Abstract

In nonlocal nonlinear media with a sine-oscillation response function, two kinds of spatial solitons with complicated structure, in-phase and out-of-phase bound-state solitons, are obtained numerically in the case of the weak nonlocality. The in-phase bound-state soliton exhibits the symmetrical profile and the nonzero central value, and the out-of-phase bound-state soliton has the antisymmetrical profile and the zero central value. The two kinds of bound-state solitons form the degenerate modes subject to the same dependance of soliton power on the propagation constant. For those solitons there exist two abnormal properties: both the soliton propagation constants and the slope of the power versus propagation constants are negative. Both the in-phase and the out-of-phase bound-state solitons are stable by the linear stability analysis, and the stability of the two kinds of solitons obey an inverted Vakhitov-Kolokolov stability criterion. We also discuss the way how to obtain a set of the soliton solutions from one numerical soliton solution via the transform invariance of the nonlocal nonlinear Schrödinger equation and the nonlinear Schrödinger equation.

© 2016 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]

2014 (2)

Y. Q. Zhang, M. R. BeliÄĞ, H. Zheng, H. Chen, C. Li, Y. Li, and Y. Zhang, “Interactions of Airy beams, nonlinear accelerating beams, and induced solitons in Kerr and saturable nonlinear media,” Opt. Express. 22, 7160 (2014).
[Crossref] [PubMed]

J. Wang, Y. Li, Q. Guo, and W. Hu, “Stabilization of nonlocal solitons by boundary conditions,” Opt. Lett. 39, 405 (2014).
[Crossref] [PubMed]

2013 (2)

Y. Q. Zhang, M. BeliÄĞ, Z. Wu, H. Zheng, K. Lu, Y. Li, and Y. Zhang, “Soliton pair generation in the interactions of Airy and nonlinear accelerating beams,” Opt. Lett. 38, 4585 (2013).
[Crossref] [PubMed]

Z. K. Wu, Y. Zhang, C. Yuan, F. Wen, H. Zheng, Y. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A. 88, 063828 (2013).
[Crossref]

2012 (1)

B. K. Esbensen, M. Bache, W. Królikowski, and O. Bang, “Quadratic solitons for negative effective second-harmonic diffraction as nonlocal solitons with periodic nonlocal response function,” Phys. Rev. A 86, 023849 (2012).
[Crossref]

2011 (1)

Y. P. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-Wave Mixing Dipole Soliton in Laser-Induced Atomic Gratings,” Phys. Rev. Lett. 106, 093904 (2011).
[Crossref] [PubMed]

2010 (1)

H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81, 013624 (2010).
[Crossref]

2008 (1)

2007 (1)

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Laguerre and Hermite Soliton Clusters in Nonlocal Nonlinear Media,” Phys. Rev. Lett. 98, 053901 (2007).
[Crossref] [PubMed]

2006 (5)

A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, “Dynamics of two-dimensional coherent structures in non-local nonlinear media,” Phys. Rev. E 73, 066605 (2006).
[Crossref]

S. Ouyang, Q. Guo, and W. Hu, “Perturbative analysis of generally nonlocal spatial optical solitons,” Phys. Rev. E 74, 036622 (2006).
[Crossref]

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2, 769 (2006).
[Crossref]

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

W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, “Nonlocality-controlled interaction of spatial solitons in nematic liquid crystals,” Appl. Phys. Lett. 89, 071111 (2006).
[Crossref]

2005 (4)

P. D. Rasmussen, O. Bang, and W. Królikowski, “Theory of nonlocal soliton interaction in nematic liquid crystals,” Phys. Rev. E 72, 066611 (2005).
[Crossref]

A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, “Bose-Einstein Condensation of Chromium,” Phys. Rev. Lett. 94, 160401 (2005).
[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 conherent elliptic solitons and vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
[Crossref]

Z. Xu, Y. V. Kartashov, and L. Torner, “Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media,” Opt. Lett. 30, 3171 (2005).
[Crossref] [PubMed]

2004 (4)

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

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, “Single-component higher-order mode solitons in liquid crystals,” Opt. Commun. 233, 211 (2004).
[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]

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature (London) 432, 733 (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]

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

2002 (2)

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

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

2001 (1)

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

2000 (2)

