Abstract

We apply the variational approach to solitons in highly nonlocal nonlinear media in D = 1, 2, 3 dimensions. We compare results obtained by the variational approach with those obtained by the accessible soliton approximation, by considering the same system of equations in the same spatial region and under the same boundary conditions. To assess the accuracy of these approximations, we also compare them with the numerical solution of the equations. We discover that the accessible soliton approximation suffers from systematic errors, when compared to the variational approach and the numerical solution. The errors increase with the dimension of the system. The variational highly nonlocal approximation provides more accurate results in any dimension and as such is more appropriate solution than the accessible soliton approximation.

© 2014 Optical Society of America

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References

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  3. J. Henninot, J. Blach, and M. Warenghem, “Experimental study of the nonlocality of spatial optical solitons excited in nematic liquid crystal,” J. Opt. A: Pure and Applied Opt. 9, 20 (2007).
    [Crossref]
  4. X. Hutsebaut, C. Cambournac, M. Haelterman, J. Beeckman, and K. Neyts, “Measurement of the self-induced waveguide of a solitonlike optical beam in a nematic liquid crystal,” J. Opt. Soc. Am. B 22, 1424–1431 (2005).
    [Crossref]
  5. J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Simulations and experiments on self-focusing conditions in nematic liquid-crystal planar cells,” Opt. Express 12, 1011–1018 (2004).
    [Crossref] [PubMed]
  6. A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71, 065603 (2005).
    [Crossref]
  7. A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
    [Crossref]
  8. C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
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  9. C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
    [Crossref] [PubMed]
  10. J. Henninot, J. Blach, and M. Warenghem, “The investigation of an electrically stabilized optical spatial soliton induced in a nematic liquid crystal,” Journal of Optics A: Pure and Applied Optics 10, 085104 (2008).
    [Crossref]
  11. N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Solitons in highly nonlocal nematic liquid crystals: Variational approach,” Phys. Rev. A 85, 033826 (2012).
    [Crossref]
  12. D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Boundary effects on the dynamics of higher-order optical spatial solitons in nonlocal thermal media,” J. Opt. A: Pure Appl. Opt. 11, 094014 (2009).
    [Crossref]
  13. A. Litvak, “Self-focusing of powerful light beams by thermal effects,” Soviet Journal of Experimental and Theoretical Physics Letters 4, 230 (1966).
  14. F. Dabby and J. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13, 284–286 (1968).
    [Crossref]
  15. A. Litvak, V. Mironov, G. Fraiman, and A. Iunakovskii, “Thermal self-effect of wave beams in a plasma with a nonlocal nonlinearity,” Fizika Plazmy 1, 60–71 (1975).
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  30. B. Aleksić, N. Aleksić, V. Skarka, and M. Belić, “Using graphical processing units to solve the multidimensional ginzburg-landau equation,” Phys. Scr. T149, 014036 (2012).
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2014 (1)

B. N. Aleksić, N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Variational approach versus accessible soliton approximation in nonlocal nonlinear media,” Phys. Scr. T162, 014003 (2014).
[Crossref]

2013 (3)

M. S. Petrović, N. B. Aleksić, A. I. Strinić, and M. R. Belić, “Destruction of shape-invariant solitons in nematic liquid crystals by noise,” Phys. Rev. A 87, 043825 (2013).
[Crossref]

G. Assanto and N. F. Smyth, “Comment on solitons in highly nonlocal nematic liquid crystals: Variational approach,” Phys. Rev. A 87, 047801 (2013).
[Crossref]

N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Reply to comment on solitons in highly nonlocal nematic liquid crystals: Variational approach,” Phys. Rev. A 87, 047802 (2013).
[Crossref]

2012 (3)

N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Solitons in highly nonlocal nematic liquid crystals: Variational approach,” Phys. Rev. A 85, 033826 (2012).
[Crossref]

X. Shi, B. A. Malomed, F. Ye, and X. Chen, “Symmetric and asymmetric solitons in a nonlocal nonlinear coupler,” Phys. Rev. A 85, 053839 (2012).
[Crossref]

B. Aleksić, N. Aleksić, V. Skarka, and M. Belić, “Using graphical processing units to solve the multidimensional ginzburg-landau equation,” Phys. Scr. T149, 014036 (2012).
[Crossref]

2009 (1)

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Boundary effects on the dynamics of higher-order optical spatial solitons in nonlocal thermal media,” J. Opt. A: Pure Appl. Opt. 11, 094014 (2009).
[Crossref]

2008 (3)

