Abstract

The propagation behavior of bright spatial solitons in periodically poled photovoltaic (PV) photorefractive (PR) crystals is investigated by considering the diffusion effect. Our analysis indicates that the diffusion effect is modulated by the periodic domain inversion structure (PDIS) of PV PR crystal and therefore can be managed by designing PDIS. By employing numerical integration and perturbation analysis, we find that the self-deflection of the bright soliton beam arising from diffusion effect can be suppressed in PV PR crystals with an appropriate PDIS.

© 2008 Optical Society of America

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  1. R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479-482 (1964).
    [CrossRef]
  2. S. Desyatnikov and Y. S. Kivshar, “Spatial optical solitons and soliton clusters carrying an angular momentum,” J. Opt. B: Quantum Semiclassical Opt. 4, S58-S65 (2002).
    [CrossRef]
  3. M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73, 3211-3214 (1994).
    [CrossRef] [PubMed]
  4. D. N. Christodoulides and M. I. Carvalho, “Bright, dark, and gray spatial soliton states in photorefractive media,” J. Opt. Soc. Am. B 12, 1628-1633 (1995).
    [CrossRef]
  5. M. I. Carvalho, S. R. Singh, and D. N. Christodoulides, “Self-deflection of steady-state bright spatial solitons in biased photorefractive crystals,” Opt. Commun. 120, 311-315 (1995).
    [CrossRef]
  6. M. I. Carvalho, M. Facão, and D. N. Christodoulides, “Self-bending of dark and gray photorefractive solitons,” Phys. Rev. E 76, 016602 (2007).
    [CrossRef]
  7. M. F. Shih, P. Leach, M. Segev, M. H. Garrett, G. Salamo, and G. C. Valley, “Two-dimensional steady-state photorefractive screening solitons,” Opt. Lett. 21, 324-326 (1996).
    [CrossRef] [PubMed]
  8. J. Petter, C. Weilnau, C. Denz, A. Stepken, and F. Kaiser, “Self-bending of photorefractive solitons,” Opt. Commun. 170, 291-297 (1999).
    [CrossRef]
  9. M. Chauvet, V. Coda, H. Maillotte, E. Fazio, and G. Salamo, “Large self-deflection of soliton beams in LiNbO3,” Opt. Lett. 30, 1977-1979 (2005).
    [CrossRef] [PubMed]
  10. K. Pismennaya, O. Kashin, V. Matusevich, A. Kiessling, and R. Kowarschik, “Beam self-trapping and self-bending dynamics in a strontium barium niobate crystal,” J. Opt. Soc. Am. B 25, 136-139 (2008).
    [CrossRef]
  11. Z. Chen, M. Mitchell, M. F. Shih, M. Segev, M. H. Garrett, and G. C. Valley, “Steady-state dark photorefractive screening solitons,” Opt. Lett. 21, 629-631 (1996).
    [CrossRef] [PubMed]
  12. M. Segev, G. C. Valley, M. C. Bashaw, M. Taya, and M. M. Fejer, “Photovoltaic spatial solitons,” J. Opt. Soc. Am. B 14, 1772-1781 (1997).
    [CrossRef]
  13. W. L. She, K. K. Lee, and W. K. Lee, “Observation of two-dimensional bright photovoltaic spatial solitons,” Phys. Rev. Lett. 83, 3182-3185 (1999).
    [CrossRef]
  14. W. L. She, K. K. Lee, and W. K. Lee, “All optical quasi-steady-state photorefractive spatial solitons,” Phys. Rev. Lett. 85, 2498-2501 (2000).
    [CrossRef] [PubMed]
  15. W. L. She, C. C. Xu, B. Guo, and W. K. Lee, “Formation of photovoltaic bright spatial soliton in photorefractive LiNbO3 crystal by a defocused laser beam induced by a background laser beam,” J. Opt. Soc. Am. B 23, 2121-2126 (2006).
    [CrossRef]
  16. J. S. Liu and K. Q. Lu, “Screening-photovoltaic spatial solitons in biased photovoltaic-photorefractive crystals and their self-deflection,” J. Opt. Soc. Am. B 16, 550-555 (1999).
    [CrossRef]
  17. E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. I. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193-2195 (2004).
    [CrossRef]
  18. X. S. Wang and W. L. She, “Spontaneous one- and two-dimensional optical spatial solitons supported by photoisomerization nonlinearity in a bulk polymer,” Phys. Rev. E 71, 026601 (2005).
    [CrossRef]
  19. M. Belić, Ph. Jander, A. Strinić, A. Desyatnikov, and C. Denz, “Self-trapped bidirectional waveguides in a saturable photorefractive medium,” Phys. Rev. E 68, 025601 (2003).
    [CrossRef]
  20. K. Q. Lu, W. Zhao, Y. L. Yang, Y. Yang, X. M. Liu, Y. P. Zhang, and J. J. Xu, “Soliton-induced waveguides in photorefractive photovoltaic materials,” J. Mod. Opt. 53, 2137-2151 (2006).
    [CrossRef]
  21. W. Królikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, “Self-bending photorefractive solitons,” Phys. Rev. E 54, 5761-5765 (1996).
    [CrossRef]
  22. F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67, 013605 (2003).
    [CrossRef]
  23. V. Zharnitsky and D. Pelinovsky, “Averaging of nonlinearity-managed pulses,” Chaos 15, 037105 (2005).
    [CrossRef]
  24. C. N. Liu, T. Morishita, and S. Watanabe, “Time-dependent hyperspherical studies for a two-dimensional attractive Bose-Einstein condensate,” Phys. Rev. A 75, 023604 (2007).
    [CrossRef]
  25. M. Centurion, M. A. Porter, P. G. Kevrekidis, and D. Psaltis, “Nonlinearity management in optics: experiment, theory, and simulation,” Phys. Rev. Lett. 97, 033903 (2006).
    [CrossRef] [PubMed]
  26. N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electron. Lett. 32, 54-55 (1996).
    [CrossRef]
  27. B. A. Malomed and A. Berntson, “Propagation of an optical pulse in a fiber link with random-dispersion management,” J. Opt. Soc. Am. B 18, 1243-1251 (2001).
    [CrossRef]
  28. E. Poutrina and G. P. Agrawal, “Design rules for dispersion-managed soliton systems,” Opt. Commun. 206, 193-200 (2002).
    [CrossRef]
  29. J. Atai and B. A. Malomed, “Spatial solitons in a medium composed of self-focusing and self-defocusing layers,” Phys. Lett. A 298, 140-148 (2002).
    [CrossRef]
  30. H. Sakaguchi and B. A. Malomed, “Resonant nonlinearity management for nonlinear Schrödinger solitons,” Phys. Rev. E 70, 066613 (2004).
    [CrossRef]
  31. A. Ciattoni, E. DelRe, C. Rizza, and A. Marini, “Tailoring wave nonlinearity through spatial composites,” arXiv:0804.3687v1 (2008).
  32. X. N. Shen, J. H. Zhao, Z. X. Cheng, S. J. Zhang, and H. C. Chen, “Domain fixing process in Cu, Mn-doped KNSBN crystals,” Proc. SPIE 3801, 233-239 (1999).
    [CrossRef]
  33. B. Sturman, M. Aguilar, F. Agulló-López, V. Pruneri, and P. G. Kazansky, “Photorefractive nonlinearity of periodically poled ferroelectrics,” J. Opt. Soc. Am. B 14, 2641-2649 (1997).
    [CrossRef]
  34. S. Odoulov, T. Tarabrova, A. Shumelyuk, I. I. Naumova, and T. O. Chaplina, “Photorefractive response of bulk periodically poled LiNbO3:Y:Fe at high and low spatial frequencies,” Phys. Rev. Lett. 84, 3294-3297 (2000).
    [CrossRef] [PubMed]

