Abstract

Using a variational approach, we obtained the interaction potential between two discrete solitons in optical waveguide arrays. The resulting potential bears the two features of soliton-soliton and soliton-waveguide interaction potentials where the former is similar to that of the continuum case and the latter is similar to the effective Pierls-Nabarro potential. The interplay between the two interaction potentials is investigated by studying its effect on the soliton molecule formation. It is found that the two solitons bind if their initial separation equals an odd number of waveguides, while they do not bind if their separation is an even number, which is a consequence of the two solitons being both either at the intersites (unstable) or being onsite (stable). We derived the equations of motion for the solitons’ centre of mass and relative separation and provided analytic solutions for some specific cases. Favourable agreement between the analytical and numerical interaction potentials is obtained. Possible applications of our results to all-optical logic gates are pointed out.

© 2016 Optical Society of America

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References

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  1. V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D. 3, 487–502 (1981)
    [Crossref]
  2. J. P. Gordon, “Interaction forces among solitons in optical fibers,” Opt. Lett. 8, 596–598 (1981)
    [Crossref]
  3. D. Anderson and M. Lisak, “Bandwidth limits due to mutual pulse interaction in optical soliton communication systems,” Opt. Lett. 11, 174–176 (1986)
    [Crossref] [PubMed]
  4. L. F. Mollenauer and J. P. Gordon, Solitons in Optical Fibers (Acadamic, 2006).
  5. U. al Khawaja and H. T. C Stoof, “Formation of matter-wave soliton molecules,” New J. Phys. 13, 085003 (2011).
    [Crossref]
  6. M. Stratmann, T. Pagel, and F. Mitschke, “Experimental Observation of Temporal Soliton Molecules,” Phys. Rev. Lett. 95, 143902 (2005).
    [Crossref] [PubMed]
  7. A. Hause, H. Hartwig, B. Seifert, H. Stolz, M. Böhm, and F. Mitschke, “Phase structure of soliton molecules,” Phys. Rev. A 75, 063836 (2007).
    [Crossref]
  8. A. Hause, H. Hartwig, M. Böhm, and F. Mitschke, “Binding mechanism of temporal soliton molecules,” Phys.Rev. A 78, 063815 (2008).
  9. R. Nath, P. Pedri, and L. Santos, “Soliton-soliton scattering in dipolar Bose-Einstein condensates,” Phys. Rev. A 76, 013606 (2007).
    [Crossref]
  10. B.B. Baizakov, S. M. Al-Marzoug, and H. Bahlouli, “Interaction of solitons in one–dimensional dipolar Bose–Einstein condesnates and formation of soliton molecules,” Phys. Rev. A 92, 033605 (2015).
    [Crossref]
  11. K. E. Strecker, Guthrie B. Partridge, Andrew G. Truscott, and Randall G. Hule, “Formation and propagation of matter-wave soliton trains,” Nature 417, 150–153 (2002).
    [Crossref] [PubMed]
  12. L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
    [Crossref] [PubMed]
  13. Simon L. Cornish, Sarah T. Thompson, and Carl E. Wieman, “Formation of Bright Matter-Wave Solitons during the Collapse of Attractive Bose-Einstein Condensates,” Phys. Rev. Lett. 96, 170401 (2006).
    [Crossref] [PubMed]
  14. U. Al Khawaja, H. T. C. Stoof, R. G. Hulet, K. E. Strecker, and G. B. Partridge, “Bright Soliton Trains of Trapped Bose-Einstein Condensates,” Phys. Rev. Lett. 89, 200404 (2002).
    [Crossref] [PubMed]
  15. Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247 (2011); Erratum Rev. Mod. Phys. 83, 405 (2011).
    [Crossref]
  16. P. G. Kevrekidis, K. Ø. Rasmussen, and A. R. Bishop, “The Discrete Nonlinear Schrodinger Equation: A Survey of Recent Results,” Int. J. Mod. Phys. B 15, 2833–2900 (2001).
    [Crossref]
  17. T. Kapitula, P. G. Kevrekidis, and B. A. Malomed, “Stability of multiple pulses in discrete systems,” Phys. Rev. E63, 036604 (2001).
  18. C. Mejía-Cortés, R. A. Vicencio, and B. A. Malomed, “Mobility of solitons in one-dimensional lattices with the cubic–quintic nonlinearity,” Phys. Rev. E 88, 052901 (2013).
    [Crossref]
  19. A. B. Aceves and et al., “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys.Rev.E 53, 1172 (1996).
  20. R. Carretero-González, J. D. Talley, C. Chong, and B. A. Malomed, “Multistable solitons in the cubic–quintic discrete nonlinear Schrödinger equation,” Physica D 216, 77–89 (2006).
    [Crossref]
  21. I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, “Soliton collisions in the discrete nonlinear Schrödinger equation, ”Phys. Rev. E. 68, 046604 (2003).
    [Crossref]
  22. Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763 (1989).
    [Crossref]
  23. Y. S. Kivshar and D. K. Campbell, “Peierls-Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077–3081 (1993).
    [Crossref]
  24. L. Brizhik, A. Eremko, L. Cruzeiro-Hansson, and Y. Olkhovska, “Soliton dynamics and Peierls-Nabarro barrier in a discrete molecular chain,” Phys. Rev. B 61, 1129 (2000).
    [Crossref]
  25. P. G. Kevrekidis, I. G. Kevrekidis, A. R. Bishop, and E. S. Titi, “Continuum approach to discreteness,” Phys. Rev. E 65, 046613 (2002).
    [Crossref]
  26. U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, “Optical discrete solitons in waveguide arrays. 2. Dynamic properties,” J. Opt. Soc. Am. B 19, 2637–2644 (2002).
    [Crossref]
  27. H. S. Eisenberg, R. Morandotti, Y. Silberberg, J. M. Arnold, G. Pennelli, and J.S. Aitchison, “Optical discrete solitons in waveguide arrays. I. Soliton formation,” J. Opt. Soc. Am. B 19, 2938–2944 (2002).
    [Crossref]
  28. H. He and P. D. Drummond, “Theory of multidimensional parametric band-gap simultons,” Phys. Rev. E 58, 5025–5046 (1998).
    [Crossref]
  29. B. Malomed and M. I. Weinstein, “Soliton dynamics in the discrete nonlinear Schrödinger equation,” Phys. Lett. A 220, 91–96 (1996).
    [Crossref]
  30. B. A. Malomed, in: Progress in Optics43, 71–194 (ed. by E. Wolf, ed. North Holland, 2002).
    [Crossref]
  31. P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation Mathematical Analysis, Numerical Computations and Physical Perspectives (Springer, 2009).
    [Crossref]
  32. Y. S. Kivshar and D. K. Campbell, “Peierls-Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077 (1993).
    [Crossref]
  33. D. J. Kaup, “Variational solutions for the discrete nonlinear Schrödinger equation,” Mathematics and computers in simulation 69, 322–333 (2005).
    [Crossref]
  34. See J. Cuevas and B. A. Malomed in Ref. [31].