J. Henninot, J. Blach, and M. Warenghem, “The investigation of an electrically stabilized optical spatial soliton induced in a nematic liquid crystal,” Journal of Optics A: Pure and Applied Optics 10, 085104 (2008).
[Crossref]

I. Tikhonenkov, B. Malomed, and A. Vardi, “Vortex solitons in dipolar bose-einstein condensates,” Phys. Rev. A 78, 043614 (2008).
[Crossref]

I. Tikhonenkov, B. A. Malomed, and A. Vardi, “Anisotropic solitons in dipolar bose-einstein condensates,” Phys. Rev. Lett. 100, 090406 (2008).
[Crossref] [PubMed]

2007 (1)

J. Henninot, J. Blach, and M. Warenghem, “Experimental study of the nonlocality of spatial optical solitons excited in nematic liquid crystal,” J. Opt. A: Pure and Applied Opt. 9, 20 (2007).
[Crossref]

2005 (3)

X. Hutsebaut, C. Cambournac, M. Haelterman, J. Beeckman, and K. Neyts, “Measurement of the self-induced waveguide of a solitonlike optical beam in a nematic liquid crystal,” J. Opt. Soc. Am. B 22, 1424–1431 (2005).
[Crossref]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
[Crossref] [PubMed]

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

2004 (3)

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
[Crossref] [PubMed]

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, and M. Haelterman, “Simulations and experiments on self-focusing conditions in nematic liquid-crystal planar cells,” Opt. Express 12, 1011–1018 (2004).
[Crossref] [PubMed]

Q. Guo, B. Luo, F. Yi, S. Chi, and Y. Xie, “Large phase shift of nonlocal optical spatial solitons,” Phys. Rev. E 69, 016602 (2004).
[Crossref]

2003 (2)

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
[Crossref] [PubMed]

L. Santos, G. V. Shlyapnikov, and M. Lewenstein, “Roton-maxon spectrum and stability of trapped dipolar bose-einstein condensates,” Phys. Rev. Lett. 90, 250403 (2003).
[Crossref] [PubMed]

2002 (2)

M. R. Belić, D. Vujić, A. Stepken, F. Kaiser, G. F. Calvo, F. Agulló-López, and M. Carrascosa, “Isotropic versus anisotropic modeling of photorefractive solitons,” Phys. Rev. E 65, 066610 (2002).
[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]

1997 (1)

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[Crossref]

1975 (1)

A. Litvak, V. Mironov, G. Fraiman, and A. Iunakovskii, “Thermal self-effect of wave beams in a plasma with a nonlocal nonlinearity,” Fizika Plazmy 1, 60–71 (1975).

1968 (1)

F. Dabby and J. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13, 284–286 (1968).
[Crossref]

1966 (1)

A. Litvak, “Self-focusing of powerful light beams by thermal effects,” Soviet Journal of Experimental and Theoretical Physics Letters 4, 230 (1966).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulars, Graphs, and Mathematical Tables, vol. 55 (Dover Publications, 1964).

Agrawal, G.

Y. S. Kivshar and G. Agrawal, Optical solitons: from fibers to photonic crystals (Academic press, 2003).

Agulló-López, F.

M. R. Belić, D. Vujić, A. Stepken, F. Kaiser, G. F. Calvo, F. Agulló-López, and M. Carrascosa, “Isotropic versus anisotropic modeling of photorefractive solitons,” Phys. Rev. E 65, 066610 (2002).
[Crossref]

Aleksic, B.

B. Aleksić, N. Aleksić, V. Skarka, and M. Belić, “Using graphical processing units to solve the multidimensional ginzburg-landau equation,” Phys. Scr. T149, 014036 (2012).
[Crossref]

B. Aleksić, N. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Variational approach vs accessible soliton approximation in nonlocal nonlinear media,” arXiv preprint arXiv:1311.6840 (2013).

Aleksic, B. N.

B. N. Aleksić, N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Variational approach versus accessible soliton approximation in nonlocal nonlinear media,” Phys. Scr. T162, 014003 (2014).
[Crossref]

Aleksic, N.

B. Aleksić, N. Aleksić, V. Skarka, and M. Belić, “Using graphical processing units to solve the multidimensional ginzburg-landau equation,” Phys. Scr. T149, 014036 (2012).
[Crossref]

B. Aleksić, N. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Variational approach vs accessible soliton approximation in nonlocal nonlinear media,” arXiv preprint arXiv:1311.6840 (2013).

Aleksic, N. B.