2008 (2)

2007 (2)

C. N. Liu, T. Morishita, and S. Watanabe, “Time-dependent hyperspherical studies for a two-dimensional attractive Bose-Einstein condensate,” Phys. Rev. A 75, 023604 (2007).
[CrossRef]

M. I. Carvalho, M. Facão, and D. N. Christodoulides, “Self-bending of dark and gray photorefractive solitons,” Phys. Rev. E 76, 016602 (2007).
[CrossRef]

2006 (3)

W. L. She, C. C. Xu, B. Guo, and W. K. Lee, “Formation of photovoltaic bright spatial soliton in photorefractive LiNbO3 crystal by a defocused laser beam induced by a background laser beam,” J. Opt. Soc. Am. B 23, 2121-2126 (2006).
[CrossRef]

M. Centurion, M. A. Porter, P. G. Kevrekidis, and D. Psaltis, “Nonlinearity management in optics: experiment, theory, and simulation,” Phys. Rev. Lett. 97, 033903 (2006).
[CrossRef] [PubMed]

K. Q. Lu, W. Zhao, Y. L. Yang, Y. Yang, X. M. Liu, Y. P. Zhang, and J. J. Xu, “Soliton-induced waveguides in photorefractive photovoltaic materials,” J. Mod. Opt. 53, 2137-2151 (2006).
[CrossRef]

2005 (3)

X. S. Wang and W. L. She, “Spontaneous one- and two-dimensional optical spatial solitons supported by photoisomerization nonlinearity in a bulk polymer,” Phys. Rev. E 71, 026601 (2005).
[CrossRef]

V. Zharnitsky and D. Pelinovsky, “Averaging of nonlinearity-managed pulses,” Chaos 15, 037105 (2005).
[CrossRef]

M. Chauvet, V. Coda, H. Maillotte, E. Fazio, and G. Salamo, “Large self-deflection of soliton beams in LiNbO3,” Opt. Lett. 30, 1977-1979 (2005).
[CrossRef] [PubMed]

2004 (2)

H. Sakaguchi and B. A. Malomed, “Resonant nonlinearity management for nonlinear Schrödinger solitons,” Phys. Rev. E 70, 066613 (2004).
[CrossRef]

E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. I. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193-2195 (2004).
[CrossRef]

2003 (2)

F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67, 013605 (2003).
[CrossRef]

M. Belić, Ph. Jander, A. Strinić, A. Desyatnikov, and C. Denz, “Self-trapped bidirectional waveguides in a saturable photorefractive medium,” Phys. Rev. E 68, 025601 (2003).
[CrossRef]

2002 (3)

E. Poutrina and G. P. Agrawal, “Design rules for dispersion-managed soliton systems,” Opt. Commun. 206, 193-200 (2002).
[CrossRef]

J. Atai and B. A. Malomed, “Spatial solitons in a medium composed of self-focusing and self-defocusing layers,” Phys. Lett. A 298, 140-148 (2002).
[CrossRef]