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7 (2000).
[Crossref]

M. L. Chiofalo, S. Succi, and M. P. Tosi, “Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm,” Phys. Rev. E 62, 7438 (2000).
[Crossref]

1998 (2)

W. Królikowski, M. Saffman, B. Luther-Davies, and C. Denz, “Anomalous Interaction of Spatial Solitons in Photorefractive Media,” Phys. Rev. Lett. 80, 3240 (1998).
[Crossref]

A. Parola, L. Salasnich, and L. Reatto, “Structure and stability of bosonic clouds: Alkali-metal atoms with negative scattering length,” Phys. Rev. A 57, R3180 (1998).
[Crossref]

1997 (3)

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

Y.R. Shen, “Solitons made simple,” Science 276, 1520 (1997).
[Crossref]

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

1993 (1)

D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48, 4583 (1993).
[Crossref] [PubMed]

1992 (1)

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923 (1992).
[Crossref] [PubMed]

Adamski, A.

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, “Single-component higher-order mode solitons in liquid crystals,” Opt. Commun. 233, 211 (2004).
[Crossref]

Agrawal, G. P.

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

Alfassi, B.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2, 769 (2006).
[Crossref]

Anderson, D. Z.

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

Assanto, G.

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]

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature (London) 432, 733 (2004).
[Crossref]

C. Conti, M. Peccianti, and G. Assanto, “Route to Nonlocality and Observation of Accessible Solitons,” Phys. Rev. Lett. 91, 073901 (2003).
[Crossref] [PubMed]

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

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7 (2000).
[Crossref]

Bache, M.

B. K. Esbensen, M. Bache, W. Królikowski, and O. Bang, “Quadratic solitons for negative effective second-harmonic diffraction as nonlocal solitons with periodic nonlocal response function,” Phys. Rev. A 86, 023849 (2012).
[Crossref]

Bang, O.

B. K. Esbensen, M. Bache, W. Królikowski, and O. Bang, “Quadratic solitons for negative effective second-harmonic diffraction as nonlocal solitons with periodic nonlocal response function,” Phys. Rev. A 86, 023849 (2012).
[Crossref]

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

P. D. Rasmussen, O. Bang, and W. Królikowski, “Theory of nonlocal soliton interaction in nematic liquid crystals,” Phys. Rev. E 72, 066611 (2005).
[Crossref]

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

N. I. Nikolov, D. Neshev, O. Bang, and W. Z. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68, 036614 (2003).
[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 (2002).
[Crossref]

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

BeliÄG, M.

BeliÄG, M. R.

Y. Q. Zhang, M. R. BeliÄĞ, H. Zheng, H. Chen, C. Li, Y. Li, and Y. Zhang, “Interactions of Airy beams, nonlinear accelerating beams, and induced solitons in Kerr and saturable nonlinear media,” Opt. Express. 22, 7160 (2014).
[Crossref] [PubMed]

Blasberg, T.

D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48, 4583 (1993).
[Crossref] [PubMed]

Brzdakiewicz, K. A.

Buccoliero, D.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Spiraling multivortex solitons in nonlocal nonlinear media,” Opt. Lett. 33, 198 (2008).
[Crossref] [PubMed]

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Laguerre and Hermite Soliton Clusters in Nonlocal Nonlinear Media,” Phys. Rev. Lett. 98, 053901 (2007).
[Crossref] [PubMed]

Cambournac, C.

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, “Single-component higher-order mode solitons in liquid crystals,” Opt. Commun. 233, 211 (2004).
[Crossref]

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 conherent elliptic solitons and vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
[Crossref]

Chen, H.

Y. Q. Zhang, M. R. BeliÄĞ, H. Zheng, H. Chen, C. Li, Y. Li, and Y. Zhang, “Interactions of Airy beams, nonlinear accelerating beams, and induced solitons in Kerr and saturable nonlinear media,” Opt. Express. 22, 7160 (2014).
[Crossref] [PubMed]

Y. P. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-Wave Mixing Dipole Soliton in Laser-Induced Atomic Gratings,” Phys. Rev. Lett. 106, 093904 (2011).
[Crossref] [PubMed]

Chiofalo, M. L.