2015 (1)

B.B. Baizakov, S. M. Al-Marzoug, and H. Bahlouli, “Interaction of solitons in one–dimensional dipolar Bose–Einstein condesnates and formation of soliton molecules,” Phys. Rev. A 92, 033605 (2015).
[Crossref]

2013 (1)

C. Mejía-Cortés, R. A. Vicencio, and B. A. Malomed, “Mobility of solitons in one-dimensional lattices with the cubic–quintic nonlinearity,” Phys. Rev. E 88, 052901 (2013).
[Crossref]

2011 (2)

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247 (2011); Erratum Rev. Mod. Phys. 83, 405 (2011).
[Crossref]

U. al Khawaja and H. T. C Stoof, “Formation of matter-wave soliton molecules,” New J. Phys. 13, 085003 (2011).
[Crossref]

2008 (1)

A. Hause, H. Hartwig, M. Böhm, and F. Mitschke, “Binding mechanism of temporal soliton molecules,” Phys.Rev. A 78, 063815 (2008).

2007 (2)

R. Nath, P. Pedri, and L. Santos, “Soliton-soliton scattering in dipolar Bose-Einstein condensates,” Phys. Rev. A 76, 013606 (2007).
[Crossref]

A. Hause, H. Hartwig, B. Seifert, H. Stolz, M. Böhm, and F. Mitschke, “Phase structure of soliton molecules,” Phys. Rev. A 75, 063836 (2007).
[Crossref]

2006 (2)

Simon L. Cornish, Sarah T. Thompson, and Carl E. Wieman, “Formation of Bright Matter-Wave Solitons during the Collapse of Attractive Bose-Einstein Condensates,” Phys. Rev. Lett. 96, 170401 (2006).
[Crossref] [PubMed]

R. Carretero-González, J. D. Talley, C. Chong, and B. A. Malomed, “Multistable solitons in the cubic–quintic discrete nonlinear Schrödinger equation,” Physica D 216, 77–89 (2006).
[Crossref]

2005 (2)

D. J. Kaup, “Variational solutions for the discrete nonlinear Schrödinger equation,” Mathematics and computers in simulation 69, 322–333 (2005).
[Crossref]

M. Stratmann, T. Pagel, and F. Mitschke, “Experimental Observation of Temporal Soliton Molecules,” Phys. Rev. Lett. 95, 143902 (2005).
[Crossref] [PubMed]

2003 (1)

I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, “Soliton collisions in the discrete nonlinear Schrödinger equation, ”Phys. Rev. E. 68, 046604 (2003).
[Crossref]

2002 (6)

P. G. Kevrekidis, I. G. Kevrekidis, A. R. Bishop, and E. S. Titi, “Continuum approach to discreteness,” Phys. Rev. E 65, 046613 (2002).
[Crossref]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, “Optical discrete solitons in waveguide arrays. 2. Dynamic properties,” J. Opt. Soc. Am. B 19, 2637–2644 (2002).
[Crossref]

