Abstract

We investigate analytically and numerically the interactions of dark solitons under competing nonlocal cubic and local quintic nonlinearities. It is shown that the self-defocusing quintic nonlinearity will strengthen the attractive interaction and decrease the relative distance between solitons, whereas the self-focusing quintic nonlinearity will enhance the repulsive interaction and increase soliton separation. We demonstrate these results by approximate variational approach and direct numerical simulation.

© 2014 Optical Society of America

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  28. F. Maucher, S. Skupin, M. Shen, and W. Krolikowski, Phys. Rev. A 81, 063617 (2010).
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2013 (3)

Q. Kong, M. Shen, Z. Chen, Q. Wang, R.-K. Lee, and W. Krolikowski, Phys. Rev. A 87, 063832 (2013).
[CrossRef]

W. Chen, Q. Kong, M. Shen, Q. Wang, and J. Shi, Phys. Rev. A 87, 013809 (2013).
[CrossRef]

L. Chen, Q. Wang, M. Shen, H. Zhao, Y.-Y. Lin, C.-C. Jeng, R.-K. Lee, and W. Krolikowski, Opt. Lett. 38, 13 (2013).
[CrossRef]

2012 (1)

B. K. Esbensen, M. Bache, O. Bang, and W. Krolikowski, Phys. Rev. A 86, 033838 (2012).
[CrossRef]

2011 (5)

Y. Du, Z. Zhou, H. Tian, and D. Liu, J. Opt. 13, 015201 (2011).
[CrossRef]

B. K. Esbensen, A. Wlotzka, M. Bache, O. Bang, and W. Krolikowski, Phys. Rev. A 84, 053854 (2011).
[CrossRef]

K.-H. Kuo, Y. Y. Lin, R.-K. Lee, and B. A. Malomed, Phys. Rev. A 83, 053838 (2011).
[CrossRef]

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

Z. Zhou, Y. Du, C. Hou, H. Tian, and Y. Wang, J. Opt. Soc. Am. B 28, 1583 (2011).
[CrossRef]

2010 (3)

E. N. Tsoy, Phys. Rev. A 82, 063829 (2010).
[CrossRef]

F. Maucher, S. Skupin, M. Shen, and W. Krolikowski, Phys. Rev. A 81, 063617 (2010).
[CrossRef]

Q. Kong, Q. Wang, O. Bang, and W. Krolikowski, Phys. Rev. A 82, 013826 (2010).
[CrossRef]

2009 (3)

I. B. Burgess, M. Peccianti, G. Assanto, and R. Morandotti, Phys. Rev. Lett. 102, 203903 (2009).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 79, 013803 (2009).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 34, 1543 (2009).
[CrossRef]

2008 (2)

M. Warenghem, J. F. Blach, and J. F. Henninot, J. Opt. Soc. Am. B 25, 1882 (2008).
[CrossRef]

I. Tikhonenkov, B. A. Malomed, and A. Vardi, Phys. Rev. Lett. 100, 090406 (2008).
[CrossRef]

2006 (5)

A. Griesmaier, J. Stuhler, T. Koch, M. Fattori, T. Pfau, and S. Giovanazzi, Phys. Rev. Lett. 97, 250402 (2006).
[CrossRef]

W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, Appl. Phys. Lett. 89, 071111 (2006).
[CrossRef]

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef]

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, Nat. Phys. 2, 769 (2006).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, L. C. Crasovan, Y. V. Kartashov, L. Torner, and B. A. Malomed, Phys. Rev. E 74, 066614 (2006).
[CrossRef]

2005 (1)

P. D. Rasmussen, O. Bang, and W. Krolikowski, Phys. Rev. E 72, 066611 (2005).
[CrossRef]

2004 (2)

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B Quantum Semiclass. Opt. 6, S288 (2004).
[CrossRef]

N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, Opt. Lett. 29, 286 (2004).
[CrossRef]

2002 (1)

1999 (1)

G. I. Stegeman and M. Segev, Science 286, 1518 (1999).
[CrossRef]

1998 (1)

W. Krolikowski, M. Saffman, B. Luther-Davies, and C. Denz, Phys. Rev. Lett. 80, 3240 (1998).
[CrossRef]

1995 (1)

Yu. S. Kivshar and W. Krolikowski, Opt. Commun. 114, 353 (1995).
[CrossRef]

1991 (2)

1989 (1)

1985 (1)

K. J. Blow and N. Doran, Phys. Lett. A 107, 55 (1985).
[CrossRef]

1983 (1)

Agrawal, G. P.