M. L. Chiofalo, S. Succi, and M. P. Tosi, “Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm,” Phys. Rev. E 62, 7438 (2000).
[Crossref]

Christiansen, P. L.

Cohen, O.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2, 769 (2006).
[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 conherent elliptic solitons and vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
[Crossref]

Conti, C.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature (London) 432, 733 (2004).
[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]

Crosignani, B.

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923 (1992).
[Crossref] [PubMed]

De Luca, A.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature (London) 432, 733 (2004).
[Crossref]

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7 (2000).
[Crossref]

De Rossi, A.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7 (2000).
[Crossref]

Denz, C.

W. Królikowski, M. Saffman, B. Luther-Davies, and C. Denz, “Anomalous Interaction of Spatial Solitons in Photorefractive Media,” Phys. Rev. Lett. 80, 3240 (1998).
[Crossref]

Desyatnikov, A. S.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Spiraling multivortex solitons in nonlocal nonlinear media,” Opt. Lett. 33, 198 (2008).
[Crossref] [PubMed]

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Laguerre and Hermite Soliton Clusters in Nonlocal Nonlinear Media,” Phys. Rev. Lett. 98, 053901 (2007).
[Crossref] [PubMed]

Dreischuh, A.

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

Esbensen, B. K.

B. K. Esbensen, M. Bache, W. Królikowski, and O. Bang, “Quadratic solitons for negative effective second-harmonic diffraction as nonlocal solitons with periodic nonlocal response function,” Phys. Rev. A 86, 023849 (2012).
[Crossref]

Fischer, B.

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923 (1992).
[Crossref] [PubMed]

Griesmaier, A.

A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, “Bose-Einstein Condensation of Chromium,” Phys. Rev. Lett. 94, 160401 (2005).
[Crossref] [PubMed]

Guo, Q.

J. Wang, Y. Li, Q. Guo, and W. Hu, “Stabilization of nonlocal solitons by boundary conditions,” Opt. Lett. 39, 405 (2014).
[Crossref] [PubMed]

W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, “Nonlocality-controlled interaction of spatial solitons in nematic liquid crystals,” Appl. Phys. Lett. 89, 071111 (2006).
[Crossref]

S. Ouyang, Q. Guo, and W. Hu, “Perturbative analysis of generally nonlocal spatial optical solitons,” Phys. Rev. E 74, 036622 (2006).
[Crossref]

Haelterman, M.

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, “Single-component higher-order mode solitons in liquid crystals,” Opt. Commun. 233, 211 (2004).
[Crossref]

Hensler, S.

A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, “Bose-Einstein Condensation of Chromium,” Phys. Rev. Lett. 94, 160401 (2005).
[Crossref] [PubMed]

Hu, W.

J. Wang, Y. Li, Q. Guo, and W. Hu, “Stabilization of nonlocal solitons by boundary conditions,” Opt. Lett. 39, 405 (2014).
[Crossref] [PubMed]

W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, “Nonlocality-controlled interaction of spatial solitons in nematic liquid crystals,” Appl. Phys. Lett. 89, 071111 (2006).
[Crossref]

S. Ouyang, Q. Guo, and W. Hu, “Perturbative analysis of generally nonlocal spatial optical solitons,” Phys. Rev. E 74, 036622 (2006).
[Crossref]

Hutsebaut, X.

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, “Single-component higher-order mode solitons in liquid crystals,” Opt. Commun. 233, 211 (2004).
[Crossref]

Kartashov, Y. V.

Khoo, I. C.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7 (2000).
[Crossref]

Kivshar, Y. S.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Spiraling multivortex solitons in nonlocal nonlinear media,” Opt. Lett. 33, 198 (2008).
[Crossref] [PubMed]

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Laguerre and Hermite Soliton Clusters in Nonlocal Nonlinear Media,” Phys. Rev. Lett. 98, 053901 (2007).
[Crossref] [PubMed]

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

Krolikowski, W.

Królikowski, W.

