Abstract

A perturbation theory is developed for vector solitons, based on the derivation of the adiabatic evolution of the vector–soliton parameters under a generic perturbation. This perturbation theory was used to describe interactions between vector solitons and the related intersoliton forces. Some specific cases were analyzed that showed the dynamics under the mixture of coherent and incoherent forces characterizing the vector–soliton interactions.

© 2001 Optical Society of America

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References

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  1. R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–481 (1964).
    [CrossRef]
  2. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Appl. Phys. Lett. 45, 1095–1097 (1980).
    [CrossRef]
  3. V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Sov. Phys. JETP 34, 62–84 (1977).
  4. D. J. Kaup and N. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London, Ser. A 361, 413–446 (1978).
    [CrossRef]
  5. H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 386–392 (1990).
    [CrossRef]
  6. S. V. Manakov, “On the theory of two-dimensional stationary self focussing of electromagnetic waves,” JETP 38, 246–252 (1974).
  7. M. R. Gupta, B. K. Som, and B. Dasgupta, “Coupled nonlinear Schrodinger equations for Langmuir and electromagnetic waves and extension of their modulational instability domain,” J. Plasma Phys. 25, 499–507 (1981).
    [CrossRef]
  8. A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–494 (1970).
  9. C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
    [CrossRef]
  10. S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
    [CrossRef]
  11. M. Midrio, S. Wabnitz, and P. Franco, “Perturbation theory for coupled nonlinear Schrodinger equations,” Phys. Rev. E 54, 5743–5751 (1996).
    [CrossRef]
  12. D. J. Muraki and W. L. Kath, “Polarization dynamics for solitons in birefringent optical fibers,” Phys. Lett. A 139, 379–381 (1989).
    [CrossRef]
  13. F. S. Locati, M. Romagnoli, A. Tajani, and S. Wabnitz, “Nonlinear nonreciprocity of soliton amplification with erbium-doped fibers,” Opt. Lett. 17, 1213–1215 (1992).
    [CrossRef] [PubMed]
  14. A. B. Aceves and S. Wabnitz, “Switching dynamics of helical solitons in a periodically twisted birefringent fiber filter,” Opt. Lett. 17, 25–27 (1992).
    [CrossRef] [PubMed]
  15. E. Caglioti, S. Trillo, S. Wabnitz, B. Crosignani, and P. Di-Porto, “Finite-dimensional description of nonlinear pulse propagation in optical-fiber couplers with applications to soliton switching,” J. Opt. Soc. Am. B 7, 374–385 (1990).
    [CrossRef]
  16. J. Yang, “Multisoliton perturbation theory for the Manakov equations and its applications in nonlinear optics,” Phys. Rev. E 59, 2393–2405 (1999).
    [CrossRef]
  17. B. A. Malomed and S. Wabnitz, “Soliton annihilation and fusion from resonant inelastic collisions in birefringent optical fibers,” Opt. Lett. 16, 1388–1390 (1991).
    [CrossRef] [PubMed]
  18. T. Ueda and W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
    [CrossRef] [PubMed]
  19. B. A. Malomed, “Polarization dynamics and interactions of solitons in a birefringent optical fiber,” Phys. Rev. A 43, 410–423 (1991).
    [CrossRef] [PubMed]
  20. C. Sophocleous and D. F. Parker, “Pulse collisions and polarisation conversion for optical fibers,” Opt. Commun. 112, 214–224 (1994).
    [CrossRef]
  21. E. A. Ostrovskaya, Y. S. Kivshar, Z. Chen, and M. Segev, “Interaction between vector solitons and solitonic gluons,” Opt. Lett. 24, 327–329 (1999).
    [CrossRef]
  22. D. N. Christodoulides, S. R. Singh, M. I. Carvalho, and M. Segev, “Incoherently coupled soliton pairs in biased photorefractive crystals,” Appl. Phys. Lett. 68, 1763–1765 (1996).
    [CrossRef]
  23. Z. Chen, M. Segev, T. H. Coskun, D. N. Christodoulides, Y. S. Kivshar, and V. V. Afanasjev, “Coupled photorefractive spatial-soliton pairs,” J. Opt. Soc. Am. B 14, 3066–3077 (1997).
    [CrossRef]
  24. D. De Angelis and S. Wabnitz, “Interactions of orthogonally polarized solitons in optical fibers,” Opt. Commun. 125, 186–196 (1996).
    [CrossRef]
  25. B. A. Malomed and R. S. Tasgal, “Internal vibrations of a vector soliton in the coupled nonlinear Schrodinger equations,” Phys. Rev. E 58, 2564–2575 (1998).
    [CrossRef]
  26. J. P. Gordon, “Interaction forces among solitons in optical fibers,” Opt. Lett. 8, 596–598 (1983).
    [CrossRef] [PubMed]
  27. J. Scheuer and M. Orenstein, “Interactions and switching of coherent,” Opt. Lett. 24, 1735–1737 (1999).
    [CrossRef]
  28. M. Karlsson, D. Anderson, A. Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scr. 50, 265–270 (1994).
    [CrossRef]