H. S. Eisenberg, R. Morandotti, Y. Silberberg, J. M. Arnold, G. Pennelli, and J.S. Aitchison, “Optical discrete solitons in waveguide arrays. I. Soliton formation,” J. Opt. Soc. Am. B 19, 2938–2944 (2002).
[Crossref]

U. Al Khawaja, H. T. C. Stoof, R. G. Hulet, K. E. Strecker, and G. B. Partridge, “Bright Soliton Trains of Trapped Bose-Einstein Condensates,” Phys. Rev. Lett. 89, 200404 (2002).
[Crossref] [PubMed]

K. E. Strecker, Guthrie B. Partridge, Andrew G. Truscott, and Randall G. Hule, “Formation and propagation of matter-wave soliton trains,” Nature 417, 150–153 (2002).
[Crossref] [PubMed]

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
[Crossref] [PubMed]

2001 (2)

P. G. Kevrekidis, K. Ø. Rasmussen, and A. R. Bishop, “The Discrete Nonlinear Schrodinger Equation: A Survey of Recent Results,” Int. J. Mod. Phys. B 15, 2833–2900 (2001).
[Crossref]

T. Kapitula, P. G. Kevrekidis, and B. A. Malomed, “Stability of multiple pulses in discrete systems,” Phys. Rev. E63, 036604 (2001).

2000 (1)

L. Brizhik, A. Eremko, L. Cruzeiro-Hansson, and Y. Olkhovska, “Soliton dynamics and Peierls-Nabarro barrier in a discrete molecular chain,” Phys. Rev. B 61, 1129 (2000).
[Crossref]

1998 (1)

H. He and P. D. Drummond, “Theory of multidimensional parametric band-gap simultons,” Phys. Rev. E 58, 5025–5046 (1998).
[Crossref]

1996 (2)

B. Malomed and M. I. Weinstein, “Soliton dynamics in the discrete nonlinear Schrödinger equation,” Phys. Lett. A 220, 91–96 (1996).
[Crossref]

A. B. Aceves and et al., “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys.Rev.E 53, 1172 (1996).

1993 (2)

Y. S. Kivshar and D. K. Campbell, “Peierls-Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077 (1993).
[Crossref]

Y. S. Kivshar and D. K. Campbell, “Peierls-Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077–3081 (1993).
[Crossref]

1989 (1)

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763 (1989).
[Crossref]

1986 (1)

1981 (2)

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D. 3, 487–502 (1981)
[Crossref]

J. P. Gordon, “Interaction forces among solitons in optical fibers,” Opt. Lett. 8, 596–598 (1981)
[Crossref]

Aceves, A. B.

A. B. Aceves and et al., “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys.Rev.E 53, 1172 (1996).

Aitchison, J. S.

Aitchison, J.S.

al Khawaja, U.

U. al Khawaja and H. T. C Stoof, “Formation of matter-wave soliton molecules,” New J. Phys. 13, 085003 (2011).
[Crossref]

U. Al Khawaja, H. T. C. Stoof, R. G. Hulet, K. E. Strecker, and G. B. Partridge, “Bright Soliton Trains of Trapped Bose-Einstein Condensates,” Phys. Rev. Lett. 89, 200404 (2002).
[Crossref] [PubMed]

Al-Marzoug, S. M.

B.B. Baizakov, S. M. Al-Marzoug, and H. Bahlouli, “Interaction of solitons in one–dimensional dipolar Bose–Einstein condesnates and formation of soliton molecules,” Phys. Rev. A 92, 033605 (2015).
[Crossref]

Anderson, D.

Arnold, J. M.

Bahlouli, H.

B.B. Baizakov, S. M. Al-Marzoug, and H. Bahlouli, “Interaction of solitons in one–dimensional dipolar Bose–Einstein condesnates and formation of soliton molecules,” Phys. Rev. A 92, 033605 (2015).
[Crossref]

Baizakov, B.B.

B.B. Baizakov, S. M. Al-Marzoug, and H. Bahlouli, “Interaction of solitons in one–dimensional dipolar Bose–Einstein condesnates and formation of soliton molecules,” Phys. Rev. A 92, 033605 (2015).
[Crossref]

Bishop, A. R.

P. G. Kevrekidis, I. G. Kevrekidis, A. R. Bishop, and E. S. Titi, “Continuum approach to discreteness,” Phys. Rev. E 65, 046613 (2002).
[Crossref]

P. G. Kevrekidis, K. Ø. Rasmussen, and A. R. Bishop, “The Discrete Nonlinear Schrodinger Equation: A Survey of Recent Results,” Int. J. Mod. Phys. B 15, 2833–2900 (2001).
[Crossref]

Böhm, M.

A. Hause, H. Hartwig, M. Böhm, and F. Mitschke, “Binding mechanism of temporal soliton molecules,” Phys.Rev. A 78, 063815 (2008).

A. Hause, H. Hartwig, B. Seifert, H. Stolz, M. Böhm, and F. Mitschke, “Phase structure of soliton molecules,” Phys. Rev. A 75, 063836 (2007).
[Crossref]

Bourdel, T.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
[Crossref] [PubMed]

Brizhik, L.