B. N. Aleksić, N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Variational approach versus accessible soliton approximation in nonlocal nonlinear media,” Phys. Scr. T162, 014003 (2014).
[Crossref]

M. S. Petrović, N. B. Aleksić, A. I. Strinić, and M. R. Belić, “Destruction of shape-invariant solitons in nematic liquid crystals by noise,” Phys. Rev. A 87, 043825 (2013).
[Crossref]

N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Reply to comment on solitons in highly nonlocal nematic liquid crystals: Variational approach,” Phys. Rev. A 87, 047802 (2013).
[Crossref]

N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Solitons in highly nonlocal nematic liquid crystals: Variational approach,” Phys. Rev. A 85, 033826 (2012).
[Crossref]

Assanto, G.

G. Assanto and N. F. Smyth, “Comment on solitons in highly nonlocal nematic liquid crystals: Variational approach,” Phys. Rev. A 87, 047801 (2013).
[Crossref]

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
[Crossref] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
[Crossref] [PubMed]

Bang, O.

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]

Beeckman, J.

Belic, M.

B. Aleksić, N. Aleksić, V. Skarka, and M. Belić, “Using graphical processing units to solve the multidimensional ginzburg-landau equation,” Phys. Scr. T149, 014036 (2012).
[Crossref]

Belic, M. R.

B. N. Aleksić, N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Variational approach versus accessible soliton approximation in nonlocal nonlinear media,” Phys. Scr. T162, 014003 (2014).
[Crossref]

M. S. Petrović, N. B. Aleksić, A. I. Strinić, and M. R. Belić, “Destruction of shape-invariant solitons in nematic liquid crystals by noise,” Phys. Rev. A 87, 043825 (2013).
[Crossref]

N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Reply to comment on solitons in highly nonlocal nematic liquid crystals: Variational approach,” Phys. Rev. A 87, 047802 (2013).
[Crossref]

N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Solitons in highly nonlocal nematic liquid crystals: Variational approach,” Phys. Rev. A 85, 033826 (2012).
[Crossref]

M. R. Belić, D. Vujić, A. Stepken, F. Kaiser, G. F. Calvo, F. Agulló-López, and M. Carrascosa, “Isotropic versus anisotropic modeling of photorefractive solitons,” Phys. Rev. E 65, 066610 (2002).
[Crossref]

B. Aleksić, N. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Variational approach vs accessible soliton approximation in nonlocal nonlinear media,” arXiv preprint arXiv:1311.6840 (2013).

Blach, J.

J. Henninot, J. Blach, and M. Warenghem, “The investigation of an electrically stabilized optical spatial soliton induced in a nematic liquid crystal,” Journal of Optics A: Pure and Applied Optics 10, 085104 (2008).
[Crossref]

J. Henninot, J. Blach, and M. Warenghem, “Experimental study of the nonlocality of spatial optical solitons excited in nematic liquid crystal,” J. Opt. A: Pure and Applied Opt. 9, 20 (2007).
[Crossref]

Buccoliero, D.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Boundary effects on the dynamics of higher-order optical spatial solitons in nonlocal thermal media,” J. Opt. A: Pure Appl. Opt. 11, 094014 (2009).
[Crossref]

Calvo, G. F.

M. R. Belić, D. Vujić, A. Stepken, F. Kaiser, G. F. Calvo, F. Agulló-López, and M. Carrascosa, “Isotropic versus anisotropic modeling of photorefractive solitons,” Phys. Rev. E 65, 066610 (2002).
[Crossref]

Cambournac, C.

Carmon, T.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
[Crossref] [PubMed]

Carrascosa, M.

M. R. Belić, D. Vujić, A. Stepken, F. Kaiser, G. F. Calvo, F. Agulló-López, and M. Carrascosa, “Isotropic versus anisotropic modeling of photorefractive solitons,” Phys. Rev. E 65, 066610 (2002).
[Crossref]

Chen, X.

X. Shi, B. A. Malomed, F. Ye, and X. Chen, “Symmetric and asymmetric solitons in a nonlocal nonlinear coupler,” Phys. Rev. A 85, 053839 (2012).
[Crossref]

Chi, S.

Q. Guo, B. Luo, F. Yi, S. Chi, and Y. Xie, “Large phase shift of nonlocal optical spatial solitons,” Phys. Rev. E 69, 016602 (2004).
[Crossref]

Cohen, O.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
[Crossref] [PubMed]

Conti, C.

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
[Crossref] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
[Crossref] [PubMed]

Dabby, F.