S. Desyatnikov and Y. S. Kivshar, “Spatial optical solitons and soliton clusters carrying an angular momentum,” J. Opt. B: Quantum Semiclassical Opt. 4, S58-S65 (2002).
[CrossRef]

2001 (1)

2000 (2)

S. Odoulov, T. Tarabrova, A. Shumelyuk, I. I. Naumova, and T. O. Chaplina, “Photorefractive response of bulk periodically poled LiNbO3:Y:Fe at high and low spatial frequencies,” Phys. Rev. Lett. 84, 3294-3297 (2000).
[CrossRef] [PubMed]

W. L. She, K. K. Lee, and W. K. Lee, “All optical quasi-steady-state photorefractive spatial solitons,” Phys. Rev. Lett. 85, 2498-2501 (2000).
[CrossRef] [PubMed]

1999 (4)

W. L. She, K. K. Lee, and W. K. Lee, “Observation of two-dimensional bright photovoltaic spatial solitons,” Phys. Rev. Lett. 83, 3182-3185 (1999).
[CrossRef]

J. S. Liu and K. Q. Lu, “Screening-photovoltaic spatial solitons in biased photovoltaic-photorefractive crystals and their self-deflection,” J. Opt. Soc. Am. B 16, 550-555 (1999).
[CrossRef]

J. Petter, C. Weilnau, C. Denz, A. Stepken, and F. Kaiser, “Self-bending of photorefractive solitons,” Opt. Commun. 170, 291-297 (1999).
[CrossRef]

X. N. Shen, J. H. Zhao, Z. X. Cheng, S. J. Zhang, and H. C. Chen, “Domain fixing process in Cu, Mn-doped KNSBN crystals,” Proc. SPIE 3801, 233-239 (1999).
[CrossRef]

1997 (2)

1996 (4)

Z. Chen, M. Mitchell, M. F. Shih, M. Segev, M. H. Garrett, and G. C. Valley, “Steady-state dark photorefractive screening solitons,” Opt. Lett. 21, 629-631 (1996).
[CrossRef] [PubMed]

M. F. Shih, P. Leach, M. Segev, M. H. Garrett, G. Salamo, and G. C. Valley, “Two-dimensional steady-state photorefractive screening solitons,” Opt. Lett. 21, 324-326 (1996).
[CrossRef] [PubMed]

W. Królikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, “Self-bending photorefractive solitons,” Phys. Rev. E 54, 5761-5765 (1996).
[CrossRef]

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electron. Lett. 32, 54-55 (1996).
[CrossRef]

1995 (2)

D. N. Christodoulides and M. I. Carvalho, “Bright, dark, and gray spatial soliton states in photorefractive media,” J. Opt. Soc. Am. B 12, 1628-1633 (1995).
[CrossRef]

M. I. Carvalho, S. R. Singh, and D. N. Christodoulides, “Self-deflection of steady-state bright spatial solitons in biased photorefractive crystals,” Opt. Commun. 120, 311-315 (1995).
[CrossRef]

1994 (1)

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

Agrawal, G. P.

E. Poutrina and G. P. Agrawal, “Design rules for dispersion-managed soliton systems,” Opt. Commun. 206, 193-200 (2002).
[CrossRef]

Aguilar, M.

Agulló-López, F.

Akhmediev, N.

W. Królikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, “Self-bending photorefractive solitons,” Phys. Rev. E 54, 5761-5765 (1996).
[CrossRef]

Atai, J.

J. Atai and B. A. Malomed, “Spatial solitons in a medium composed of self-focusing and self-defocusing layers,” Phys. Lett. A 298, 140-148 (2002).
[CrossRef]

Bashaw, M. C.

Belic, M.

M. Belić, Ph. Jander, A. Strinić, A. Desyatnikov, and C. Denz, “Self-trapped bidirectional waveguides in a saturable photorefractive medium,” Phys. Rev. E 68, 025601 (2003).
[CrossRef]

Bennion, I.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electron. Lett. 32, 54-55 (1996).
[CrossRef]

Berntson, A.

Bertolotti, M.

E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. I. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193-2195 (2004).
[CrossRef]

Blow, K. J.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electron. Lett. 32, 54-55 (1996).
[CrossRef]

Caputo, J. G.

F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67, 013605 (2003).
[CrossRef]

Carvalho, M. I.

M. I. Carvalho, M. Facão, and D. N. Christodoulides, “Self-bending of dark and gray photorefractive solitons,” Phys. Rev. E 76, 016602 (2007).
[CrossRef]

D. N. Christodoulides and M. I. Carvalho, “Bright, dark, and gray spatial soliton states in photorefractive media,” J. Opt. Soc. Am. B 12, 1628-1633 (1995).
[CrossRef]

M. I. Carvalho, S. R. Singh, and D. N. Christodoulides, “Self-deflection of steady-state bright spatial solitons in biased photorefractive crystals,” Opt. Commun. 120, 311-315 (1995).
[CrossRef]

Centurion, M.

M. Centurion, M. A. Porter, P. G. Kevrekidis, and D. Psaltis, “Nonlinearity management in optics: experiment, theory, and simulation,” Phys. Rev. Lett. 97, 033903 (2006).
[CrossRef] [PubMed]

Chaplina, T. O.

S. Odoulov, T. Tarabrova, A. Shumelyuk, I. I. Naumova, and T. O. Chaplina, “Photorefractive response of bulk periodically poled LiNbO3:Y:Fe at high and low spatial frequencies,” Phys. Rev. Lett. 84, 3294-3297 (2000).
[CrossRef] [PubMed]

Chauvet, M.

