Abstract

Dark soliton propagation is studied in the presence of higher-order effects, including third-order dispersion, self-steepening, linear/nonlinear gain/loss, and Raman scattering. It is found that for certain values of the parameters a stable evolution can exist for both the soliton and the relative continuous-wave background. Using a newly developed perturbation theory we show that the perturbing effects give rise to a shelf that accompanies the soliton in its propagation. Although, the stable solitons are not affected by the shelf it remains an integral part of the dynamics otherwise not considered so far in studies of higher-order nonlinear Schrödinger models.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  25. D. J. Frantzeskakis, J. Phys. A 43, 213001 (2010).
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    [CrossRef]
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    [CrossRef]

2011 (3)

G. Assanto, T. R. Marchant, A. A. Minzoni, and N. F. Smyth, Phys. Rev. E 84, 066602 (2011).
[CrossRef]

M. J. Ablowitz, S. D. Nixon, T. P. Horikis, and D. J. Frantzeskakis, Proc. R. Soc. A 2133, 2597 (2011).

A. Piccardi, A. Alberucci, N. Tabiryan, and G. Assanto, Opt. Lett. 36, 1356 (2011).
[CrossRef]

2010 (1)

D. J. Frantzeskakis, J. Phys. A 43, 213001 (2010).
[CrossRef]

2009 (2)

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, Phys. Rev. E 79, 037601 (2009).
[CrossRef]

W.-J. Liu, B. Tian, H.-Q. Zhang, T. Xu, and H. Li, Phys. Rev. E 79, 063810 (2009).
[CrossRef]

2007 (1)

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, Phys. Rev. A 75, 033815 (2007).
[CrossRef]

2006 (1)

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, Phys. Rev. E 73, 036617 (2006).
[CrossRef]

2005 (1)

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

2004 (1)

V. M. Lashkin, Phys. Rev. E 70, 066620 (2004).
[CrossRef]

2000 (1)

N. Efremidis, K. Hizanidis, H. E. Nistazakis, D. J. Frantzeskakis, and B. A. Malomed, Phys. Rev. E 62, 7410 (2000).
[CrossRef]

1999 (2)

N. N. Huang, S. Chi, and X. J. Chen, J. Phys. A 32, 3939 (1999).
[CrossRef]

S. L. Palacios, A. Guinea, J. M. Fernández-Díaz, and R. D. Crespo, Phys. Rev. E 60, R45 (1999).
[CrossRef]

1998 (2)

X. J. Chen, Z. D. Chen, and N. N. Huang, J. Phys. A 31, 6929 (1998).
[CrossRef]

Yu. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

1997 (1)

D. Mihalache, N. Truta, and L. C. Crasovan, Phys. Rev. E 56, 1064 (1997).
[CrossRef]

1996 (4)

K. Hizanidis, D. J. Frantzeskakis, and C. Polymilis, J. Phys. A 29, 7687 (1996).
[CrossRef]

K. Porsezian and K. Nakkeeran, Phys. Rev. Lett. 76, 3955 (1996).
[CrossRef]

D. J. Frantzeskakis, J. Phys. A 29, 3631 (1996).
[CrossRef]

A. D. Kim, W. L. Kath, and C. G. Goedde, Opt. Lett. 21, 465 (1996).
[CrossRef]

1995 (3)

1994 (2)

Yu. S. Kivshar and X. Yang, Phys. Rev. E 49, 1657 (1994).
[CrossRef]

V. V. Konotop and V. E. Vekslerchik, Phys. Rev. E 49, 2397 (1994).
[CrossRef]

1993 (1)

I. M. Uzunov and V. S. Gerdjikov, Phys. Rev. A 47, 1582 (1993).
[CrossRef]

1991 (1)

Yu. S. Kivshar and V. V. Afanasjev, Phys. Rev. A 44, R1446 (1991).
[CrossRef]

1990 (2)

M. Florjańczyk and L. Gagnon, Phys. Rev. A 41, 4478 (1990).
[CrossRef]

Yu. S. Kivshar, Phys. Rev. A 42, 1757 (1990).
[CrossRef]

1974 (1)

T. Taniuti, Prog. Theor. Phys. Suppl. 55, 1 (1974).
[CrossRef]

1973 (1)

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 171 (1973).
[CrossRef]

1971 (1)

T. Tsuzuki, J. Low Temp. Phys. 4, 441 (1971).
[CrossRef]

Ablowitz, M. J.

M. J. Ablowitz, S. D. Nixon, T. P. Horikis, and D. J. Frantzeskakis, Proc. R. Soc. A 2133, 2597 (2011).

Afanasjev, V. V.