F. Dabby and J. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13, 284–286 (1968).
[Crossref]

Desyatnikov, A. S.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Boundary effects on the dynamics of higher-order optical spatial solitons in nonlocal thermal media,” J. Opt. A: Pure Appl. Opt. 11, 094014 (2009).
[Crossref]

Fraiman, G.

A. Litvak, V. Mironov, G. Fraiman, and A. Iunakovskii, “Thermal self-effect of wave beams in a plasma with a nonlocal nonlinearity,” Fizika Plazmy 1, 60–71 (1975).

Guo, Q.

Q. Guo, B. Luo, F. Yi, S. Chi, and Y. Xie, “Large phase shift of nonlocal optical spatial solitons,” Phys. Rev. E 69, 016602 (2004).
[Crossref]

Haelterman, M.

Henninot, J.

J. Henninot, J. Blach, and M. Warenghem, “The investigation of an electrically stabilized optical spatial soliton induced in a nematic liquid crystal,” Journal of Optics A: Pure and Applied Optics 10, 085104 (2008).
[Crossref]

J. Henninot, J. Blach, and M. Warenghem, “Experimental study of the nonlocality of spatial optical solitons excited in nematic liquid crystal,” J. Opt. A: Pure and Applied Opt. 9, 20 (2007).
[Crossref]

Hutsebaut, X.

Iunakovskii, A.

A. Litvak, V. Mironov, G. Fraiman, and A. Iunakovskii, “Thermal self-effect of wave beams in a plasma with a nonlocal nonlinearity,” Fizika Plazmy 1, 60–71 (1975).

Kaiser, F.

M. R. Belić, D. Vujić, A. Stepken, F. Kaiser, G. F. Calvo, F. Agulló-López, and M. Carrascosa, “Isotropic versus anisotropic modeling of photorefractive solitons,” Phys. Rev. E 65, 066610 (2002).
[Crossref]

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 (2005).
[Crossref]

Kivshar, Y. S.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Boundary effects on the dynamics of higher-order optical spatial solitons in nonlocal thermal media,” J. Opt. A: Pure Appl. Opt. 11, 094014 (2009).
[Crossref]

Y. S. Kivshar and G. Agrawal, Optical solitons: from fibers to photonic crystals (Academic press, 2003).

Krolikowski, W.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Boundary effects on the dynamics of higher-order optical spatial solitons in nonlocal thermal media,” J. Opt. A: Pure Appl. Opt. 11, 094014 (2009).
[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]

Lewenstein, M.

L. Santos, G. V. Shlyapnikov, and M. Lewenstein, “Roton-maxon spectrum and stability of trapped dipolar bose-einstein condensates,” Phys. Rev. Lett. 90, 250403 (2003).
[Crossref] [PubMed]

Litvak, A.

A. Litvak, V. Mironov, G. Fraiman, and A. Iunakovskii, “Thermal self-effect of wave beams in a plasma with a nonlocal nonlinearity,” Fizika Plazmy 1, 60–71 (1975).

A. Litvak, “Self-focusing of powerful light beams by thermal effects,” Soviet Journal of Experimental and Theoretical Physics Letters 4, 230 (1966).

Luo, B.

Q. Guo, B. Luo, F. Yi, S. Chi, and Y. Xie, “Large phase shift of nonlocal optical spatial solitons,” Phys. Rev. E 69, 016602 (2004).
[Crossref]

Malomed, B.

I. Tikhonenkov, B. Malomed, and A. Vardi, “Vortex solitons in dipolar bose-einstein condensates,” Phys. Rev. A 78, 043614 (2008).
[Crossref]

Malomed, B. A.

X. Shi, B. A. Malomed, F. Ye, and X. Chen, “Symmetric and asymmetric solitons in a nonlocal nonlinear coupler,” Phys. Rev. A 85, 053839 (2012).
[Crossref]

I. Tikhonenkov, B. A. Malomed, and A. Vardi, “Anisotropic solitons in dipolar bose-einstein condensates,” Phys. Rev. Lett. 100, 090406 (2008).
[Crossref] [PubMed]

Manela, O.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
[Crossref] [PubMed]

Mironov, V.

A. Litvak, V. Mironov, G. Fraiman, and A. Iunakovskii, “Thermal self-effect of wave beams in a plasma with a nonlocal nonlinearity,” Fizika Plazmy 1, 60–71 (1975).

Mitchell, D. J.

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[Crossref]

Neyts, K.

Peccianti, M.