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

Aitchison, J. S.

Alberucci, A.

Alfassi, B.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, Nat. Phys. 2, 769 (2006).
[CrossRef]

Assanto, G.

Bache, M.

B. K. Esbensen, M. Bache, O. Bang, and W. Krolikowski, Phys. Rev. A 86, 033838 (2012).
[CrossRef]

B. K. Esbensen, A. Wlotzka, M. Bache, O. Bang, and W. Krolikowski, Phys. Rev. A 84, 053854 (2011).
[CrossRef]

Bang, O.

B. K. Esbensen, M. Bache, O. Bang, and W. Krolikowski, Phys. Rev. A 86, 033838 (2012).
[CrossRef]

B. K. Esbensen, A. Wlotzka, M. Bache, O. Bang, and W. Krolikowski, Phys. Rev. A 84, 053854 (2011).
[CrossRef]

Q. Kong, Q. Wang, O. Bang, and W. Krolikowski, Phys. Rev. A 82, 013826 (2010).
[CrossRef]

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef]

P. D. Rasmussen, O. Bang, and W. Krolikowski, Phys. Rev. E 72, 066611 (2005).
[CrossRef]

N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, Opt. Lett. 29, 286 (2004).
[CrossRef]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B Quantum Semiclass. Opt. 6, S288 (2004).
[CrossRef]

Barthelemy, A.

Blach, J. F.

Blow, K. J.

K. J. Blow and N. Doran, Phys. Lett. A 107, 55 (1985).
[CrossRef]

Bourkoff, E.

Brzdakiewicz, K. A.

Burgess, I. B.

I. B. Burgess, M. Peccianti, G. Assanto, and R. Morandotti, Phys. Rev. Lett. 102, 203903 (2009).
[CrossRef]

Chen, L.

Chen, W.

W. Chen, Q. Kong, M. Shen, Q. Wang, and J. Shi, Phys. Rev. A 87, 013809 (2013).
[CrossRef]

Chen, Z.

Q. Kong, M. Shen, Z. Chen, Q. Wang, R.-K. Lee, and W. Krolikowski, Phys. Rev. A 87, 063832 (2013).
[CrossRef]

Christiansen, P. L.

Cohen, O.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, Nat. Phys. 2, 769 (2006).
[CrossRef]

Crasovan, L. C.

D. Mihalache, D. Mazilu, F. Lederer, L. C. Crasovan, Y. V. Kartashov, L. Torner, and B. A. Malomed, Phys. Rev. E 74, 066614 (2006).
[CrossRef]

Denz, C.

W. Krolikowski, M. Saffman, B. Luther-Davies, and C. Denz, Phys. Rev. Lett. 80, 3240 (1998).
[CrossRef]

Doran, N.

K. J. Blow and N. Doran, Phys. Lett. A 107, 55 (1985).
[CrossRef]

Dreischuh, A.

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef]

Du, Y.

Edmundson, D.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B Quantum Semiclass. Opt. 6, S288 (2004).
[CrossRef]

Esbensen, B. K.

B. K. Esbensen, M. Bache, O. Bang, and W. Krolikowski, Phys. Rev. A 86, 033838 (2012).
[CrossRef]

B. K. Esbensen, A. Wlotzka, M. Bache, O. Bang, and W. Krolikowski, Phys. Rev. A 84, 053854 (2011).
[CrossRef]

Fattori, M.

A. Griesmaier, J. Stuhler, T. Koch, M. Fattori, T. Pfau, and S. Giovanazzi, Phys. Rev. Lett. 97, 250402 (2006).
[CrossRef]

Giovanazzi, S.

A. Griesmaier, J. Stuhler, T. Koch, M. Fattori, T. Pfau, and S. Giovanazzi, Phys. Rev. Lett. 97, 250402 (2006).
[CrossRef]

Gordon, J. P.