M. Chauvet, V. Coda, H. Maillotte, E. Fazio, and G. Salamo, “Large self-deflection of soliton beams in LiNbO3,” Opt. Lett. 30, 1977-1979 (2005).
[CrossRef] [PubMed]

E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. I. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193-2195 (2004).
[CrossRef]

Chen, H. C.

X. N. Shen, J. H. Zhao, Z. X. Cheng, S. J. Zhang, and H. C. Chen, “Domain fixing process in Cu, Mn-doped KNSBN crystals,” Proc. SPIE 3801, 233-239 (1999).
[CrossRef]

Chen, Z.

Cheng, Z. X.

X. N. Shen, J. H. Zhao, Z. X. Cheng, S. J. Zhang, and H. C. Chen, “Domain fixing process in Cu, Mn-doped KNSBN crystals,” Proc. SPIE 3801, 233-239 (1999).
[CrossRef]

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

Christodoulides, D. N.

M. I. Carvalho, M. Facão, and D. N. Christodoulides, “Self-bending of dark and gray photorefractive solitons,” Phys. Rev. E 76, 016602 (2007).
[CrossRef]

D. N. Christodoulides and M. I. Carvalho, “Bright, dark, and gray spatial soliton states in photorefractive media,” J. Opt. Soc. Am. B 12, 1628-1633 (1995).
[CrossRef]

M. I. Carvalho, S. R. Singh, and D. N. Christodoulides, “Self-deflection of steady-state bright spatial solitons in biased photorefractive crystals,” Opt. Commun. 120, 311-315 (1995).
[CrossRef]

Ciattoni, A.

A. Ciattoni, E. DelRe, C. Rizza, and A. Marini, “Tailoring wave nonlinearity through spatial composites,” arXiv:0804.3687v1 (2008).

Coda, V.

Cronin-Golomb, M.

W. Królikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, “Self-bending photorefractive solitons,” Phys. Rev. E 54, 5761-5765 (1996).
[CrossRef]

Crosignani, B.

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

DelRe, E.

A. Ciattoni, E. DelRe, C. Rizza, and A. Marini, “Tailoring wave nonlinearity through spatial composites,” arXiv:0804.3687v1 (2008).

Denz, C.

M. Belić, Ph. Jander, A. Strinić, A. Desyatnikov, and C. Denz, “Self-trapped bidirectional waveguides in a saturable photorefractive medium,” Phys. Rev. E 68, 025601 (2003).
[CrossRef]

J. Petter, C. Weilnau, C. Denz, A. Stepken, and F. Kaiser, “Self-bending of photorefractive solitons,” Opt. Commun. 170, 291-297 (1999).
[CrossRef]

Desyatnikov, A.

M. Belić, Ph. Jander, A. Strinić, A. Desyatnikov, and C. Denz, “Self-trapped bidirectional waveguides in a saturable photorefractive medium,” Phys. Rev. E 68, 025601 (2003).
[CrossRef]

Desyatnikov, S.

S. Desyatnikov and Y. S. Kivshar, “Spatial optical solitons and soliton clusters carrying an angular momentum,” J. Opt. B: Quantum Semiclassical Opt. 4, S58-S65 (2002).
[CrossRef]

DiPorto, P.

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

Doran, N. J.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electron. Lett. 32, 54-55 (1996).
[CrossRef]

Facão, M.

M. I. Carvalho, M. Facão, and D. N. Christodoulides, “Self-bending of dark and gray photorefractive solitons,” Phys. Rev. E 76, 016602 (2007).
[CrossRef]

Fazio, E.

M. Chauvet, V. Coda, H. Maillotte, E. Fazio, and G. Salamo, “Large self-deflection of soliton beams in LiNbO3,” Opt. Lett. 30, 1977-1979 (2005).
[CrossRef] [PubMed]

E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. I. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193-2195 (2004).
[CrossRef]

Fejer, M. M.

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

Garrett, M. H.

Guo, B.

Kaiser, F.

J. Petter, C. Weilnau, C. Denz, A. Stepken, and F. Kaiser, “Self-bending of photorefractive solitons,” Opt. Commun. 170, 291-297 (1999).
[CrossRef]

Kashin, O.

Kazansky, P. G.

Kevrekidis, P. G.

M. Centurion, M. A. Porter, P. G. Kevrekidis, and D. Psaltis, “Nonlinearity management in optics: experiment, theory, and simulation,” Phys. Rev. Lett. 97, 033903 (2006).
[CrossRef] [PubMed]

Kh. Abdullaev, F.

F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67, 013605 (2003).
[CrossRef]

Kiessling, A.

Kivshar, Y. S.

S. Desyatnikov and Y. S. Kivshar, “Spatial optical solitons and soliton clusters carrying an angular momentum,” J. Opt. B: Quantum Semiclassical Opt. 4, S58-S65 (2002).
[CrossRef]

Knox, F. M.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electron. Lett. 32, 54-55 (1996).
[CrossRef]

Kowarschik, R.

Kraenkel, R. A.

F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67, 013605 (2003).
[CrossRef]

Królikowski, W.

W. Królikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, “Self-bending photorefractive solitons,” Phys. Rev. E 54, 5761-5765 (1996).
[CrossRef]

Leach, P.

Lee, K. K.