L. Brizhik, A. Eremko, L. Cruzeiro-Hansson, and Y. Olkhovska, “Soliton dynamics and Peierls-Nabarro barrier in a discrete molecular chain,” Phys. Rev. B 61, 1129 (2000).
[Crossref]

Campbell, D. K.

Y. S. Kivshar and D. K. Campbell, “Peierls-Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077–3081 (1993).
[Crossref]

Y. S. Kivshar and D. K. Campbell, “Peierls-Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077 (1993).
[Crossref]

Carr, L. D.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
[Crossref] [PubMed]

Carretero-González, R.

R. Carretero-González, J. D. Talley, C. Chong, and B. A. Malomed, “Multistable solitons in the cubic–quintic discrete nonlinear Schrödinger equation,” Physica D 216, 77–89 (2006).
[Crossref]

Castin, Y.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
[Crossref] [PubMed]

Chong, C.

R. Carretero-González, J. D. Talley, C. Chong, and B. A. Malomed, “Multistable solitons in the cubic–quintic discrete nonlinear Schrödinger equation,” Physica D 216, 77–89 (2006).
[Crossref]

Cornish, Simon L.

Simon L. Cornish, Sarah T. Thompson, and Carl E. Wieman, “Formation of Bright Matter-Wave Solitons during the Collapse of Attractive Bose-Einstein Condensates,” Phys. Rev. Lett. 96, 170401 (2006).
[Crossref] [PubMed]

Cruzeiro-Hansson, L.

L. Brizhik, A. Eremko, L. Cruzeiro-Hansson, and Y. Olkhovska, “Soliton dynamics and Peierls-Nabarro barrier in a discrete molecular chain,” Phys. Rev. B 61, 1129 (2000).
[Crossref]

Cubizolles, J.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
[Crossref] [PubMed]

Drummond, P. D.

H. He and P. D. Drummond, “Theory of multidimensional parametric band-gap simultons,” Phys. Rev. E 58, 5025–5046 (1998).
[Crossref]

Eisenberg, H. S.

Eremko, A.

L. Brizhik, A. Eremko, L. Cruzeiro-Hansson, and Y. Olkhovska, “Soliton dynamics and Peierls-Nabarro barrier in a discrete molecular chain,” Phys. Rev. B 61, 1129 (2000).
[Crossref]

Ferrari, G.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
[Crossref] [PubMed]

Frantzeskakis, D. J.

I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, “Soliton collisions in the discrete nonlinear Schrödinger equation, ”Phys. Rev. E. 68, 046604 (2003).
[Crossref]

Gordon, J. P.

J. P. Gordon, “Interaction forces among solitons in optical fibers,” Opt. Lett. 8, 596–598 (1981)
[Crossref]

L. F. Mollenauer and J. P. Gordon, Solitons in Optical Fibers (Acadamic, 2006).

Hartwig, H.

A. Hause, H. Hartwig, M. Böhm, and F. Mitschke, “Binding mechanism of temporal soliton molecules,” Phys.Rev. A 78, 063815 (2008).

A. Hause, H. Hartwig, B. Seifert, H. Stolz, M. Böhm, and F. Mitschke, “Phase structure of soliton molecules,” Phys. Rev. A 75, 063836 (2007).
[Crossref]

Hause, A.

A. Hause, H. Hartwig, M. Böhm, and F. Mitschke, “Binding mechanism of temporal soliton molecules,” Phys.Rev. A 78, 063815 (2008).

A. Hause, H. Hartwig, B. Seifert, H. Stolz, M. Böhm, and F. Mitschke, “Phase structure of soliton molecules,” Phys. Rev. A 75, 063836 (2007).
[Crossref]

He, H.

H. He and P. D. Drummond, “Theory of multidimensional parametric band-gap simultons,” Phys. Rev. E 58, 5025–5046 (1998).
[Crossref]

Hule, Randall G.

K. E. Strecker, Guthrie B. Partridge, Andrew G. Truscott, and Randall G. Hule, “Formation and propagation of matter-wave soliton trains,” Nature 417, 150–153 (2002).
[Crossref] [PubMed]

Hulet, R. G.

U. Al Khawaja, H. T. C. Stoof, R. G. Hulet, K. E. Strecker, and G. B. Partridge, “Bright Soliton Trains of Trapped Bose-Einstein Condensates,” Phys. Rev. Lett. 89, 200404 (2002).
[Crossref] [PubMed]

Kapitula, T.

T. Kapitula, P. G. Kevrekidis, and B. A. Malomed, “Stability of multiple pulses in discrete systems,” Phys. Rev. E63, 036604 (2001).

Karpman, V. I.

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D. 3, 487–502 (1981)
[Crossref]

Kartashov, Y. V.

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247 (2011); Erratum Rev. Mod. Phys. 83, 405 (2011).
[Crossref]

Kaup, D. J.

D. J. Kaup, “Variational solutions for the discrete nonlinear Schrödinger equation,” Mathematics and computers in simulation 69, 322–333 (2005).
[Crossref]

Kevrekidis, I. G.