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
[Crossref] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
[Crossref] [PubMed]

Petrovic, M. S.

B. N. Aleksić, N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Variational approach versus accessible soliton approximation in nonlocal nonlinear media,” Phys. Scr. T162, 014003 (2014).
[Crossref]

N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Reply to comment on solitons in highly nonlocal nematic liquid crystals: Variational approach,” Phys. Rev. A 87, 047802 (2013).
[Crossref]

M. S. Petrović, N. B. Aleksić, A. I. Strinić, and M. R. Belić, “Destruction of shape-invariant solitons in nematic liquid crystals by noise,” Phys. Rev. A 87, 043825 (2013).
[Crossref]

N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Solitons in highly nonlocal nematic liquid crystals: Variational approach,” Phys. Rev. A 85, 033826 (2012).
[Crossref]

B. Aleksić, N. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Variational approach vs accessible soliton approximation in nonlocal nonlinear media,” arXiv preprint arXiv:1311.6840 (2013).

Polyanin, A. D.

V. F. Zaitsev and A. D. Polyanin, Handbook of Exact Solutions for Ordinary Differential Equations (CRC Press, 2012).

Press, W. H.

W. H. Press, Numerical recipes 3rd edition: The art of scientific computing (Cambridge University, 2007).

Rasmussen, J. J.

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]

Rotschild, C.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
[Crossref] [PubMed]

Santos, L.

L. Santos, G. V. Shlyapnikov, and M. Lewenstein, “Roton-maxon spectrum and stability of trapped dipolar bose-einstein condensates,” Phys. Rev. Lett. 90, 250403 (2003).
[Crossref] [PubMed]

Segev, M.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
[Crossref] [PubMed]

Shi, X.

X. Shi, B. A. Malomed, F. Ye, and X. Chen, “Symmetric and asymmetric solitons in a nonlocal nonlinear coupler,” Phys. Rev. A 85, 053839 (2012).
[Crossref]

Shlyapnikov, G. V.

L. Santos, G. V. Shlyapnikov, and M. Lewenstein, “Roton-maxon spectrum and stability of trapped dipolar bose-einstein condensates,” Phys. Rev. Lett. 90, 250403 (2003).
[Crossref] [PubMed]

Skarka, V.

B. Aleksić, N. Aleksić, V. Skarka, and M. Belić, “Using graphical processing units to solve the multidimensional ginzburg-landau equation,” Phys. Scr. T149, 014036 (2012).
[Crossref]

Smyth, N. F.

G. Assanto and N. F. Smyth, “Comment on solitons in highly nonlocal nematic liquid crystals: Variational approach,” Phys. Rev. A 87, 047801 (2013).
[Crossref]

Snyder, A. W.

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[Crossref]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulars, Graphs, and Mathematical Tables, vol. 55 (Dover Publications, 1964).

Stepken, A.

M. R. Belić, D. Vujić, A. Stepken, F. Kaiser, G. F. Calvo, F. Agulló-López, and M. Carrascosa, “Isotropic versus anisotropic modeling of photorefractive solitons,” Phys. Rev. E 65, 066610 (2002).
[Crossref]

Strinic, A. I.

B. N. Aleksić, N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Variational approach versus accessible soliton approximation in nonlocal nonlinear media,” Phys. Scr. T162, 014003 (2014).
[Crossref]

N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Reply to comment on solitons in highly nonlocal nematic liquid crystals: Variational approach,” Phys. Rev. A 87, 047802 (2013).
[Crossref]

M. S. Petrović, N. B. Aleksić, A. I. Strinić, and M. R. Belić, “Destruction of shape-invariant solitons in nematic liquid crystals by noise,” Phys. Rev. A 87, 043825 (2013).
[Crossref]

N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Solitons in highly nonlocal nematic liquid crystals: Variational approach,” Phys. Rev. A 85, 033826 (2012).
[Crossref]

B. Aleksić, N. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Variational approach vs accessible soliton approximation in nonlocal nonlinear media,” arXiv preprint arXiv:1311.6840 (2013).

Tikhonenkov, I.

I. Tikhonenkov, B. A. Malomed, and A. Vardi, “Anisotropic solitons in dipolar bose-einstein condensates,” Phys. Rev. Lett. 100, 090406 (2008).
[Crossref] [PubMed]

I. Tikhonenkov, B. Malomed, and A. Vardi, “Vortex solitons in dipolar bose-einstein condensates,” Phys. Rev. A 78, 043614 (2008).
[Crossref]

Vardi, A.