W. L. She, K. K. Lee, and W. K. Lee, “All optical quasi-steady-state photorefractive spatial solitons,” Phys. Rev. Lett. 85, 2498-2501 (2000).
[CrossRef] [PubMed]

W. L. She, K. K. Lee, and W. K. Lee, “Observation of two-dimensional bright photovoltaic spatial solitons,” Phys. Rev. Lett. 83, 3182-3185 (1999).
[CrossRef]

Lee, W. K.

W. L. She, C. C. Xu, B. Guo, and W. K. Lee, “Formation of photovoltaic bright spatial soliton in photorefractive LiNbO3 crystal by a defocused laser beam induced by a background laser beam,” J. Opt. Soc. Am. B 23, 2121-2126 (2006).
[CrossRef]

W. L. She, K. K. Lee, and W. K. Lee, “All optical quasi-steady-state photorefractive spatial solitons,” Phys. Rev. Lett. 85, 2498-2501 (2000).
[CrossRef] [PubMed]

W. L. She, K. K. Lee, and W. K. Lee, “Observation of two-dimensional bright photovoltaic spatial solitons,” Phys. Rev. Lett. 83, 3182-3185 (1999).
[CrossRef]

Liu, C. N.

C. N. Liu, T. Morishita, and S. Watanabe, “Time-dependent hyperspherical studies for a two-dimensional attractive Bose-Einstein condensate,” Phys. Rev. A 75, 023604 (2007).
[CrossRef]

Liu, J. S.

Liu, X. M.

K. Q. Lu, W. Zhao, Y. L. Yang, Y. Yang, X. M. Liu, Y. P. Zhang, and J. J. Xu, “Soliton-induced waveguides in photorefractive photovoltaic materials,” J. Mod. Opt. 53, 2137-2151 (2006).
[CrossRef]

Lu, K. Q.

K. Q. Lu, W. Zhao, Y. L. Yang, Y. Yang, X. M. Liu, Y. P. Zhang, and J. J. Xu, “Soliton-induced waveguides in photorefractive photovoltaic materials,” J. Mod. Opt. 53, 2137-2151 (2006).
[CrossRef]

J. S. Liu and K. Q. Lu, “Screening-photovoltaic spatial solitons in biased photovoltaic-photorefractive crystals and their self-deflection,” J. Opt. Soc. Am. B 16, 550-555 (1999).
[CrossRef]

Luther-Davies, B.

W. Królikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, “Self-bending photorefractive solitons,” Phys. Rev. E 54, 5761-5765 (1996).
[CrossRef]

Maillotte, H.

Malomed, B. A.

H. Sakaguchi and B. A. Malomed, “Resonant nonlinearity management for nonlinear Schrödinger solitons,” Phys. Rev. E 70, 066613 (2004).
[CrossRef]

F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67, 013605 (2003).
[CrossRef]

J. Atai and B. A. Malomed, “Spatial solitons in a medium composed of self-focusing and self-defocusing layers,” Phys. Lett. A 298, 140-148 (2002).
[CrossRef]

B. A. Malomed and A. Berntson, “Propagation of an optical pulse in a fiber link with random-dispersion management,” J. Opt. Soc. Am. B 18, 1243-1251 (2001).
[CrossRef]

Marini, A.

A. Ciattoni, E. DelRe, C. Rizza, and A. Marini, “Tailoring wave nonlinearity through spatial composites,” arXiv:0804.3687v1 (2008).

Matusevich, V.

Mitchell, M.

Morishita, T.

C. N. Liu, T. Morishita, and S. Watanabe, “Time-dependent hyperspherical studies for a two-dimensional attractive Bose-Einstein condensate,” Phys. Rev. A 75, 023604 (2007).
[CrossRef]

Naumova, I. I.

S. Odoulov, T. Tarabrova, A. Shumelyuk, I. I. Naumova, and T. O. Chaplina, “Photorefractive response of bulk periodically poled LiNbO3:Y:Fe at high and low spatial frequencies,” Phys. Rev. Lett. 84, 3294-3297 (2000).
[CrossRef] [PubMed]

Odoulov, S.

S. Odoulov, T. Tarabrova, A. Shumelyuk, I. I. Naumova, and T. O. Chaplina, “Photorefractive response of bulk periodically poled LiNbO3:Y:Fe at high and low spatial frequencies,” Phys. Rev. Lett. 84, 3294-3297 (2000).
[CrossRef] [PubMed]

Pelinovsky, D.

V. Zharnitsky and D. Pelinovsky, “Averaging of nonlinearity-managed pulses,” Chaos 15, 037105 (2005).
[CrossRef]

Petris, A.

E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. I. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193-2195 (2004).
[CrossRef]

Petter, J.

J. Petter, C. Weilnau, C. Denz, A. Stepken, and F. Kaiser, “Self-bending of photorefractive solitons,” Opt. Commun. 170, 291-297 (1999).
[CrossRef]

Pismennaya, K.

Porter, M. A.

M. Centurion, M. A. Porter, P. G. Kevrekidis, and D. Psaltis, “Nonlinearity management in optics: experiment, theory, and simulation,” Phys. Rev. Lett. 97, 033903 (2006).
[CrossRef] [PubMed]

Poutrina, E.

E. Poutrina and G. P. Agrawal, “Design rules for dispersion-managed soliton systems,” Opt. Commun. 206, 193-200 (2002).
[CrossRef]

Pruneri, V.

Psaltis, D.