P. G. Kevrekidis, I. G. Kevrekidis, A. R. Bishop, and E. S. Titi, “Continuum approach to discreteness,” Phys. Rev. E 65, 046613 (2002).
[Crossref]

Kevrekidis, P. G.

I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, “Soliton collisions in the discrete nonlinear Schrödinger equation, ”Phys. Rev. E. 68, 046604 (2003).
[Crossref]

P. G. Kevrekidis, I. G. Kevrekidis, A. R. Bishop, and E. S. Titi, “Continuum approach to discreteness,” Phys. Rev. E 65, 046613 (2002).
[Crossref]

P. G. Kevrekidis, K. Ø. Rasmussen, and A. R. Bishop, “The Discrete Nonlinear Schrodinger Equation: A Survey of Recent Results,” Int. J. Mod. Phys. B 15, 2833–2900 (2001).
[Crossref]

T. Kapitula, P. G. Kevrekidis, and B. A. Malomed, “Stability of multiple pulses in discrete systems,” Phys. Rev. E63, 036604 (2001).

P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation Mathematical Analysis, Numerical Computations and Physical Perspectives (Springer, 2009).
[Crossref]

Khaykovich, L.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
[Crossref] [PubMed]

Kivshar, Y. S.

Y. S. Kivshar and D. K. Campbell, “Peierls-Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077–3081 (1993).
[Crossref]

Y. S. Kivshar and D. K. Campbell, “Peierls-Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077 (1993).
[Crossref]

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763 (1989).
[Crossref]

Lederer, F.

Lisak, M.

Malomed, B.

B. Malomed and M. I. Weinstein, “Soliton dynamics in the discrete nonlinear Schrödinger equation,” Phys. Lett. A 220, 91–96 (1996).
[Crossref]

Malomed, B. A.

C. Mejía-Cortés, R. A. Vicencio, and B. A. Malomed, “Mobility of solitons in one-dimensional lattices with the cubic–quintic nonlinearity,” Phys. Rev. E 88, 052901 (2013).
[Crossref]

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247 (2011); Erratum Rev. Mod. Phys. 83, 405 (2011).
[Crossref]

R. Carretero-González, J. D. Talley, C. Chong, and B. A. Malomed, “Multistable solitons in the cubic–quintic discrete nonlinear Schrödinger equation,” Physica D 216, 77–89 (2006).
[Crossref]

I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, “Soliton collisions in the discrete nonlinear Schrödinger equation, ”Phys. Rev. E. 68, 046604 (2003).
[Crossref]

T. Kapitula, P. G. Kevrekidis, and B. A. Malomed, “Stability of multiple pulses in discrete systems,” Phys. Rev. E63, 036604 (2001).

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763 (1989).
[Crossref]

B. A. Malomed, in: Progress in Optics43, 71–194 (ed. by E. Wolf, ed. North Holland, 2002).
[Crossref]

Mejía-Cortés, C.

C. Mejía-Cortés, R. A. Vicencio, and B. A. Malomed, “Mobility of solitons in one-dimensional lattices with the cubic–quintic nonlinearity,” Phys. Rev. E 88, 052901 (2013).
[Crossref]

Mitschke, F.

A. Hause, H. Hartwig, M. Böhm, and F. Mitschke, “Binding mechanism of temporal soliton molecules,” Phys.Rev. A 78, 063815 (2008).

A. Hause, H. Hartwig, B. Seifert, H. Stolz, M. Böhm, and F. Mitschke, “Phase structure of soliton molecules,” Phys. Rev. A 75, 063836 (2007).
[Crossref]

M. Stratmann, T. Pagel, and F. Mitschke, “Experimental Observation of Temporal Soliton Molecules,” Phys. Rev. Lett. 95, 143902 (2005).
[Crossref] [PubMed]

Mollenauer, L. F.

L. F. Mollenauer and J. P. Gordon, Solitons in Optical Fibers (Acadamic, 2006).

Morandotti, R.

Nath, R.

R. Nath, P. Pedri, and L. Santos, “Soliton-soliton scattering in dipolar Bose-Einstein condensates,” Phys. Rev. A 76, 013606 (2007).
[Crossref]

Olkhovska, Y.

L. Brizhik, A. Eremko, L. Cruzeiro-Hansson, and Y. Olkhovska, “Soliton dynamics and Peierls-Nabarro barrier in a discrete molecular chain,” Phys. Rev. B 61, 1129 (2000).
[Crossref]

Pagel, T.

M. Stratmann, T. Pagel, and F. Mitschke, “Experimental Observation of Temporal Soliton Molecules,” Phys. Rev. Lett. 95, 143902 (2005).
[Crossref] [PubMed]

Papacharalampous, I. E.

I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, “Soliton collisions in the discrete nonlinear Schrödinger equation, ”Phys. Rev. E. 68, 046604 (2003).
[Crossref]

Partridge, G. B.