B. K. Esbensen, M. Bache, W. Królikowski, and O. Bang, “Quadratic solitons for negative effective second-harmonic diffraction as nonlocal solitons with periodic nonlocal response function,” Phys. Rev. A 86, 023849 (2012).
[Crossref]

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

P. D. Rasmussen, O. Bang, and W. Królikowski, “Theory of nonlocal soliton interaction in nematic liquid crystals,” Phys. Rev. E 72, 066611 (2005).
[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 (2002).
[Crossref]

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

W. Królikowski, M. Saffman, B. Luther-Davies, and C. Denz, “Anomalous Interaction of Spatial Solitons in Photorefractive Media,” Phys. Rev. Lett. 80, 3240 (1998).
[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 (2003).
[Crossref]

Lan, S.

W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, “Nonlocality-controlled interaction of spatial solitons in nematic liquid crystals,” Appl. Phys. Lett. 89, 071111 (2006).
[Crossref]

Lashkin, V. M.

A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, “Dynamics of two-dimensional coherent structures in non-local nonlinear media,” Phys. Rev. E 73, 066605 (2006).
[Crossref]

Li, C.

Y. Q. Zhang, M. R. BeliÄĞ, H. Zheng, H. Chen, C. Li, Y. Li, and Y. Zhang, “Interactions of Airy beams, nonlinear accelerating beams, and induced solitons in Kerr and saturable nonlinear media,” Opt. Express. 22, 7160 (2014).
[Crossref] [PubMed]

Y. P. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-Wave Mixing Dipole Soliton in Laser-Induced Atomic Gratings,” Phys. Rev. Lett. 106, 093904 (2011).
[Crossref] [PubMed]

Li, Y.

Lu, K.

Y. Q. Zhang, M. BeliÄĞ, Z. Wu, H. Zheng, K. Lu, Y. Li, and Y. Zhang, “Soliton pair generation in the interactions of Airy and nonlinear accelerating beams,” Opt. Lett. 38, 4585 (2013).
[Crossref] [PubMed]

Y. P. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-Wave Mixing Dipole Soliton in Laser-Induced Atomic Gratings,” Phys. Rev. Lett. 106, 093904 (2011).
[Crossref] [PubMed]

Luther-Davies, B.

W. Królikowski, M. Saffman, B. Luther-Davies, and C. Denz, “Anomalous Interaction of Spatial Solitons in Photorefractive Media,” Phys. Rev. Lett. 80, 3240 (1998).
[Crossref]

Malomed, B. A.

H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81, 013624 (2010).
[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, R1110 (1997).
[Crossref]

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 conherent elliptic solitons and vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
[Crossref]

Mezentsev, V. K.

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

Mitchell, D.J.

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

Neshev, D.

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

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

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

Neyts, K.

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, “Single-component higher-order mode solitons in liquid crystals,” Opt. Commun. 233, 211 (2004).
[Crossref]

Nie, Z.

Y. P. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-Wave Mixing Dipole Soliton in Laser-Induced Atomic Gratings,” Phys. Rev. Lett. 106, 093904 (2011).
[Crossref] [PubMed]

Nikolov, N. I.

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

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

Ouyang, S.

S. Ouyang, Q. Guo, and W. Hu, “Perturbative analysis of generally nonlocal spatial optical solitons,” Phys. Rev. E 74, 036622 (2006).
[Crossref]

Parola, A.

A. Parola, L. Salasnich, and L. Reatto, “Structure and stability of bosonic clouds: Alkali-metal atoms with negative scattering length,” Phys. Rev. A 57, R3180 (1998).
[Crossref]

Peccianti, M.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature (London) 432, 733 (2004).
[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]

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

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7 (2000).
[Crossref]

Petersen, D. E.

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

Pfau, T.

A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, “Bose-Einstein Condensation of Chromium,” Phys. Rev. Lett. 94, 160401 (2005).
[Crossref] [PubMed]

Prikhodko, O. O.

A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, “Dynamics of two-dimensional coherent structures in non-local nonlinear media,” Phys. Rev. E 73, 066605 (2006).
[Crossref]

Rasmussen, J. J.

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

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

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

Rasmussen, P. D.

P. D. Rasmussen, O. Bang, and W. Królikowski, “Theory of nonlocal soliton interaction in nematic liquid crystals,” Phys. Rev. E 72, 066611 (2005).
[Crossref]

Reatto, L.

A. Parola, L. Salasnich, and L. Reatto, “Structure and stability of bosonic clouds: Alkali-metal atoms with negative scattering length,” Phys. Rev. A 57, R3180 (1998).
[Crossref]

Rotschild, C.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2, 769 (2006).
[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 conherent elliptic solitons and vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
[Crossref]

Saffman, M.