M. Centurion, M. A. Porter, P. G. Kevrekidis, and D. Psaltis, “Nonlinearity management in optics: experiment, theory, and simulation,” Phys. Rev. Lett. 97, 033903 (2006).
[CrossRef] [PubMed]

Ramadan, W.

E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. I. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193-2195 (2004).
[CrossRef]

Renzi, F.

E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. I. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193-2195 (2004).
[CrossRef]

Rinaldi, R.

E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. I. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193-2195 (2004).
[CrossRef]

Rizza, C.

A. Ciattoni, E. DelRe, C. Rizza, and A. Marini, “Tailoring wave nonlinearity through spatial composites,” arXiv:0804.3687v1 (2008).

Sakaguchi, H.

H. Sakaguchi and B. A. Malomed, “Resonant nonlinearity management for nonlinear Schrödinger solitons,” Phys. Rev. E 70, 066613 (2004).
[CrossRef]

Salamo, G.

Segev, M.

She, W. L.

W. L. She, C. C. Xu, B. Guo, and W. K. Lee, “Formation of photovoltaic bright spatial soliton in photorefractive LiNbO3 crystal by a defocused laser beam induced by a background laser beam,” J. Opt. Soc. Am. B 23, 2121-2126 (2006).
[CrossRef]

X. S. Wang and W. L. She, “Spontaneous one- and two-dimensional optical spatial solitons supported by photoisomerization nonlinearity in a bulk polymer,” Phys. Rev. E 71, 026601 (2005).
[CrossRef]

W. L. She, K. K. Lee, and W. K. Lee, “All optical quasi-steady-state photorefractive spatial solitons,” Phys. Rev. Lett. 85, 2498-2501 (2000).
[CrossRef] [PubMed]

W. L. She, K. K. Lee, and W. K. Lee, “Observation of two-dimensional bright photovoltaic spatial solitons,” Phys. Rev. Lett. 83, 3182-3185 (1999).
[CrossRef]

Shen, X. N.

X. N. Shen, J. H. Zhao, Z. X. Cheng, S. J. Zhang, and H. C. Chen, “Domain fixing process in Cu, Mn-doped KNSBN crystals,” Proc. SPIE 3801, 233-239 (1999).
[CrossRef]

Shih, M. F.

Shumelyuk, A.

S. Odoulov, T. Tarabrova, A. Shumelyuk, I. I. Naumova, and T. O. Chaplina, “Photorefractive response of bulk periodically poled LiNbO3:Y:Fe at high and low spatial frequencies,” Phys. Rev. Lett. 84, 3294-3297 (2000).
[CrossRef] [PubMed]

Singh, S. R.

M. I. Carvalho, S. R. Singh, and D. N. Christodoulides, “Self-deflection of steady-state bright spatial solitons in biased photorefractive crystals,” Opt. Commun. 120, 311-315 (1995).
[CrossRef]

Smith, N. J.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electron. Lett. 32, 54-55 (1996).
[CrossRef]

Stepken, A.

J. Petter, C. Weilnau, C. Denz, A. Stepken, and F. Kaiser, “Self-bending of photorefractive solitons,” Opt. Commun. 170, 291-297 (1999).
[CrossRef]

Strinic, A.

M. Belić, Ph. Jander, A. Strinić, A. Desyatnikov, and C. Denz, “Self-trapped bidirectional waveguides in a saturable photorefractive medium,” Phys. Rev. E 68, 025601 (2003).
[CrossRef]

Sturman, B.

Tarabrova, T.

S. Odoulov, T. Tarabrova, A. Shumelyuk, I. I. Naumova, and T. O. Chaplina, “Photorefractive response of bulk periodically poled LiNbO3:Y:Fe at high and low spatial frequencies,” Phys. Rev. Lett. 84, 3294-3297 (2000).
[CrossRef] [PubMed]

Taya, M.

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

Valley, G. C.

Vlad, V. I.

E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. I. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193-2195 (2004).
[CrossRef]

Wang, X. S.

X. S. Wang and W. L. She, “Spontaneous one- and two-dimensional optical spatial solitons supported by photoisomerization nonlinearity in a bulk polymer,” Phys. Rev. E 71, 026601 (2005).
[CrossRef]

Watanabe, S.

C. N. Liu, T. Morishita, and S. Watanabe, “Time-dependent hyperspherical studies for a two-dimensional attractive Bose-Einstein condensate,” Phys. Rev. A 75, 023604 (2007).
[CrossRef]

Weilnau, C.

J. Petter, C. Weilnau, C. Denz, A. Stepken, and F. Kaiser, “Self-bending of photorefractive solitons,” Opt. Commun. 170, 291-297 (1999).
[CrossRef]

Xu, C. C.

Xu, J. J.

K. Q. Lu, W. Zhao, Y. L. Yang, Y. Yang, X. M. Liu, Y. P. Zhang, and J. J. Xu, “Soliton-induced waveguides in photorefractive photovoltaic materials,” J. Mod. Opt. 53, 2137-2151 (2006).
[CrossRef]

Yang, Y.

K. Q. Lu, W. Zhao, Y. L. Yang, Y. Yang, X. M. Liu, Y. P. Zhang, and J. J. Xu, “Soliton-induced waveguides in photorefractive photovoltaic materials,” J. Mod. Opt. 53, 2137-2151 (2006).
[CrossRef]

Yang, Y. L.