U. Al Khawaja, H. T. C. Stoof, R. G. Hulet, K. E. Strecker, and G. B. Partridge, “Bright Soliton Trains of Trapped Bose-Einstein Condensates,” Phys. Rev. Lett. 89, 200404 (2002).
[Crossref] [PubMed]

Partridge, Guthrie B.

K. E. Strecker, Guthrie B. Partridge, Andrew G. Truscott, and Randall G. Hule, “Formation and propagation of matter-wave soliton trains,” Nature 417, 150–153 (2002).
[Crossref] [PubMed]

Pedri, P.

R. Nath, P. Pedri, and L. Santos, “Soliton-soliton scattering in dipolar Bose-Einstein condensates,” Phys. Rev. A 76, 013606 (2007).
[Crossref]

Pennelli, G.

Pertsch, T.

Peschel, U.

Rasmussen, K. Ø.

P. G. Kevrekidis, K. Ø. Rasmussen, and A. R. Bishop, “The Discrete Nonlinear Schrodinger Equation: A Survey of Recent Results,” Int. J. Mod. Phys. B 15, 2833–2900 (2001).
[Crossref]

Salomon, C.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
[Crossref] [PubMed]

Santos, L.

R. Nath, P. Pedri, and L. Santos, “Soliton-soliton scattering in dipolar Bose-Einstein condensates,” Phys. Rev. A 76, 013606 (2007).
[Crossref]

Schreck, F.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
[Crossref] [PubMed]

Seifert, B.

A. Hause, H. Hartwig, B. Seifert, H. Stolz, M. Böhm, and F. Mitschke, “Phase structure of soliton molecules,” Phys. Rev. A 75, 063836 (2007).
[Crossref]

Silberberg, Y.

Solov’ev, V. V.

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D. 3, 487–502 (1981)
[Crossref]

Stolz, H.

A. Hause, H. Hartwig, B. Seifert, H. Stolz, M. Böhm, and F. Mitschke, “Phase structure of soliton molecules,” Phys. Rev. A 75, 063836 (2007).
[Crossref]

Stoof, H. T. C

U. al Khawaja and H. T. C Stoof, “Formation of matter-wave soliton molecules,” New J. Phys. 13, 085003 (2011).
[Crossref]

Stoof, H. T. C.

U. Al Khawaja, H. T. C. Stoof, R. G. Hulet, K. E. Strecker, and G. B. Partridge, “Bright Soliton Trains of Trapped Bose-Einstein Condensates,” Phys. Rev. Lett. 89, 200404 (2002).
[Crossref] [PubMed]

Stratmann, M.

M. Stratmann, T. Pagel, and F. Mitschke, “Experimental Observation of Temporal Soliton Molecules,” Phys. Rev. Lett. 95, 143902 (2005).
[Crossref] [PubMed]

Strecker, K. E.

K. E. Strecker, Guthrie B. Partridge, Andrew G. Truscott, and Randall G. Hule, “Formation and propagation of matter-wave soliton trains,” Nature 417, 150–153 (2002).
[Crossref] [PubMed]

U. Al Khawaja, H. T. C. Stoof, R. G. Hulet, K. E. Strecker, and G. B. Partridge, “Bright Soliton Trains of Trapped Bose-Einstein Condensates,” Phys. Rev. Lett. 89, 200404 (2002).
[Crossref] [PubMed]

Talley, J. D.

R. Carretero-González, J. D. Talley, C. Chong, and B. A. Malomed, “Multistable solitons in the cubic–quintic discrete nonlinear Schrödinger equation,” Physica D 216, 77–89 (2006).
[Crossref]

Thompson, Sarah T.

Simon L. Cornish, Sarah T. Thompson, and Carl E. Wieman, “Formation of Bright Matter-Wave Solitons during the Collapse of Attractive Bose-Einstein Condensates,” Phys. Rev. Lett. 96, 170401 (2006).
[Crossref] [PubMed]

Titi, E. S.

P. G. Kevrekidis, I. G. Kevrekidis, A. R. Bishop, and E. S. Titi, “Continuum approach to discreteness,” Phys. Rev. E 65, 046613 (2002).
[Crossref]

Torner, L.

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247 (2011); Erratum Rev. Mod. Phys. 83, 405 (2011).
[Crossref]

Truscott, Andrew G.

K. E. Strecker, Guthrie B. Partridge, Andrew G. Truscott, and Randall G. Hule, “Formation and propagation of matter-wave soliton trains,” Nature 417, 150–153 (2002).
[Crossref] [PubMed]

Vicencio, R. A.

C. Mejía-Cortés, R. A. Vicencio, and B. A. Malomed, “Mobility of solitons in one-dimensional lattices with the cubic–quintic nonlinearity,” Phys. Rev. E 88, 052901 (2013).
[Crossref]

Weinstein, M. I.

B. Malomed and M. I. Weinstein, “Soliton dynamics in the discrete nonlinear Schrödinger equation,” Phys. Lett. A 220, 91–96 (1996).
[Crossref]

Wieman, Carl E.