W. Królikowski, M. Saffman, B. Luther-Davies, and C. Denz, “Anomalous Interaction of Spatial Solitons in Photorefractive Media,” Phys. Rev. Lett. 80, 3240 (1998).
[Crossref]

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

Sakaguchi, H.

H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81, 013624 (2010).
[Crossref]

Salasnich, L.

A. Parola, L. Salasnich, and L. Reatto, “Structure and stability of bosonic clouds: Alkali-metal atoms with negative scattering length,” Phys. Rev. A 57, R3180 (1998).
[Crossref]

Segev, M.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2, 769 (2006).
[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 conherent elliptic solitons and vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
[Crossref]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923 (1992).
[Crossref] [PubMed]

Shen, Y.R.

Y.R. Shen, “Solitons made simple,” Science 276, 1520 (1997).
[Crossref]

Snyder, A.W.

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

Stuhler, J.

A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, “Bose-Einstein Condensation of Chromium,” Phys. Rev. Lett. 94, 160401 (2005).
[Crossref] [PubMed]

Succi, S.

M. L. Chiofalo, S. Succi, and M. P. Tosi, “Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm,” Phys. Rev. E 62, 7438 (2000).
[Crossref]

Suter, D.

D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48, 4583 (1993).
[Crossref] [PubMed]

Torner, L.

Tosi, M. P.

M. L. Chiofalo, S. Succi, and M. P. Tosi, “Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm,” Phys. Rev. E 62, 7438 (2000).
[Crossref]

Umeton, C.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature (London) 432, 733 (2004).
[Crossref]

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7 (2000).
[Crossref]

Wang, J.

Wang, Z.

Y. P. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-Wave Mixing Dipole Soliton in Laser-Induced Atomic Gratings,” Phys. Rev. Lett. 106, 093904 (2011).
[Crossref] [PubMed]

Wen, F.

Z. K. Wu, Y. Zhang, C. Yuan, F. Wen, H. Zheng, Y. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A. 88, 063828 (2013).
[Crossref]

Werner, J.

A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, “Bose-Einstein Condensation of Chromium,” Phys. Rev. Lett. 94, 160401 (2005).
[Crossref] [PubMed]

Wu, Z.

Wu, Z. K.

Z. K. Wu, Y. Zhang, C. Yuan, F. Wen, H. Zheng, Y. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A. 88, 063828 (2013).
[Crossref]

Wyller, J.

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

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

Xiao, M.

Z. K. Wu, Y. Zhang, C. Yuan, F. Wen, H. Zheng, Y. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A. 88, 063828 (2013).
[Crossref]

Y. P. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-Wave Mixing Dipole Soliton in Laser-Induced Atomic Gratings,” Phys. Rev. Lett. 106, 093904 (2011).
[Crossref] [PubMed]

Xu, Z.

Xuan, L.

W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, “Nonlocality-controlled interaction of spatial solitons in nematic liquid crystals,” Appl. Phys. Lett. 89, 071111 (2006).
[Crossref]

Yakimenko, A. I.

A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, “Dynamics of two-dimensional coherent structures in non-local nonlinear media,” Phys. Rev. E 73, 066605 (2006).
[Crossref]

Yariv, A.

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68, 923 (1992).
[Crossref] [PubMed]

Yuan, C.

Z. K. Wu, Y. Zhang, C. Yuan, F. Wen, H. Zheng, Y. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A. 88, 063828 (2013).
[Crossref]

Zhang, T.

W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, “Nonlocality-controlled interaction of spatial solitons in nematic liquid crystals,” Appl. Phys. Lett. 89, 071111 (2006).
[Crossref]

Zhang, Y.