Yu. S. Kivshar and V. V. Afanasjev, Phys. Rev. A 44, R1446 (1991).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2012).

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

Akozbek, N.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

Alberucci, A.

Assanto, G.

A. Piccardi, A. Alberucci, N. Tabiryan, and G. Assanto, Opt. Lett. 36, 1356 (2011).
[CrossRef]

G. Assanto, T. R. Marchant, A. A. Minzoni, and N. F. Smyth, Phys. Rev. E 84, 066602 (2011).
[CrossRef]

Bloemer, M. J.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

Chen, X. J.

N. N. Huang, S. Chi, and X. J. Chen, J. Phys. A 32, 3939 (1999).
[CrossRef]

X. J. Chen, Z. D. Chen, and N. N. Huang, J. Phys. A 31, 6929 (1998).
[CrossRef]

Chen, Z. D.

X. J. Chen, Z. D. Chen, and N. N. Huang, J. Phys. A 31, 6929 (1998).
[CrossRef]

Chi, S.

N. N. Huang, S. Chi, and X. J. Chen, J. Phys. A 32, 3939 (1999).
[CrossRef]

Crasovan, L. C.

D. Mihalache, N. Truta, and L. C. Crasovan, Phys. Rev. E 56, 1064 (1997).
[CrossRef]

Crespo, R. D.

S. L. Palacios, A. Guinea, J. M. Fernández-Díaz, and R. D. Crespo, Phys. Rev. E 60, R45 (1999).
[CrossRef]

D’Aguanno, G.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

Dai, X.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Efremidis, N.

N. Efremidis, K. Hizanidis, H. E. Nistazakis, D. J. Frantzeskakis, and B. A. Malomed, Phys. Rev. E 62, 7410 (2000).
[CrossRef]

Fan, D.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, Phys. Rev. A 75, 033815 (2007).
[CrossRef]

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, Phys. Rev. E 73, 036617 (2006).
[CrossRef]

Fernández-Díaz, J. M.

S. L. Palacios, A. Guinea, J. M. Fernández-Díaz, and R. D. Crespo, Phys. Rev. E 60, R45 (1999).
[CrossRef]

Florjanczyk, M.

M. Florjańczyk and L. Gagnon, Phys. Rev. A 41, 4478 (1990).
[CrossRef]

Frantzeskakis, D. J.

M. J. Ablowitz, S. D. Nixon, T. P. Horikis, and D. J. Frantzeskakis, Proc. R. Soc. A 2133, 2597 (2011).

D. J. Frantzeskakis, J. Phys. A 43, 213001 (2010).
[CrossRef]

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, Phys. Rev. E 79, 037601 (2009).
[CrossRef]

N. Efremidis, K. Hizanidis, H. E. Nistazakis, D. J. Frantzeskakis, and B. A. Malomed, Phys. Rev. E 62, 7410 (2000).
[CrossRef]

K. Hizanidis, D. J. Frantzeskakis, and C. Polymilis, J. Phys. A 29, 7687 (1996).
[CrossRef]

D. J. Frantzeskakis, J. Phys. A 29, 3631 (1996).
[CrossRef]

Fu, X.

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, Phys. Rev. E 73, 036617 (2006).
[CrossRef]

Gagnon, L.

M. Florjańczyk and L. Gagnon, Phys. Rev. A 41, 4478 (1990).
[CrossRef]

Gerdjikov, V. S.

I. M. Uzunov and V. S. Gerdjikov, Phys. Rev. A 47, 1582 (1993).
[CrossRef]

Goedde, C. G.

Guinea, A.

S. L. Palacios, A. Guinea, J. M. Fernández-Díaz, and R. D. Crespo, Phys. Rev. E 60, R45 (1999).
[CrossRef]

Hasegawa, A.

H. Ikeda, M. Matsumoto, and A. Hasegawa, Opt. Lett. 20, 1113 (1995).
[CrossRef]

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 171 (1973).
[CrossRef]

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

Hizanidis, K.

N. Efremidis, K. Hizanidis, H. E. Nistazakis, D. J. Frantzeskakis, and B. A. Malomed, Phys. Rev. E 62, 7410 (2000).
[CrossRef]

K. Hizanidis, D. J. Frantzeskakis, and C. Polymilis, J. Phys. A 29, 7687 (1996).
[CrossRef]

Horikis, T. P.

M. J. Ablowitz, S. D. Nixon, T. P. Horikis, and D. J. Frantzeskakis, Proc. R. Soc. A 2133, 2597 (2011).

Huang, N. N.