Simon L. Cornish, Sarah T. Thompson, and Carl E. Wieman, “Formation of Bright Matter-Wave Solitons during the Collapse of Attractive Bose-Einstein Condensates,” Phys. Rev. Lett. 96, 170401 (2006).
[Crossref] [PubMed]

Int. J. Mod. Phys. B (1)

P. G. Kevrekidis, K. Ø. Rasmussen, and A. R. Bishop, “The Discrete Nonlinear Schrodinger Equation: A Survey of Recent Results,” Int. J. Mod. Phys. B 15, 2833–2900 (2001).
[Crossref]

J. Opt. Soc. Am. B (2)

Mathematics and computers in simulation (1)

D. J. Kaup, “Variational solutions for the discrete nonlinear Schrödinger equation,” Mathematics and computers in simulation 69, 322–333 (2005).
[Crossref]

Nature (1)

K. E. Strecker, Guthrie B. Partridge, Andrew G. Truscott, and Randall G. Hule, “Formation and propagation of matter-wave soliton trains,” Nature 417, 150–153 (2002).
[Crossref] [PubMed]

New J. Phys. (1)

U. al Khawaja and H. T. C Stoof, “Formation of matter-wave soliton molecules,” New J. Phys. 13, 085003 (2011).
[Crossref]

Opt. Lett. (2)

Phys. Lett. A (1)

B. Malomed and M. I. Weinstein, “Soliton dynamics in the discrete nonlinear Schrödinger equation,” Phys. Lett. A 220, 91–96 (1996).
[Crossref]

Phys. Rev. (1)

T. Kapitula, P. G. Kevrekidis, and B. A. Malomed, “Stability of multiple pulses in discrete systems,” Phys. Rev. E63, 036604 (2001).

Phys. Rev. A (3)

A. Hause, H. Hartwig, B. Seifert, H. Stolz, M. Böhm, and F. Mitschke, “Phase structure of soliton molecules,” Phys. Rev. A 75, 063836 (2007).
[Crossref]

R. Nath, P. Pedri, and L. Santos, “Soliton-soliton scattering in dipolar Bose-Einstein condensates,” Phys. Rev. A 76, 013606 (2007).
[Crossref]

B.B. Baizakov, S. M. Al-Marzoug, and H. Bahlouli, “Interaction of solitons in one–dimensional dipolar Bose–Einstein condesnates and formation of soliton molecules,” Phys. Rev. A 92, 033605 (2015).
[Crossref]

Phys. Rev. B (1)

L. Brizhik, A. Eremko, L. Cruzeiro-Hansson, and Y. Olkhovska, “Soliton dynamics and Peierls-Nabarro barrier in a discrete molecular chain,” Phys. Rev. B 61, 1129 (2000).
[Crossref]

Phys. Rev. E (5)

P. G. Kevrekidis, I. G. Kevrekidis, A. R. Bishop, and E. S. Titi, “Continuum approach to discreteness,” Phys. Rev. E 65, 046613 (2002).
[Crossref]

H. He and P. D. Drummond, “Theory of multidimensional parametric band-gap simultons,” Phys. Rev. E 58, 5025–5046 (1998).
[Crossref]

Y. S. Kivshar and D. K. Campbell, “Peierls-Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077–3081 (1993).
[Crossref]

Y. S. Kivshar and D. K. Campbell, “Peierls-Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077 (1993).
[Crossref]

C. Mejía-Cortés, R. A. Vicencio, and B. A. Malomed, “Mobility of solitons in one-dimensional lattices with the cubic–quintic nonlinearity,” Phys. Rev. E 88, 052901 (2013).
[Crossref]

Phys. Rev. E. (1)

I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, “Soliton collisions in the discrete nonlinear Schrödinger equation, ”Phys. Rev. E. 68, 046604 (2003).
[Crossref]

Phys. Rev. Lett. (3)

Simon L. Cornish, Sarah T. Thompson, and Carl E. Wieman, “Formation of Bright Matter-Wave Solitons during the Collapse of Attractive Bose-Einstein Condensates,” Phys. Rev. Lett. 96, 170401 (2006).
[Crossref] [PubMed]

U. Al Khawaja, H. T. C. Stoof, R. G. Hulet, K. E. Strecker, and G. B. Partridge, “Bright Soliton Trains of Trapped Bose-Einstein Condensates,” Phys. Rev. Lett. 89, 200404 (2002).
[Crossref] [PubMed]

M. Stratmann, T. Pagel, and F. Mitschke, “Experimental Observation of Temporal Soliton Molecules,” Phys. Rev. Lett. 95, 143902 (2005).
[Crossref] [PubMed]

Phys.Rev. A (1)

A. Hause, H. Hartwig, M. Böhm, and F. Mitschke, “Binding mechanism of temporal soliton molecules,” Phys.Rev. A 78, 063815 (2008).