Griesmaier, A.

A. Griesmaier, J. Stuhler, T. Koch, M. Fattori, T. Pfau, and S. Giovanazzi, Phys. Rev. Lett. 97, 250402 (2006).
[CrossRef]

Guo, Q.

W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, Appl. Phys. Lett. 89, 071111 (2006).
[CrossRef]

Henninot, J. F.

Hou, C.

Hu, W.

W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, Appl. Phys. Lett. 89, 071111 (2006).
[CrossRef]

Jackel, J. L.

Jeng, C.-C.

Kartashov, Y. V.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 79, 013803 (2009).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 34, 1543 (2009).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, L. C. Crasovan, Y. V. Kartashov, L. Torner, and B. A. Malomed, Phys. Rev. E 74, 066614 (2006).
[CrossRef]

Kivshar, Y. S.

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

Kivshar, Yu. S.

Yu. S. Kivshar and W. Krolikowski, Opt. Commun. 114, 353 (1995).
[CrossRef]

Koch, T.

A. Griesmaier, J. Stuhler, T. Koch, M. Fattori, T. Pfau, and S. Giovanazzi, Phys. Rev. Lett. 97, 250402 (2006).
[CrossRef]

Kong, Q.

Q. Kong, M. Shen, Z. Chen, Q. Wang, R.-K. Lee, and W. Krolikowski, Phys. Rev. A 87, 063832 (2013).
[CrossRef]

W. Chen, Q. Kong, M. Shen, Q. Wang, and J. Shi, Phys. Rev. A 87, 013809 (2013).
[CrossRef]

Q. Kong, Q. Wang, O. Bang, and W. Krolikowski, Phys. Rev. A 82, 013826 (2010).
[CrossRef]

Krolikowski, W.

L. Chen, Q. Wang, M. Shen, H. Zhao, Y.-Y. Lin, C.-C. Jeng, R.-K. Lee, and W. Krolikowski, Opt. Lett. 38, 13 (2013).
[CrossRef]

Q. Kong, M. Shen, Z. Chen, Q. Wang, R.-K. Lee, and W. Krolikowski, Phys. Rev. A 87, 063832 (2013).
[CrossRef]

B. K. Esbensen, M. Bache, O. Bang, and W. Krolikowski, Phys. Rev. A 86, 033838 (2012).
[CrossRef]

B. K. Esbensen, A. Wlotzka, M. Bache, O. Bang, and W. Krolikowski, Phys. Rev. A 84, 053854 (2011).
[CrossRef]

Q. Kong, Q. Wang, O. Bang, and W. Krolikowski, Phys. Rev. A 82, 013826 (2010).
[CrossRef]

F. Maucher, S. Skupin, M. Shen, and W. Krolikowski, Phys. Rev. A 81, 063617 (2010).
[CrossRef]

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef]

P. D. Rasmussen, O. Bang, and W. Krolikowski, Phys. Rev. E 72, 066611 (2005).
[CrossRef]

N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, Opt. Lett. 29, 286 (2004).
[CrossRef]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B Quantum Semiclass. Opt. 6, S288 (2004).
[CrossRef]

W. Krolikowski, M. Saffman, B. Luther-Davies, and C. Denz, Phys. Rev. Lett. 80, 3240 (1998).
[CrossRef]

Yu. S. Kivshar and W. Krolikowski, Opt. Commun. 114, 353 (1995).
[CrossRef]

Kuo, K.-H.

K.-H. Kuo, Y. Y. Lin, R.-K. Lee, and B. A. Malomed, Phys. Rev. A 83, 053838 (2011).
[CrossRef]

Lan, S.

W. Hu, T. Zhang, Q. Guo, L. Xuan, and S. Lan, Appl. Phys. Lett. 89, 071111 (2006).
[CrossRef]

Leaird, D. E.

Lederer, F.

D. Mihalache, D. Mazilu, F. Lederer, L. C. Crasovan, Y. V. Kartashov, L. Torner, and B. A. Malomed, Phys. Rev. E 74, 066614 (2006).
[CrossRef]

Lee, R.-K.