N. N. Huang, S. Chi, and X. J. Chen, J. Phys. A 32, 3939 (1999).
[CrossRef]

X. J. Chen, Z. D. Chen, and N. N. Huang, J. Phys. A 31, 6929 (1998).
[CrossRef]

Ikeda, H.

Kath, W. L.

A. D. Kim, W. L. Kath, and C. G. Goedde, Opt. Lett. 21, 465 (1996).
[CrossRef]

W. L. Kath and N. F. Smyth, Phys. Rev. E 51, 1484 (1995).
[CrossRef]

Kevrekidis, P. G.

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, Phys. Rev. E 79, 037601 (2009).
[CrossRef]

Kim, A. D.

Kivshar, Yu. S.

Yu. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

Yu. S. Kivshar and X. Yang, Phys. Rev. E 49, 1657 (1994).
[CrossRef]

Yu. S. Kivshar and V. V. Afanasjev, Phys. Rev. A 44, R1446 (1991).
[CrossRef]

Yu. S. Kivshar, Phys. Rev. A 42, 1757 (1990).
[CrossRef]

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

Kodama, Y.

A. Maruta and Y. Kodama, Opt. Lett. 20, 1752 (1995).
[CrossRef]

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

Konotop, V. V.

V. V. Konotop and V. E. Vekslerchik, Phys. Rev. E 49, 2397 (1994).
[CrossRef]

Kourakis, I.

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, Phys. Rev. E 79, 037601 (2009).
[CrossRef]

Lashkin, V. M.

V. M. Lashkin, Phys. Rev. E 70, 066620 (2004).
[CrossRef]

Li, H.

W.-J. Liu, B. Tian, H.-Q. Zhang, T. Xu, and H. Li, Phys. Rev. E 79, 063810 (2009).
[CrossRef]

Liu, W.-J.

W.-J. Liu, B. Tian, H.-Q. Zhang, T. Xu, and H. Li, Phys. Rev. E 79, 063810 (2009).
[CrossRef]

Luther-Davies, B.

Yu. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

Malomed, B. A.

N. Efremidis, K. Hizanidis, H. E. Nistazakis, D. J. Frantzeskakis, and B. A. Malomed, Phys. Rev. E 62, 7410 (2000).
[CrossRef]

Marchant, T. R.

G. Assanto, T. R. Marchant, A. A. Minzoni, and N. F. Smyth, Phys. Rev. E 84, 066602 (2011).
[CrossRef]

Maruta, A.

Matsumoto, M.

Mattiucci, N.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

Mihalache, D.

D. Mihalache, N. Truta, and L. C. Crasovan, Phys. Rev. E 56, 1064 (1997).
[CrossRef]

Minzoni, A. A.

G. Assanto, T. R. Marchant, A. A. Minzoni, and N. F. Smyth, Phys. Rev. E 84, 066602 (2011).
[CrossRef]

Nakkeeran, K.

K. Porsezian and K. Nakkeeran, Phys. Rev. Lett. 76, 3955 (1996).
[CrossRef]

Nistazakis, H. E.

N. Efremidis, K. Hizanidis, H. E. Nistazakis, D. J. Frantzeskakis, and B. A. Malomed, Phys. Rev. E 62, 7410 (2000).
[CrossRef]

Nixon, S. D.

M. J. Ablowitz, S. D. Nixon, T. P. Horikis, and D. J. Frantzeskakis, Proc. R. Soc. A 2133, 2597 (2011).

Palacios, S. L.

S. L. Palacios, A. Guinea, J. M. Fernández-Díaz, and R. D. Crespo, Phys. Rev. E 60, R45 (1999).
[CrossRef]

Piccardi, A.

Poliakov, E. Y.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

Polymilis, C.

K. Hizanidis, D. J. Frantzeskakis, and C. Polymilis, J. Phys. A 29, 7687 (1996).
[CrossRef]

Porsezian, K.

K. Porsezian and K. Nakkeeran, Phys. Rev. Lett. 76, 3955 (1996).
[CrossRef]

Rompotis, N.

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, Phys. Rev. E 79, 037601 (2009).
[CrossRef]

Scalora, M.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

Smyth, N. F.

G. Assanto, T. R. Marchant, A. A. Minzoni, and N. F. Smyth, Phys. Rev. E 84, 066602 (2011).
[CrossRef]

W. L. Kath and N. F. Smyth, Phys. Rev. E 51, 1484 (1995).
[CrossRef]

Su, W.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, Phys. Rev. A 75, 033815 (2007).
[CrossRef]

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, Phys. Rev. E 73, 036617 (2006).
[CrossRef]

Syrchin, M. S.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

Tabiryan, N.