Y. Q. Zhang, M. R. BeliÄĞ, H. Zheng, H. Chen, C. Li, Y. Li, and Y. Zhang, “Interactions of Airy beams, nonlinear accelerating beams, and induced solitons in Kerr and saturable nonlinear media,” Opt. Express. 22, 7160 (2014).
[Crossref] [PubMed]

Z. K. Wu, Y. Zhang, C. Yuan, F. Wen, H. Zheng, Y. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A. 88, 063828 (2013).
[Crossref]

Y. Q. Zhang, M. BeliÄĞ, Z. Wu, H. Zheng, K. Lu, Y. Li, and Y. Zhang, “Soliton pair generation in the interactions of Airy and nonlinear accelerating beams,” Opt. Lett. 38, 4585 (2013).
[Crossref] [PubMed]

Z. K. Wu, Y. Zhang, C. Yuan, F. Wen, H. Zheng, Y. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A. 88, 063828 (2013).
[Crossref]

Zhang, Y. P.

Y. P. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-Wave Mixing Dipole Soliton in Laser-Induced Atomic Gratings,” Phys. Rev. Lett. 106, 093904 (2011).
[Crossref] [PubMed]

Zhang, Y. Q.

Y. Q. Zhang, M. R. BeliÄĞ, H. Zheng, H. Chen, C. Li, Y. Li, and Y. Zhang, “Interactions of Airy beams, nonlinear accelerating beams, and induced solitons in Kerr and saturable nonlinear media,” Opt. Express. 22, 7160 (2014).
[Crossref] [PubMed]

Y. Q. Zhang, M. BeliÄĞ, Z. Wu, H. Zheng, K. Lu, Y. Li, and Y. Zhang, “Soliton pair generation in the interactions of Airy and nonlinear accelerating beams,” Opt. Lett. 38, 4585 (2013).
[Crossref] [PubMed]

Zheng, H.

Y. Q. Zhang, M. R. BeliÄĞ, H. Zheng, H. Chen, C. Li, Y. Li, and Y. Zhang, “Interactions of Airy beams, nonlinear accelerating beams, and induced solitons in Kerr and saturable nonlinear media,” Opt. Express. 22, 7160 (2014).
[Crossref] [PubMed]

Y. Q. Zhang, M. BeliÄĞ, Z. Wu, H. Zheng, K. Lu, Y. Li, and Y. Zhang, “Soliton pair generation in the interactions of Airy and nonlinear accelerating beams,” Opt. Lett. 38, 4585 (2013).
[Crossref] [PubMed]

Z. K. Wu, Y. Zhang, C. Yuan, F. Wen, H. Zheng, Y. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A. 88, 063828 (2013).
[Crossref]

Zozulya, A. A.

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

Appl. Phys. Lett. (2)

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7 (2000).
[Crossref]

W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, “Nonlocality-controlled interaction of spatial solitons in nematic liquid crystals,” Appl. Phys. Lett. 89, 071111 (2006).
[Crossref]

Nat. Phys. (1)

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2, 769 (2006).
[Crossref]

Nature (London) (1)

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature (London) 432, 733 (2004).
[Crossref]

Opt. Commun. (1)

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, “Single-component higher-order mode solitons in liquid crystals,” Opt. Commun. 233, 211 (2004).
[Crossref]

Opt. Express. (1)

Y. Q. Zhang, M. R. BeliÄĞ, H. Zheng, H. Chen, C. Li, Y. Li, and Y. Zhang, “Interactions of Airy beams, nonlinear accelerating beams, and induced solitons in Kerr and saturable nonlinear media,” Opt. Express. 22, 7160 (2014).
[Crossref] [PubMed]

Opt. Lett. (6)

Phys. Rev. A (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, R1110 (1997).
[Crossref]

B. K. Esbensen, M. Bache, W. Królikowski, and O. Bang, “Quadratic solitons for negative effective second-harmonic diffraction as nonlocal solitons with periodic nonlocal response function,” Phys. Rev. A 86, 023849 (2012).
[Crossref]

H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81, 013624 (2010).
[Crossref]

A. Parola, L. Salasnich, and L. Reatto, “Structure and stability of bosonic clouds: Alkali-metal atoms with negative scattering length,” Phys. Rev. A 57, R3180 (1998).
[Crossref]

D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48, 4583 (1993).
[Crossref] [PubMed]

Phys. Rev. A. (1)

Z. K. Wu, Y. Zhang, C. Yuan, F. Wen, H. Zheng, Y. Zhang, and M. Xiao, “Cubic-quintic condensate solitons in four-wave mixing,” Phys. Rev. A. 88, 063828 (2013).
[Crossref]