1999 (3)

1998 (1)

B. A. Malomed and R. S. Tasgal, “Internal vibrations of a vector soliton in the coupled nonlinear Schrodinger equations,” Phys. Rev. E 58, 2564–2575 (1998).
[CrossRef]

1997 (1)

1996 (3)

D. De Angelis and S. Wabnitz, “Interactions of orthogonally polarized solitons in optical fibers,” Opt. Commun. 125, 186–196 (1996).
[CrossRef]

D. N. Christodoulides, S. R. Singh, M. I. Carvalho, and M. Segev, “Incoherently coupled soliton pairs in biased photorefractive crystals,” Appl. Phys. Lett. 68, 1763–1765 (1996).
[CrossRef]

M. Midrio, S. Wabnitz, and P. Franco, “Perturbation theory for coupled nonlinear Schrodinger equations,” Phys. Rev. E 54, 5743–5751 (1996).
[CrossRef]

1994 (2)

M. Karlsson, D. Anderson, A. Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scr. 50, 265–270 (1994).
[CrossRef]

C. Sophocleous and D. F. Parker, “Pulse collisions and polarisation conversion for optical fibers,” Opt. Commun. 112, 214–224 (1994).
[CrossRef]

1992 (3)

1991 (2)

1990 (3)

1989 (1)

D. J. Muraki and W. L. Kath, “Polarization dynamics for solitons in birefringent optical fibers,” Phys. Lett. A 139, 379–381 (1989).
[CrossRef]

1987 (1)

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
[CrossRef]

1983 (1)

1981 (1)

M. R. Gupta, B. K. Som, and B. Dasgupta, “Coupled nonlinear Schrodinger equations for Langmuir and electromagnetic waves and extension of their modulational instability domain,” J. Plasma Phys. 25, 499–507 (1981).
[CrossRef]

1980 (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Appl. Phys. Lett. 45, 1095–1097 (1980).
[CrossRef]

1978 (1)

D. J. Kaup and N. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London, Ser. A 361, 413–446 (1978).
[CrossRef]

1977 (1)

V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Sov. Phys. JETP 34, 62–84 (1977).

1974 (1)

S. V. Manakov, “On the theory of two-dimensional stationary self focussing of electromagnetic waves,” JETP 38, 246–252 (1974).

1970 (1)

A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–494 (1970).

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–481 (1964).
[CrossRef]

Aceves, A. B.

Afanasjev, V. V.

Anderson, D.

M. Karlsson, D. Anderson, A. Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scr. 50, 265–270 (1994).
[CrossRef]

Bergano, N. S.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

Berkhoer, A. L.

A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–494 (1970).

Caglioti, E.

Carvalho, M. I.

D. N. Christodoulides, S. R. Singh, M. I. Carvalho, and M. Segev, “Incoherently coupled soliton pairs in biased photorefractive crystals,” Appl. Phys. Lett. 68, 1763–1765 (1996).
[CrossRef]

Chen, Z.

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–481 (1964).
[CrossRef]

Christodoulides, D. N.