K. Q. Lu, W. Zhao, Y. L. Yang, Y. Yang, X. M. Liu, Y. P. Zhang, and J. J. Xu, “Soliton-induced waveguides in photorefractive photovoltaic materials,” J. Mod. Opt. 53, 2137-2151 (2006).
[CrossRef]

Yariv, A.

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

Zhang, S. J.

X. N. Shen, J. H. Zhao, Z. X. Cheng, S. J. Zhang, and H. C. Chen, “Domain fixing process in Cu, Mn-doped KNSBN crystals,” Proc. SPIE 3801, 233-239 (1999).
[CrossRef]

Zhang, Y. P.

K. Q. Lu, W. Zhao, Y. L. Yang, Y. Yang, X. M. Liu, Y. P. Zhang, and J. J. Xu, “Soliton-induced waveguides in photorefractive photovoltaic materials,” J. Mod. Opt. 53, 2137-2151 (2006).
[CrossRef]

Zhao, J. H.

X. N. Shen, J. H. Zhao, Z. X. Cheng, S. J. Zhang, and H. C. Chen, “Domain fixing process in Cu, Mn-doped KNSBN crystals,” Proc. SPIE 3801, 233-239 (1999).
[CrossRef]

Zhao, W.

K. Q. Lu, W. Zhao, Y. L. Yang, Y. Yang, X. M. Liu, Y. P. Zhang, and J. J. Xu, “Soliton-induced waveguides in photorefractive photovoltaic materials,” J. Mod. Opt. 53, 2137-2151 (2006).
[CrossRef]

Zharnitsky, V.

V. Zharnitsky and D. Pelinovsky, “Averaging of nonlinearity-managed pulses,” Chaos 15, 037105 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. I. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193-2195 (2004).
[CrossRef]

Chaos (1)

V. Zharnitsky and D. Pelinovsky, “Averaging of nonlinearity-managed pulses,” Chaos 15, 037105 (2005).
[CrossRef]

Electron. Lett. (1)

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electron. Lett. 32, 54-55 (1996).
[CrossRef]

J. Mod. Opt. (1)

K. Q. Lu, W. Zhao, Y. L. Yang, Y. Yang, X. M. Liu, Y. P. Zhang, and J. J. Xu, “Soliton-induced waveguides in photorefractive photovoltaic materials,” J. Mod. Opt. 53, 2137-2151 (2006).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (1)

S. Desyatnikov and Y. S. Kivshar, “Spatial optical solitons and soliton clusters carrying an angular momentum,” J. Opt. B: Quantum Semiclassical Opt. 4, S58-S65 (2002).
[CrossRef]

J. Opt. Soc. Am. B (7)

Opt. Commun. (3)

E. Poutrina and G. P. Agrawal, “Design rules for dispersion-managed soliton systems,” Opt. Commun. 206, 193-200 (2002).
[CrossRef]

M. I. Carvalho, S. R. Singh, and D. N. Christodoulides, “Self-deflection of steady-state bright spatial solitons in biased photorefractive crystals,” Opt. Commun. 120, 311-315 (1995).
[CrossRef]

J. Petter, C. Weilnau, C. Denz, A. Stepken, and F. Kaiser, “Self-bending of photorefractive solitons,” Opt. Commun. 170, 291-297 (1999).
[CrossRef]

Opt. Lett. (3)

Phys. Lett. A (1)

J. Atai and B. A. Malomed, “Spatial solitons in a medium composed of self-focusing and self-defocusing layers,” Phys. Lett. A 298, 140-148 (2002).
[CrossRef]

Phys. Rev. A (2)

C. N. Liu, T. Morishita, and S. Watanabe, “Time-dependent hyperspherical studies for a two-dimensional attractive Bose-Einstein condensate,” Phys. Rev. A 75, 023604 (2007).
[CrossRef]

F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67, 013605 (2003).
[CrossRef]

Phys. Rev. E (5)

W. Królikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, “Self-bending photorefractive solitons,” Phys. Rev. E 54, 5761-5765 (1996).
[CrossRef]

H. Sakaguchi and B. A. Malomed, “Resonant nonlinearity management for nonlinear Schrödinger solitons,” Phys. Rev. E 70, 066613 (2004).
[CrossRef]

X. S. Wang and W. L. She, “Spontaneous one- and two-dimensional optical spatial solitons supported by photoisomerization nonlinearity in a bulk polymer,” Phys. Rev. E 71, 026601 (2005).
[CrossRef]

M. Belić, Ph. Jander, A. Strinić, A. Desyatnikov, and C. Denz, “Self-trapped bidirectional waveguides in a saturable photorefractive medium,” Phys. Rev. E 68, 025601 (2003).
[CrossRef]

M. I. Carvalho, M. Facão, and D. N. Christodoulides, “Self-bending of dark and gray photorefractive solitons,” Phys. Rev. E 76, 016602 (2007).
[CrossRef]

Phys. Rev. Lett. (6)

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

W. L. She, K. K. Lee, and W. K. Lee, “Observation of two-dimensional bright photovoltaic spatial solitons,” Phys. Rev. Lett. 83, 3182-3185 (1999).
[CrossRef]

W. L. She, K. K. Lee, and W. K. Lee, “All optical quasi-steady-state photorefractive spatial solitons,” Phys. Rev. Lett. 85, 2498-2501 (2000).
[CrossRef] [PubMed]

M. Centurion, M. A. Porter, P. G. Kevrekidis, and D. Psaltis, “Nonlinearity management in optics: experiment, theory, and simulation,” Phys. Rev. Lett. 97, 033903 (2006).
[CrossRef] [PubMed]