I. Tikhonenkov, B. A. Malomed, and A. Vardi, “Anisotropic solitons in dipolar bose-einstein condensates,” Phys. Rev. Lett. 100, 090406 (2008).
[Crossref] [PubMed]

I. Tikhonenkov, B. Malomed, and A. Vardi, “Vortex solitons in dipolar bose-einstein condensates,” Phys. Rev. A 78, 043614 (2008).
[Crossref]

Vujic, D.

M. R. Belić, D. Vujić, A. Stepken, F. Kaiser, G. F. Calvo, F. Agulló-López, and M. Carrascosa, “Isotropic versus anisotropic modeling of photorefractive solitons,” Phys. Rev. E 65, 066610 (2002).
[Crossref]

Warenghem, M.

J. Henninot, J. Blach, and M. Warenghem, “The investigation of an electrically stabilized optical spatial soliton induced in a nematic liquid crystal,” Journal of Optics A: Pure and Applied Optics 10, 085104 (2008).
[Crossref]

J. Henninot, J. Blach, and M. Warenghem, “Experimental study of the nonlocality of spatial optical solitons excited in nematic liquid crystal,” J. Opt. A: Pure and Applied Opt. 9, 20 (2007).
[Crossref]

Whinnery, J.

F. Dabby and J. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13, 284–286 (1968).
[Crossref]

Wyller, J.

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]

Xie, Y.

Q. Guo, B. Luo, F. Yi, S. Chi, and Y. Xie, “Large phase shift of nonlocal optical spatial solitons,” Phys. Rev. E 69, 016602 (2004).
[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 (2005).
[Crossref]

Ye, F.

X. Shi, B. A. Malomed, F. Ye, and X. Chen, “Symmetric and asymmetric solitons in a nonlocal nonlinear coupler,” Phys. Rev. A 85, 053839 (2012).
[Crossref]

Yi, F.

Q. Guo, B. Luo, F. Yi, S. Chi, and Y. Xie, “Large phase shift of nonlocal optical spatial solitons,” Phys. Rev. E 69, 016602 (2004).
[Crossref]

Zaitsev, V. F.

V. F. Zaitsev and A. D. Polyanin, Handbook of Exact Solutions for Ordinary Differential Equations (CRC Press, 2012).

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 (2005).
[Crossref]

Appl. Phys. Lett. (1)

F. Dabby and J. Whinnery, “Thermal self-focusing of laser beams in lead glasses,” Appl. Phys. Lett. 13, 284–286 (1968).
[Crossref]

Fizika Plazmy (1)

A. Litvak, V. Mironov, G. Fraiman, and A. Iunakovskii, “Thermal self-effect of wave beams in a plasma with a nonlocal nonlinearity,” Fizika Plazmy 1, 60–71 (1975).

J. Opt. A: Pure and Applied Opt. (1)

J. Henninot, J. Blach, and M. Warenghem, “Experimental study of the nonlocality of spatial optical solitons excited in nematic liquid crystal,” J. Opt. A: Pure and Applied Opt. 9, 20 (2007).
[Crossref]

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

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Boundary effects on the dynamics of higher-order optical spatial solitons in nonlocal thermal media,” J. Opt. A: Pure Appl. Opt. 11, 094014 (2009).
[Crossref]

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

Journal of Optics A: Pure and Applied Optics (1)

J. Henninot, J. Blach, and M. Warenghem, “The investigation of an electrically stabilized optical spatial soliton induced in a nematic liquid crystal,” Journal of Optics A: Pure and Applied Optics 10, 085104 (2008).
[Crossref]

Opt. Express (1)

Phys. Rev. A (6)

N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Solitons in highly nonlocal nematic liquid crystals: Variational approach,” Phys. Rev. A 85, 033826 (2012).
[Crossref]

M. S. Petrović, N. B. Aleksić, A. I. Strinić, and M. R. Belić, “Destruction of shape-invariant solitons in nematic liquid crystals by noise,” Phys. Rev. A 87, 043825 (2013).
[Crossref]

G. Assanto and N. F. Smyth, “Comment on solitons in highly nonlocal nematic liquid crystals: Variational approach,” Phys. Rev. A 87, 047801 (2013).
[Crossref]

N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Reply to comment on solitons in highly nonlocal nematic liquid crystals: Variational approach,” Phys. Rev. A 87, 047802 (2013).
[Crossref]

I. Tikhonenkov, B. Malomed, and A. Vardi, “Vortex solitons in dipolar bose-einstein condensates,” Phys. Rev. A 78, 043614 (2008).
[Crossref]