Z. Chen, M. Segev, T. H. Coskun, D. N. Christodoulides, Y. S. Kivshar, and V. V. Afanasjev, “Coupled photorefractive spatial-soliton pairs,” J. Opt. Soc. Am. B 14, 3066–3077 (1997).
[CrossRef]

D. N. Christodoulides, S. R. Singh, M. I. Carvalho, and M. Segev, “Incoherently coupled soliton pairs in biased photorefractive crystals,” Appl. Phys. Lett. 68, 1763–1765 (1996).
[CrossRef]

Coskun, T. H.

Crosignani, B.

Dasgupta, B.

M. R. Gupta, B. K. Som, and B. Dasgupta, “Coupled nonlinear Schrodinger equations for Langmuir and electromagnetic waves and extension of their modulational instability domain,” J. Plasma Phys. 25, 499–507 (1981).
[CrossRef]

De Angelis, D.

D. De Angelis and S. Wabnitz, “Interactions of orthogonally polarized solitons in optical fibers,” Opt. Commun. 125, 186–196 (1996).
[CrossRef]

Di-Porto, P.

Evangelides, S. G.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

Franco, P.

M. Midrio, S. Wabnitz, and P. Franco, “Perturbation theory for coupled nonlinear Schrodinger equations,” Phys. Rev. E 54, 5743–5751 (1996).
[CrossRef]

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–481 (1964).
[CrossRef]

Gordon, J. P.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

J. P. Gordon, “Interaction forces among solitons in optical fibers,” Opt. Lett. 8, 596–598 (1983).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Appl. Phys. Lett. 45, 1095–1097 (1980).
[CrossRef]

Gupta, M. R.

M. R. Gupta, B. K. Som, and B. Dasgupta, “Coupled nonlinear Schrodinger equations for Langmuir and electromagnetic waves and extension of their modulational instability domain,” J. Plasma Phys. 25, 499–507 (1981).
[CrossRef]

Haus, H. A.

Höök, A.

M. Karlsson, D. Anderson, A. Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scr. 50, 265–270 (1994).
[CrossRef]

Karlsson, M.

M. Karlsson, D. Anderson, A. Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scr. 50, 265–270 (1994).
[CrossRef]

Karpman, V. I.

V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Sov. Phys. JETP 34, 62–84 (1977).

Kath, W. L.

T. Ueda and W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
[CrossRef] [PubMed]

D. J. Muraki and W. L. Kath, “Polarization dynamics for solitons in birefringent optical fibers,” Phys. Lett. A 139, 379–381 (1989).
[CrossRef]

Kaup, D. J.

D. J. Kaup and N. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London, Ser. A 361, 413–446 (1978).
[CrossRef]

Kivshar, Y. S.

Lai, Y.

Lisak, M.

M. Karlsson, D. Anderson, A. Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scr. 50, 265–270 (1994).
[CrossRef]

Locati, F. S.

Malomed, B. A.

B. A. Malomed and R. S. Tasgal, “Internal vibrations of a vector soliton in the coupled nonlinear Schrodinger equations,” Phys. Rev. E 58, 2564–2575 (1998).
[CrossRef]

B. A. Malomed and S. Wabnitz, “Soliton annihilation and fusion from resonant inelastic collisions in birefringent optical fibers,” Opt. Lett. 16, 1388–1390 (1991).
[CrossRef] [PubMed]

B. A. Malomed, “Polarization dynamics and interactions of solitons in a birefringent optical fiber,” Phys. Rev. A 43, 410–423 (1991).
[CrossRef] [PubMed]

Manakov, S. V.

S. V. Manakov, “On the theory of two-dimensional stationary self focussing of electromagnetic waves,” JETP 38, 246–252 (1974).

Maslov, E. M.

V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Sov. Phys. JETP 34, 62–84 (1977).

Menyuk, C. R.

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
[CrossRef]

Midrio, M.

M. Midrio, S. Wabnitz, and P. Franco, “Perturbation theory for coupled nonlinear Schrodinger equations,” Phys. Rev. E 54, 5743–5751 (1996).
[CrossRef]

Mollenauer, L. F.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Appl. Phys. Lett. 45, 1095–1097 (1980).
[CrossRef]

Muraki, D. J.