S. Odoulov, T. Tarabrova, A. Shumelyuk, I. I. Naumova, and T. O. Chaplina, “Photorefractive response of bulk periodically poled LiNbO3:Y:Fe at high and low spatial frequencies,” Phys. Rev. Lett. 84, 3294-3297 (2000).
[CrossRef] [PubMed]

Proc. SPIE (1)

X. N. Shen, J. H. Zhao, Z. X. Cheng, S. J. Zhang, and H. C. Chen, “Domain fixing process in Cu, Mn-doped KNSBN crystals,” Proc. SPIE 3801, 233-239 (1999).
[CrossRef]

Other (1)

A. Ciattoni, E. DelRe, C. Rizza, and A. Marini, “Tailoring wave nonlinearity through spatial composites,” arXiv:0804.3687v1 (2008).

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Figures (9)

Fig. 1
Fig. 1

Intensity FWHM of bright spatial solitons versus ρ for r = 2.5 and β = 123.248 .

Fig. 2
Fig. 2

Schematic diagram of a simple structure with 50% duty cycle and uniform periodicity. The arrows indicate the polarized directions.

Fig. 3
Fig. 3

Propagation of bright solitons in the structure shown in Fig. 2. (a) Intensity profile evolution of a soliton with ρ = 8.9 . (b) Trajectories of the soliton beam center for ρ = 1.5 (dot-dashed curve), ρ = 2.6 (dashed curve), and ρ = 8.9 (solid curve); the alternating unshaded and shaded strips represent the positive and negative domains, respectively.

Fig. 4
Fig. 4

Two fundamental two-component domain inversion structures designed for the self-deflection suppression of the soliton beam. The arrows indicate the polarized directions. (a) Two building blocks, B and B ; each of them is composed of a pair of antiparallel domains. (b), (c) the schematic diagram of a two-component structure constructed by blocks B and B .

Fig. 5
Fig. 5

Propagation of bright solitons in the structure shown in Fig. 4b. (a) Intensity profile evolution of a soliton with ρ = 8.9 . (b) Trajectories of the soliton beam center for ρ = 1.5 (dot-dashed curve), ρ = 2.6 (dashed curve), and ρ = 8.9 (solid curve); the alternating unshaded and shaded strips represent the blocks B and B , respectively.

Fig. 6
Fig. 6

Propagation of bright solitons in the structure shown in Fig. 4c. (a) Intensity profile evolution of a soliton with ρ = 8.9 . (b) Trajectories of the soliton beam center for ρ = 1.5 (dot-dashed curve), ρ = 2.6 (dashed curve), and ρ = 8.9 (solid curve); the alternating shaded and unshaded strips represent the blocks B and B, respectively.

Fig. 7
Fig. 7

Schematic diagram of a relatively complex structure constructed by DM cells C and C . The arrows indicate the polarized directions.

Fig. 8
Fig. 8

Propagation of bright solitons in the structure shown in Fig. 7. (a) Intensity profile evolution of a soliton with ρ = 8.9 . (b) Trajectories of the soliton beam center for ρ = 1.5 (dot-dashed curve), ρ = 2.6 (dashed curve), and ρ = 8.9 (solid curve); the alternating shaded and unshaded strips represent the DM cells C and C , respectively.

Fig. 9
Fig. 9

Trajectory of the soliton beam center with the comparison of numerical and analytical results in the structure shown in Fig. 4b when ρ = 8.9 .

Equations (18)

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

i u ξ = 1 2 2 u ζ 2 β 1 + r u 2 1 + u 2 u γ ln ( 1 + u 2 ) ζ u ,
d 2 y d ζ 2 = 2 ( μ β 1 + r ρ y 2 1 + ρ y 2 ) y .
μ = β r β r 1 ρ ln ( 1 + ρ ) ,
( d y d ζ ) 2 = 2 β ( r 1 ) ρ [ ln ( 1 + ρ y 2 ) y 2 ln ( 1 + ρ ) ] .
u ( ζ , ξ ) = ρ 1 2 y [ ζ S ( ξ ) ] exp ( i φ ( ζ , ξ ) ) ,
d S ( ξ ) d ξ = ω ( ξ ) ,
d σ ( ξ ) d ξ = ω 2 ( ξ ) 2 ,
d ω ( ξ ) d ξ = 4 β γ ( r 1 ) K ( ρ ) ,
K ( ρ ) = + 2 y 2 1 + ρ y 2 [ ln ( 1 + ρ y 2 ) y 2 ln ( 1 + ρ ) ] d ζ + y 2 d ζ .
ω ( ξ ) = 4 β γ ( r 1 ) K ( ρ ) ξ ,
S ( ξ ) = 2 β γ ( r 1 ) K ( ρ ) ξ 2 ,
σ ( ξ ) = 8 [ β γ ( r 1 ) K ( ρ ) ] 2 ξ 3 3 .
a = 2 S ( ξ ) ξ 2 = 4 β γ ( r 1 ) K ( ρ ) .
a n = a f ( ξ ) ,
ν 1 i = S 1 i = 0 ,
ν n i = ν n 1 i + a n 1 l ,
S n i = S n 1 i + ν n 1 i l + 1 2 a n 1 l 2 ,
S n ( ξ ) = S n i + ν n i [ ξ ( n 1 ) l ] + 1 2 a n [ ξ ( n 1 ) l ] 2 ,

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