Q. Kong, M. Shen, Z. Chen, Q. Wang, R.-K. Lee, and W. Krolikowski, Phys. Rev. A 87, 063832 (2013).
[CrossRef]

L. Chen, Q. Wang, M. Shen, H. Zhao, Y.-Y. Lin, C.-C. Jeng, R.-K. Lee, and W. Krolikowski, Opt. Lett. 38, 13 (2013).
[CrossRef]

K.-H. Kuo, Y. Y. Lin, R.-K. Lee, and B. A. Malomed, Phys. Rev. A 83, 053838 (2011).
[CrossRef]

Lin, Y. Y.

K.-H. Kuo, Y. Y. Lin, R.-K. Lee, and B. A. Malomed, Phys. Rev. A 83, 053838 (2011).
[CrossRef]

Lin, Y.-Y.

Liu, D.

Y. Du, Z. Zhou, H. Tian, and D. Liu, J. Opt. 13, 015201 (2011).
[CrossRef]

Luther-Davies, B.

W. Krolikowski, M. Saffman, B. Luther-Davies, and C. Denz, Phys. Rev. Lett. 80, 3240 (1998).
[CrossRef]

Malomed, B. A.

K.-H. Kuo, Y. Y. Lin, R.-K. Lee, and B. A. Malomed, Phys. Rev. A 83, 053838 (2011).
[CrossRef]

I. Tikhonenkov, B. A. Malomed, and A. Vardi, Phys. Rev. Lett. 100, 090406 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, L. C. Crasovan, Y. V. Kartashov, L. Torner, and B. A. Malomed, Phys. Rev. E 74, 066614 (2006).
[CrossRef]

Maucher, F.

F. Maucher, S. Skupin, M. Shen, and W. Krolikowski, Phys. Rev. A 81, 063617 (2010).
[CrossRef]

Mazilu, D.

D. Mihalache, D. Mazilu, F. Lederer, L. C. Crasovan, Y. V. Kartashov, L. Torner, and B. A. Malomed, Phys. Rev. E 74, 066614 (2006).
[CrossRef]

Mihalache, D.

D. Mihalache, D. Mazilu, F. Lederer, L. C. Crasovan, Y. V. Kartashov, L. Torner, and B. A. Malomed, Phys. Rev. E 74, 066614 (2006).
[CrossRef]

Morandotti, R.

I. B. Burgess, M. Peccianti, G. Assanto, and R. Morandotti, Phys. Rev. Lett. 102, 203903 (2009).
[CrossRef]

Neshev, D.

N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, Opt. Lett. 29, 286 (2004).
[CrossRef]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B Quantum Semiclass. Opt. 6, S288 (2004).
[CrossRef]

Neshev, D. N.

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef]

Nikolov, N. I.

N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, Opt. Lett. 29, 286 (2004).
[CrossRef]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B Quantum Semiclass. Opt. 6, S288 (2004).
[CrossRef]

Oliver, M. K.

Peccianti, M.

I. B. Burgess, M. Peccianti, G. Assanto, and R. Morandotti, Phys. Rev. Lett. 102, 203903 (2009).
[CrossRef]

M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, Opt. Lett. 27, 1460 (2002).
[CrossRef]

Petersen, D. E.

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef]

Pfau, T.

A. Griesmaier, J. Stuhler, T. Koch, M. Fattori, T. Pfau, and S. Giovanazzi, Phys. Rev. Lett. 97, 250402 (2006).
[CrossRef]

Piccardi, A.

Rasmussen, J. J.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B Quantum Semiclass. Opt. 6, S288 (2004).
[CrossRef]

N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, Opt. Lett. 29, 286 (2004).
[CrossRef]

Rasmussen, P. D.

P. D. Rasmussen, O. Bang, and W. Krolikowski, Phys. Rev. E 72, 066611 (2005).
[CrossRef]

Rotschild, C.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, Nat. Phys. 2, 769 (2006).
[CrossRef]

Saffman, M.

W. Krolikowski, M. Saffman, B. Luther-Davies, and C. Denz, Phys. Rev. Lett. 80, 3240 (1998).
[CrossRef]

Segev, M.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, Nat. Phys. 2, 769 (2006).
[CrossRef]

G. I. Stegeman and M. Segev, Science 286, 1518 (1999).
[CrossRef]

Shalaby, M.