D. J. Muraki and W. L. Kath, “Polarization dynamics for solitons in birefringent optical fibers,” Phys. Lett. A 139, 379–381 (1989).
[CrossRef]

Newell, N. C.

D. J. Kaup and N. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London, Ser. A 361, 413–446 (1978).
[CrossRef]

Orenstein, M.

Ostrovskaya, E. A.

Parker, D. F.

C. Sophocleous and D. F. Parker, “Pulse collisions and polarisation conversion for optical fibers,” Opt. Commun. 112, 214–224 (1994).
[CrossRef]

Romagnoli, M.

Scheuer, J.

Segev, M.

Singh, S. R.

D. N. Christodoulides, S. R. Singh, M. I. Carvalho, and M. Segev, “Incoherently coupled soliton pairs in biased photorefractive crystals,” Appl. Phys. Lett. 68, 1763–1765 (1996).
[CrossRef]

Som, B. K.

M. R. Gupta, B. K. Som, and B. Dasgupta, “Coupled nonlinear Schrodinger equations for Langmuir and electromagnetic waves and extension of their modulational instability domain,” J. Plasma Phys. 25, 499–507 (1981).
[CrossRef]

Sophocleous, C.

C. Sophocleous and D. F. Parker, “Pulse collisions and polarisation conversion for optical fibers,” Opt. Commun. 112, 214–224 (1994).
[CrossRef]

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Appl. Phys. Lett. 45, 1095–1097 (1980).
[CrossRef]

Tajani, A.

Tasgal, R. S.

B. A. Malomed and R. S. Tasgal, “Internal vibrations of a vector soliton in the coupled nonlinear Schrodinger equations,” Phys. Rev. E 58, 2564–2575 (1998).
[CrossRef]

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–481 (1964).
[CrossRef]

Trillo, S.

Ueda, T.

T. Ueda and W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
[CrossRef] [PubMed]

Wabnitz, S.

Yang, J.

J. Yang, “Multisoliton perturbation theory for the Manakov equations and its applications in nonlinear optics,” Phys. Rev. E 59, 2393–2405 (1999).
[CrossRef]

Zakharov, V. E.

A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–494 (1970).

Appl. Phys. Lett. (2)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Appl. Phys. Lett. 45, 1095–1097 (1980).
[CrossRef]

D. N. Christodoulides, S. R. Singh, M. I. Carvalho, and M. Segev, “Incoherently coupled soliton pairs in biased photorefractive crystals,” Appl. Phys. Lett. 68, 1763–1765 (1996).
[CrossRef]

IEEE J. Quantum Electron. (1)

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
[CrossRef]

J. Lightwave Technol. (1)

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

J. Opt. Soc. Am. B (3)

J. Plasma Phys. (1)

M. R. Gupta, B. K. Som, and B. Dasgupta, “Coupled nonlinear Schrodinger equations for Langmuir and electromagnetic waves and extension of their modulational instability domain,” J. Plasma Phys. 25, 499–507 (1981).
[CrossRef]

JETP (1)

S. V. Manakov, “On the theory of two-dimensional stationary self focussing of electromagnetic waves,” JETP 38, 246–252 (1974).

Opt. Commun. (2)

C. Sophocleous and D. F. Parker, “Pulse collisions and polarisation conversion for optical fibers,” Opt. Commun. 112, 214–224 (1994).
[CrossRef]

D. De Angelis and S. Wabnitz, “Interactions of orthogonally polarized solitons in optical fibers,” Opt. Commun. 125, 186–196 (1996).
[CrossRef]

Opt. Lett. (6)

Phys. Lett. A (1)

D. J. Muraki and W. L. Kath, “Polarization dynamics for solitons in birefringent optical fibers,” Phys. Lett. A 139, 379–381 (1989).
[CrossRef]

Phys. Rev. A (2)

T. Ueda and W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
[CrossRef] [PubMed]

B. A. Malomed, “Polarization dynamics and interactions of solitons in a birefringent optical fiber,” Phys. Rev. A 43, 410–423 (1991).
[CrossRef] [PubMed]