Tang, Z.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Taniuti, T.

T. Taniuti, Prog. Theor. Phys. Suppl. 55, 1 (1974).
[CrossRef]

Tappert, F.

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 171 (1973).
[CrossRef]

Tian, B.

W.-J. Liu, B. Tian, H.-Q. Zhang, T. Xu, and H. Li, Phys. Rev. E 79, 063810 (2009).
[CrossRef]

Truta, N.

D. Mihalache, N. Truta, and L. C. Crasovan, Phys. Rev. E 56, 1064 (1997).
[CrossRef]

Tsitsas, N. L.

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, Phys. Rev. E 79, 037601 (2009).
[CrossRef]

Tsuzuki, T.

T. Tsuzuki, J. Low Temp. Phys. 4, 441 (1971).
[CrossRef]

Uzunov, I. M.

I. M. Uzunov and V. S. Gerdjikov, Phys. Rev. A 47, 1582 (1993).
[CrossRef]

Vekslerchik, V. E.

V. V. Konotop and V. E. Vekslerchik, Phys. Rev. E 49, 2397 (1994).
[CrossRef]

Wang, Y.

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, Phys. Rev. E 73, 036617 (2006).
[CrossRef]

Wen, S.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, Phys. Rev. A 75, 033815 (2007).
[CrossRef]

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, Phys. Rev. E 73, 036617 (2006).
[CrossRef]

Xiang, Y.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, Phys. Rev. A 75, 033815 (2007).
[CrossRef]

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, Phys. Rev. E 73, 036617 (2006).
[CrossRef]

Xu, T.

W.-J. Liu, B. Tian, H.-Q. Zhang, T. Xu, and H. Li, Phys. Rev. E 79, 063810 (2009).
[CrossRef]

Yang, X.

Yu. S. Kivshar and X. Yang, Phys. Rev. E 49, 1657 (1994).
[CrossRef]

Zhang, H.-Q.

W.-J. Liu, B. Tian, H.-Q. Zhang, T. Xu, and H. Li, Phys. Rev. E 79, 063810 (2009).
[CrossRef]

Zheltikov, A. M.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 171 (1973).
[CrossRef]

J. Low Temp. Phys. (1)

T. Tsuzuki, J. Low Temp. Phys. 4, 441 (1971).
[CrossRef]

J. Phys. A (5)

D. J. Frantzeskakis, J. Phys. A 43, 213001 (2010).
[CrossRef]

K. Hizanidis, D. J. Frantzeskakis, and C. Polymilis, J. Phys. A 29, 7687 (1996).
[CrossRef]

D. J. Frantzeskakis, J. Phys. A 29, 3631 (1996).
[CrossRef]

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[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

Top panel: contour plot showing the evolution of a unit-amplitude black soliton; analytical predictions are depicted by the solid (white) lines, for the propagation of the shelf’s edge, and the dotted (white) line, for the evolution of the soliton center. Bottom panel: evolution of u(z) and A(z); solid lines (circles/squares) correspond to analytical (numerical) results.

Fig. 2.
Fig. 2.

Panels are similar to those in Fig. 1, but now for the stable gray soliton, characterized by the parameters given in Eq. (14).

Fig. 3.
Fig. 3.

Similar to top panel of Fig. 1, but for the stable black soliton (for σR=0), characterized by the parameters given in Eq. (15), for γ=δ=0.02.

Equations (18)

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iuz12utt+|u|2u=iγu+iδ|u|2u+iβuttt+iμ(|u|2u)t+(iν+σR)u(|u|2)t,
us(z,t)=[A+iBtanh(BT)]eiσ0,
iu0z|u0|2u0=iγu0+iδ|u0|2u0.
u=(γ+δu2)u
2BAz=Re{F[us](us*)tdt},
Bt0z=Im{t(F[u]uF[us]us*)dt},
uσ0z=Im{(F[u]uF[us]us*)dt}+Re{F[u]},
BBz=uuzAAz,
u2Δϕ0z=2ABz2BAz,
q1±=12σ0z±Δϕ0zuA,ϕ1t±=2q1±.
A=415σRA4+23δA3815σRu2A2+(γ+δ3u2)A+415σRu4,
t0=(2βμ2ν3)A2(2β+2μ+4ν3)u2,
σ0=Bzu2B3u(3γ+4u2δ+2δA2).
u2=γδ,A=5δ2+25δ416γδσR24δσR.
u2=γδ,A=23(δA2γ)A,
u=0,A=415σR(A2u2)2.
u2(z)=γu02e2γzγ+δu02δu02e2γz,
z=12γln(1+γδu02).

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