Shen, M.

Q. Kong, M. Shen, Z. Chen, Q. Wang, R.-K. Lee, and W. Krolikowski, Phys. Rev. A 87, 063832 (2013).
[CrossRef]

L. Chen, Q. Wang, M. Shen, H. Zhao, Y.-Y. Lin, C.-C. Jeng, R.-K. Lee, and W. Krolikowski, Opt. Lett. 38, 13 (2013).
[CrossRef]

W. Chen, Q. Kong, M. Shen, Q. Wang, and J. Shi, Phys. Rev. A 87, 013809 (2013).
[CrossRef]

F. Maucher, S. Skupin, M. Shen, and W. Krolikowski, Phys. Rev. A 81, 063617 (2010).
[CrossRef]

Shi, J.

W. Chen, Q. Kong, M. Shen, Q. Wang, and J. Shi, Phys. Rev. A 87, 013809 (2013).
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Figures (2)

Fig. 1.
Fig. 1.

“Effective” interaction potential of two identical dark solitons in media with competing nonlinearities for a few values of the strength of the local quintic nonlinearity, α2. (a) α2=0, soliton interaction in a medium with nonlocal nonlinearity only [16], (b) self-defocusing local quintic nonlinearity with α2=0.3, and (c) self-focusing local quintic nonlinearity with α2=0.2.

Fig. 2.
Fig. 2.

Trajectories of two initially parallel dark solitons interacting in nonlinear media with competing cubic nonlocal and quintic local nonlinearities. The degree of nonlocality, σ, and strength of quintic nonlinearity, α2, are used as control parameters. The initial separation of solitons x0=1.8.

Equations (10)

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iuz+122ux2uR(xξ)|u(ξ,z)|2dξ+α2|u(x,z)|4u=0,
R(x)={12σσxσ,0otherwise.
L=i2(u*uzuu*z)(11|u|2)12|ux|212(|u|21)R(xξ)(|u(ξ,z)|21)dξ+α23(|u|63|u|2+2).
u(x,z)=(Btanhz+iA)(Btanhz+iA),
L=dBdz4B2ADtanh(2Dx0)+4dx0dz[AB+tan1BA]43B2D[4B2+1224Dx0]exp(4Dx0)43B2D+2B4D[csch2(Dσ)coth(Dσ)Dσ]+4B4D[2cosh(2Dσ)(4Dx01)sinh(2Dσ)Dσ+8B2+4B2csch2(Dσ)4B2coth(Dσ)Dσ]exp(4Dx0)+8B43Dα2(14B215)+32B4Dα2(2Dx01)exp(4Dx0).
dAdz=23BD2(24Dx04B212)exp(4Dx0)+2B3[(14Dx0)sinh(2Dσ)Dσ+2cosh(2Dσ)+8B2+4B2csch(Dσ)4B2coth(Dσ)Dσ]exp(4Dx0)+8B4α2(4x03)exp(4Dx0),
coth(Dσ)Dσ[1D2σ2csch2(Dσ)]2α23D2(14B215)=13B2,
dx0dz=AD3BABD[csch2(Dσ)coth(Dσ)Dσ]+AD3B(8B2+1224Dx0)exp(4Dx+0)dDdz12D2tanh(2Dx0)+ABD[2(4Dx01)dDsinh(2Dσ)Dσ4cosh(2Dσ)12B2csch2(Dσ)+12B2coth(Dσ)Dσ24B2]exp(4Dx0)16ABDα2(2Dx01)exp(4Dx0)43ABDα2(12B25).
d2x0dz2={D3BBD[csch2(Dσ)coth(Dσ)Dσ]4B3Dα2(12B25)}dAdz=dV(x0)dx0,
V(x0)={D23B2[csch2(Dσ)coth(Dσ)Dσ]4B23α2(12B25)}{[1B22D2sinh(2Dσ)Dσ+2B3D2α2](1+4Dx0)23(B2+3)6B3D2α2+B22D2[sinh(2Dσ)Dσ+2cosh(2Dσ)+8B2+4B2csch2(Dσ)4B2coth(Dσ)Dσ]}exp(4Dx0).

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