Phys. Rev. E (3)

B. A. Malomed and R. S. Tasgal, “Internal vibrations of a vector soliton in the coupled nonlinear Schrodinger equations,” Phys. Rev. E 58, 2564–2575 (1998).
[CrossRef]

M. Midrio, S. Wabnitz, and P. Franco, “Perturbation theory for coupled nonlinear Schrodinger equations,” Phys. Rev. E 54, 5743–5751 (1996).
[CrossRef]

J. Yang, “Multisoliton perturbation theory for the Manakov equations and its applications in nonlinear optics,” Phys. Rev. E 59, 2393–2405 (1999).
[CrossRef]

Phys. Rev. Lett. (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–481 (1964).
[CrossRef]

Phys. Scr. (1)

M. Karlsson, D. Anderson, A. Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scr. 50, 265–270 (1994).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

D. J. Kaup and N. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London, Ser. A 361, 413–446 (1978).
[CrossRef]

Sov. Phys. JETP (2)

V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Sov. Phys. JETP 34, 62–84 (1977).

A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–494 (1970).

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Figures (3)

Fig. 1
Fig. 1

Force acting between two solitons as a function of their separation: coherent pair (solid curve), semicoherent pair (dashed–dotted curve), and coherent pair (dashed curve).

Fig. 2
Fig. 2

Trajectories of two vector solitons with φ=36.87°, φ1=-45°, and δθA=δθB=0. The initial distance between the vector solitons is 10 NU. (a) BPM simulation; (b) comparison between the BPM (squares, circles) and the perturbation theory (solid, dotted curve) results.

Fig. 3
Fig. 3

Comparison between the BPM (squares, circles) and the perturbation theory (solid, dotted curves) calculated trajectories for interacting vector solitons with φ=36.87 °, φ1=45°, δθA=π, and δθB=0. The initial distance between the vector solitons is 10 NU.

Equations (45)