X. Shi, B. A. Malomed, F. Ye, and X. Chen, “Symmetric and asymmetric solitons in a nonlocal nonlinear coupler,” Phys. Rev. A 85, 053839 (2012).
[Crossref]

Phys. Rev. E (4)

Q. Guo, B. Luo, F. Yi, S. Chi, and Y. Xie, “Large phase shift of nonlocal optical spatial solitons,” Phys. Rev. E 69, 016602 (2004).
[Crossref]

M. R. Belić, D. Vujić, A. Stepken, F. Kaiser, G. F. Calvo, F. Agulló-López, and M. Carrascosa, “Isotropic versus anisotropic modeling of photorefractive solitons,” Phys. Rev. E 65, 066610 (2002).
[Crossref]

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

Phys. Rev. Lett. (5)

C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. 91, 073901 (2003).
[Crossref] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
[Crossref] [PubMed]

L. Santos, G. V. Shlyapnikov, and M. Lewenstein, “Roton-maxon spectrum and stability of trapped dipolar bose-einstein condensates,” Phys. Rev. Lett. 90, 250403 (2003).
[Crossref] [PubMed]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
[Crossref] [PubMed]

I. Tikhonenkov, B. A. Malomed, and A. Vardi, “Anisotropic solitons in dipolar bose-einstein condensates,” Phys. Rev. Lett. 100, 090406 (2008).
[Crossref] [PubMed]

Phys. Scr. (2)

B. Aleksić, N. Aleksić, V. Skarka, and M. Belić, “Using graphical processing units to solve the multidimensional ginzburg-landau equation,” Phys. Scr. T149, 014036 (2012).
[Crossref]

B. N. Aleksić, N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Variational approach versus accessible soliton approximation in nonlocal nonlinear media,” Phys. Scr. T162, 014003 (2014).
[Crossref]

Science (1)

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[Crossref]

Soviet Journal of Experimental and Theoretical Physics Letters (1)

A. Litvak, “Self-focusing of powerful light beams by thermal effects,” Soviet Journal of Experimental and Theoretical Physics Letters 4, 230 (1966).

Other (5)

Y. S. Kivshar and G. Agrawal, Optical solitons: from fibers to photonic crystals (Academic press, 2003).

W. H. Press, Numerical recipes 3rd edition: The art of scientific computing (Cambridge University, 2007).

V. F. Zaitsev and A. D. Polyanin, Handbook of Exact Solutions for Ordinary Differential Equations (CRC Press, 2012).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulars, Graphs, and Mathematical Tables, vol. 55 (Dover Publications, 1964).

B. Aleksić, N. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Variational approach vs accessible soliton approximation in nonlocal nonlinear media,” arXiv preprint arXiv:1311.6840 (2013).

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Figures (4)

Fig. 1:
Fig. 1: Comparison of both VA and AS approximations (solid lines) and numerical results (dots) for one dimensional (D = 1) nonlocal solitons. Beam width R (a), amplitude A (b), and the period of small oscillations Λ (c), are shown as functions of power P. (d) Distribution of |E| as a function of r for power P = 20. Other parameters are d = 40, z = 100.
Fig. 2:
Fig. 2: Same as Fig. 1, but for D = 2. The only difference is in the propagation length, z = 103.
Fig. 3:
Fig. 3: Same as Fig. 1, but for D = 3 and d = 150. The propagation distance is now z = 104 and P = 60.
Fig. 4:
Fig. 4: Demonstrating stability of numerical and VA solutions. Propagation constant is shown as a function of the beam power, for D = 1 (a), D = 2 (b), and D = 3 (c). According to the Vakhitov-Kolokolov criterion, solitons are stable as long as the slope of these curves is positive.

Equations (44)

