(z,x)=λ|Fx|tanh(λF+θ)ei(12λλzF2λωzFz+φ),
λ(z)=λ0,θ(z)=acos(ω0z),
θ(z)=θ0,λ(z)=λ0,

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