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

i Az+12 2Ax2+(|A|2+|B|2)A=0,
i Bz+12 2Bx2+(|A|2+|B|2)B=0.
i zΨ+12 2x2Ψ+(ΨΨ)Ψ=0,
Ψ0=A0cos(φ) exp(iθA)sin(φ) exp(iθB)×sech{[A0(x-x0-kxz)]}×expikxx+A02-kx22z.
i zΨ+12 2x2Ψ+(ΨΨ)Ψ=P,
Ψ=Ψ0+ΔΨ exp(iA02z/2).
ΔΨ=mΔm(z)Vm(x);m=A0,φ,x˜,kx,θA,θB,
ReUi+Vjdx=δi,ji, j=A0,φ, x˜, kx,θA,θB,
zΔΨ=A0(VθA+VθB)ΔA0+Vx˜Δkx-iP.
Ψ1(0, x)=A1B1=A0cos(φ1)exp(iθA1)sin(φ1)exp(iθB1)×sech[A0(x-δx˜)].
i Az+12 2Ax2+(|A+A1|2+|B+B1|2)(A+A1)=0,
i Bz+12 2Bx2+(|A+A1|2+|B+B1|2)(B+B1)=0.
P=PAPB
=-(2|A|2+|B|2)A1+A2A1*+ABB1*+AB*B1+(2|A1|2+|B1|2)A+A*A12+A1BB1*+A1B*B1(2|B|2+|A|2)B1+B2B1*+BAA1*+BA*A1+(2|B1|2+|A1|2)B+B*B12+AA1*B1+B1A*A1.
2Δx˜z2=Δkxz=A0 Re PA cos(φ)exp(-iθA)+PB sin(φ)exp(-iθB)tanh(A0x)sech(A0x)dx,
2Δθmz2=A0 ΔA0z=-Re iA02[PA cos(φ)exp(-iθA)+PB sin(φ)exp(-iθB)]sech(A0x)dx,
2Δx˜z2=-{4A03[cos(φ)cos(φ1)cos(δθA)+sin(φ)sin(φ1)cos(δθB)]exp(-A0δx˜)+16A04[1+sin(2φ)sin(2φ1)cos(δθA)cos(δθB)+2 cos2(φ)cos2(φ1)cos2(δθA)+2 sin2(φ)sin2(φ1)cos2(δθB)]δx˜ exp(-2A0δx˜)},
2Δθmz2=4A04[cos(φ)cos(φ1)sin(δθA)+sin(φ)sin(φ1)sin(δθB)]exp(-A0δx˜)-16A05[cos2(φ)cos2(φ1)sin(2δθA)+sin2(φ)sin2(φ1)sin(2δθB)+sin(2φ)sin(2φ1)sin(δθA+δθB)/2]δx˜ exp(-2A0δx˜),
2Δx˜z2=-[4A03 cos(δθA)exp(-A0δx˜)+16A05 cos(2δθA)δx˜ exp(-2A0δx˜)],
2Δθmz2=4A04 sin(δθA)exp(-A0δx˜)-16A05 sin(2δθA)δx˜ exp(-2A0δx˜).
2Δx˜z2=-16A04 δx˜ exp(-2A0δx˜),2Δθmz2=0.
Az+12 2Ax2+(|A+A1|2+|B|2
+|B1|2)(A+A1)=0,
i Bz+12 2Bx2+(|A+A1|2+|B|2+|B1|2)B=0,
R=RARB=-(2|A|2+|B|2)A1+(2|A1|2+|B1|2)A+A2A1*+A*A12(|B1|2+|A1|2+AA1*+A*A1)B.
2Δx˜z2=-{4A03 cos(φ)cos(φ1)cos(δθA)exp(-A0δx˜)+16A04[1+2 cos2(φ)cos2(φ1)cos2(δθA)]×δx˜ exp(-2A0δx˜)},
2Δθmz2=4A04 cos(φ)cos(φ1)sin(δθA)exp(-A0δx˜)-16A05 cos2(φ)cos2(φ1)sin(2δθA)×δx exp(-2A0δx˜).
VA0=cos(φ) exp (iθA)sin(φ) exp (iθB) [1-A0x tanh(A0x)] sech(A0x),
Vx˜=cos(φ) exp(iθA)sin(φ) exp(iθB)A02x tanh(A0x)sech(A0x),
Vkx=cos(φ) exp(iθA)sin(φ) exp(iθB)iA0x sech(A0x),
Vφ=-sin(φ) exp(iθA)cos(φ) exp(iθB)A0 sech(A0x),
VθA=cos(φ) exp(iθA)0iA0x sech(A0x),
VθB=0sin(φ) exp(iθB)iA0x sech(A0x),
UA0=cos(φ) exp(iθA)sin(φ) exp(iθB)A0 sech(A0x),
Ux˜=cos(φ) exp(iθA)sin(φ) exp(iθB)x sech(A0x),
Ukx=cos(φ) exp(iθA)sin(φ) exp(iθB)iA0 tanh(A0x)sech(A0x),
Uφ=-sin(φ) exp(iθA)cos(φ) exp(iθB)A0 sech(A0x),
UθA=1cos (φ) exp(iθA)0i[1-A0x tanh(A0x)]×sech(A0x),
UθB=01sin (φ) exp(iθB)i[1-A0x tanh(A0x)]×sech(A0x).
ΔA0z=-ReiA0[PA cos(φ)exp(-iθA)+PA sin(φ)×exp(-iθB)]sech(A0x)dx,
Δkxz=ReA0[PA cos(φ) exp(-iθA)+PB sin(φ)×exp(-iθB)]tanh(A0x)sech(A0x)dx,
Δx˜z=Δkx-Rei[PA cos(φ) exp(-iθA)+PB sin(φ)×exp(-iθB)]x sech(A0x)dx,
ΔθAz=A0 ΔA0+ReiA0 PA[1-A0x tanh(A0x)]cos(φ) exp(iθA)×sech(A0x)dx,
ΔθBz=A0ΔA0+ReiA0 PB[1-A0x tanh(A0x)]sin(φ) exp(iθB)×sech(A0x)dx ,
Δφz=-Rei[-PA sin(φ) exp(-iθA)+PB cos(φ)×exp(-iθB)]A0 sech(A0x)dx.

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