Phys. Rev. E (7)

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Figures (6)

Fig. 1
Fig. 1 Normalized intensity of Gaussian beam ψ = exp(−x2/2) (solid red line) and its induced (normalized) NRI for small σ (dashed blue line) and large σ (dot-dashed green line) when s = 1.
Fig. 2
Fig. 2 Profiles of in-phase bound-state solitons (solid red lines) and their induced NRI (dashed black lines) for the case that s = 1. (a) σ = 0.472, (b) σ = 0.700, (c) σ = 0.773, and (d) σ = 0.778. The corresponding powers are P = 0.1, P = 1, P = 50 and P = 500 respectively.
Fig. 3
Fig. 3 Dependance of power P and propagation constant λ on the degree of nonlocality σ (a) and P versus λ diagram of the in-phase and out-of-phase bound-state solitons for the case that s = 1.
Fig. 4
Fig. 4 Profiles of the numerical in-phase bound-state solitons (dashed red lines) and the profiles at about 10 Rayleigh distances (dot-dashed green lines) when s = 1 for different degree of the nonlocality. (a) σ = 0.700(P = 1), (b) σ = 0.773(P = 500). The numerical propagations of the solitons for about 10 Rayleigh distances are shown in (c) and (d), respectively.
Fig. 5
Fig. 5 Profile of out-of-phase bound-state solitons (solid red lines) and the induced NRI (dashed black lines) for the case that s = 1. (a) σ = 0.453, (b) σ = 0.731, (c) σ = 0.777, and (d) σ = 0.778. The corresponding power are P = 0.5, P = 10,P = 500 and P = 1000 respectively.
Fig. 6
Fig. 6 Profiles of the numerical out-of-phase bound-state solitons (dashed red lines) and the profiles at about 10 Rayleigh distances (dashed green lines) when s = 1 for different degree of the nonlocality. (a) σ = 0.73(P = 1), (b) σ = 0.78(P = 500). The numerical propagations of the solitons for about 10 Rayleigh distances are shown in (c) and (d), respectively.

Tables (2)

Tables Icon

Table 1 Real part of the perturbation growth rate δ for the in-phase bound-state soli-tons versus the degree of nonlocality σ

Tables Icon

Table 2 Real part of the perturbation growth rate δ for the out-of-phase bound-state solitons versus the degree of nonlocality σ

Equations (14)

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Δ n ( I ) = s R ( x x ) I ( x ) d x ,
i ψ z + 1 2 2 ψ x 2 + Δ n ψ = 0 ,
R ( x ) = 1 2 w m sin ( | x | w m ) ,
w m 2 d 2 Δ n d x 2 + Δ n s | ψ | 2 = 0 .
δ μ = 1 2 d 2 ν d x 2 + λ ν n ν ,
δ ν = 1 2 d 2 μ d x 2 λ μ + n μ + ϕ Δ N .
x ¯ = x κ , z ¯ = z κ 2 , ψ ¯ ( x ¯ , z ¯ ) = κ ψ ( x , z ) , R ¯ ( x ¯ ) = κ R ( x ) , Δ n ¯ ( x ¯ , z ¯ ) = κ 2 Δ n ( x , z ) , w ¯ m = w m κ ,
1 = { ψ ( x , z ) , Δ n ( x , z ) , λ , P , w r , w m } ,
2 = { κ ψ ( x κ , z κ 2 ) , κ 2 Δ n ( x κ , z κ 2 ) , κ 2 λ , κ P , w r κ , w m κ } .
2 = { κ ψ ( x κ , z κ 2 ) , κ 2 Δ n ( x κ , z κ 2 ) , κ 2 λ , κ P , w r κ , w m 2 } | κ = w m 1 / w m 2
1 = { ψ ( x , z ) , Δ n ( x , z ) , λ , P , w r , w m 1 } ,
i ψ z + 1 2 2 ψ x 2 + s | ψ | 2 ψ = 0 ,
1 = { ψ ( x , z ) , | ψ ( x , z ) | 2 , λ , P , w r } ,
2 = { κ ψ ( x κ , z κ 2 ) , κ 2 | ψ ( x κ , z κ 2 ) | 2 , κ 2 λ , P ¯ , w r κ } | κ = P ¯ / P ,

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