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

2 i E z + Δ E + θ E = 0 ,
2 Δ θ α θ + | E | 2 = 0 ,
θ D ( r ) = 1 2 G D ( r , ρ ) | E ( ρ ) | 2 ρ D 1 d ρ ,
G D ( r , ρ ) = { ( r 2 D d 2 D ) / ( D 2 ) , ρ < r < d ( ρ 2 D d 2 D ) / ( D 2 ) , r < ρ < d } .
G 2 ( r , ρ ) = { ln ( r / d ) , ρ < r < d ln ( ρ / d ) , r < ρ < d } .
| E ( r ) | 2 = Q D T D π D / 2 exp ( r 2 T 2 ) ,
θ D ( r ) = Q D 4 π D / 2 ( 2 D ) { r 2 D Γ ( D / 2 , r 2 / T 2 ) T 2 D exp ( r 2 / T 2 ) + [ d 2 D r 2 D ] Γ ( D / 2 ) } + O ( δ ) ,
θ 1 ( r ) = Q 1 4 [ d r erf ( r T ) T exp ( r 2 / T 2 ) π ] + O ( δ ) ,
θ 2 ( r ) = Q 2 8 π [ E i ( r 2 T 2 ) ln ( r 2 d 2 ) ] + O ( δ ) ,
θ 3 ( r ) = Q 3 8 π [ erf ( r / T ) r 1 d ] + O ( δ ) ,
D = i ( E * z E E z E * ) + | E | 2 θ D | E | 2 + ( θ D ) 2 .
δ D r D 1 d r d z = 0
E = A exp [ r 2 2 R 2 + i C r 2 + i ψ ] ,
L D = 2 P D ψ + D P D R 2 ( C + 2 C 2 ) + D P D 2 R 2 + U D ( Q D , T , P D , R ) + V D ( Q D , T ) ,
U D ( Q D , T , P D , R ) = ν θ D | E | 2 r D 1 d r = Q D P D 4 ( 2 D ) π D / 2 ( ( R 2 + T 2 ) ( 2 D ) / 2 d 2 D Γ ( D / 2 ) ) + O ( δ ) ,
V D ( Q D , T ) = ν ( θ ) 2 r D 1 d r = Q D 2 8 π D / 2 Γ ( D / 2 ) d 2 D ( 2 T ) 2 D 2 D + O ( δ ) .
U 2 = Q 2 P 2 8 π ln R 2 + T 2 e γ d 2 + O ( δ ) ,
V 2 = Q 2 2 16 π ln ( e γ d 2 2 T 2 ) + O ( δ ) ,
d P D d z = 0 ,
C = 1 2 R d R d z ,
d 2 R d z 2 = 1 R 3 1 D P D ( U D R ) T = R , Q = P ,
d ψ d z = D 2 R 2 + 1 2 ( R 2 P U D R U D P ) T = R , Q = P .
R VA = ( 2 2 + D / 2 π D / 2 D P D ) 1 / ( 4 D ) + O ( δ ) ,
A VA = 2 1 + D / 4 D R VA 2 = ( P D 2 2 D ( 1 + D / 4 ) D D / 2 π D ) 1 / ( 4 D ) + O ( δ )
Λ VA = 2 π 4 D R VA 2 = 2 π 4 D ( 2 2 + D / 2 π D / 2 D P D ) 2 / ( 4 D ) + O ( δ ) .
μ D = 6 D 2 ( D 2 ) ( D ( 2 D ) P D 2 2 8 π D ) 1 / ( 4 D ) + d ( 2 D ) Γ ( D / 2 ) 8 ( 2 D ) π D / 2 P D + O ( δ ) .
μ 2 = P 2 16 π ln ( e γ 1 / 2 32 π d 2 P 2 ) + O ( δ ) .
θ D ( r ) θ D max Θ D r 2 ,
Θ D = Q D 4 D π D / 2 T D ,
θ D max = Q D 4 π D / 2 d 2 D Γ ( D / 2 ) T 2 D 2 D + O ( δ ) .
θ 2 max = Q 2 8 π ln ( e γ d 2 T 2 ) + O ( δ ) ,
d A d z = D C A ,
C = 1 2 R d R d z ,
d 2 R d z 2 = ( 1 R 3 Θ D R ) ,
d ψ d z = D 2 R 2 + θ D max 2 .
d 2 R d z 2 = 1 R 3 P D 4 D π D / 2 R D 1 .
R AS = ( 4 D π D / 2 P D ) 1 / ( 4 D )
A AS = 2 D R AS 2 = 2 ( P D 2 16 π D D D / 2 ) 1 / ( 4 D ) .
Λ AS = 2 π 4 D R AS 2 = 2 π 4 D ( 4 D π D / 2 P D ) 2 / ( 4 D ) .
R = R * cos 2 ( π z / Λ ) + P * P sin 2 ( π z / Λ ) ,
C = π Λ ( P * / P 1 ) sin ( 2 π z / Λ ) 4 ( cos 2 ( π z / Λ ) + ( P * / P ) sin 2 ( π z / Λ ) ) .
Λ * = π R * 2 .
( E r ) r = 0 = 0 , E r = d = 0 ,
( θ r ) r = 0 = 0 , θ r = d = 0 .

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