Phys.Rev.E (1)

A. B. Aceves and et al., “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys.Rev.E 53, 1172 (1996).

Physica D (1)

R. Carretero-González, J. D. Talley, C. Chong, and B. A. Malomed, “Multistable solitons in the cubic–quintic discrete nonlinear Schrödinger equation,” Physica D 216, 77–89 (2006).
[Crossref]

Physica D. (1)

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D. 3, 487–502 (1981)
[Crossref]

Rev. Mod. Phys. (2)

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763 (1989).
[Crossref]

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247 (2011); Erratum Rev. Mod. Phys. 83, 405 (2011).
[Crossref]

Science (1)

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a Matter-Wave Bright Soliton,” Science 296, 1290–1293 (2002).
[Crossref] [PubMed]

Other (4)

L. F. Mollenauer and J. P. Gordon, Solitons in Optical Fibers (Acadamic, 2006).

B. A. Malomed, in: Progress in Optics43, 71–194 (ed. by E. Wolf, ed. North Holland, 2002).
[Crossref]

P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation Mathematical Analysis, Numerical Computations and Physical Perspectives (Springer, 2009).
[Crossref]

See J. Cuevas and B. A. Malomed in Ref. [31].

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Figures (9)

Fig. 1
Fig. 1 Left: Initial stationary two solitons’ profile as obtained by numerical solution of the DNLSE, Eq. (1), (black lines with dots) and the variational trial function (red line). Right: Spacio-temporal density plot of the evolution of the initial state on the left panel. The parameters used are: γ = 1, d = 0.38, initial separation Δn(0) = 13, P = 2
Fig. 2
Fig. 2 Force (upper), potential (middle), and phase difference (lower panel) between the two solitons considered in Fig. 1.
Fig. 3
Fig. 3 Phase difference between two solitons for three different values of initial separation and nonlinearity strength. The parameters for the lowest curve are: Δni = 13, γ = 1, for the middle curve: Δni = 15, γ = 0.9, and for the upper curve: Δni = 19, γ = 0.7. For all curves: d = 0.38 and P = 2. Blue lines correspond to the numerical calculation and the red lines correspond to the power law fit Δϕ = c/(Δn)4, where c for each curve was determined by the condition (15).
Fig. 4
Fig. 4 Interaction potential between two discrete solitons. Black solid curve corresponds to the full variational calculation, Eq. (7), dashed curve corresponds to the approximate formula Eq. (10), and the red curve corresponds to the numerical solution. Parameters used are: γ = 1, d = 0.5, P = 2.
Fig. 5
Fig. 5 Zoom-in of Fig. 4 in the tail region. Red dashed curve corresponds to the PN potential.
Fig. 6
Fig. 6 Spatio-temporal density plots showing the two solitons time evolution for different initial separations. Parameters used are: γ = 1, d = 0.38, P = 2.
Fig. 7
Fig. 7 Two solitons located initially at the minima of the PN potential (n1 = 54, n2 = 66). We have used γ = 1.0, d = 0.4.
Fig. 8
Fig. 8 Two solitons located at different initial separations and the right soliton in each subfigure was given an initial velocity kick of 0.115. We have used γ = 1.0, d = 0.4.
Fig. 9
Fig. 9 Two solitons located initially at the maxima of the PN potential (n1 = 53.5, n2 = 66.5).γ = 1.0, d = 0.4.

Equations (30)

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i t ψ n + d ( ψ n + 1 + ψ n 1 2 ψ n ) + γ | ψ n | 2 ψ n = 0 ,
L = n = [ i 2 ( ψ n * ψ ˙ n ψ n ψ ˙ n * ) + d ψ n * ( ψ n + 1 + ψ n 1 2 ψ n ) + γ 2 | ψ n | 4 ] ,
E = n = [ d ψ n * ( ψ n + 1 + ψ n 1 2 ψ n ) + γ 2 | ψ n | 4 ] ,
ψ n = A e ( n n 1 ) 2 η 2 + i ϕ 1 + A e ( n n 2 ) 2 η 2 + i ϕ 2 ,
P = n = ψ n * ψ n
A = P 2 π η [ ϑ 3 ( Δ n π / 2 , q ) + ϑ 3 ( 0 , q ) exp ( ( Δ n ) 2 2 η 2 ) cos ( Δ ϕ ) ] ,
V = d E K γ 2 E I ,
E K = 4 A 2 π 2 η [ ϑ 3 ( π ( Δ n 1 ) / 2 , q ) exp ( 1 2 η 2 ) ϑ 3 ( π Δ n / 2 , q ) + ( ϑ 3 ( π / 2 , q ) exp ( 1 2 η 2 ) cosh ( Δ n η 2 ) ϑ 3 ( 0 , q ) ) exp ( ( Δ n ) 2 2 η 2 ) cos ( Δ ϕ ) ]
E I = A 4 π η [ ϑ 3 ( π Δ n / 2 , q ) + ϑ 3 ( 0 , q ) exp ( ( Δ n ) 2 η 2 ) ( 2 + cos ( 2 Δ ϕ ) ) + 4 ϑ 3 ( π Δ n / 4 , q ) exp ( 3 ( Δ n ) 2 4 η 2 ) cos ( Δ ϕ ) ] .
V [ Δ n , Δ ϕ ( Δ n ) ] = V 0 + P 4 π η [ P γ + 8 π η y d ( 1 cosh ( Δ n / η 2 ) ) ] exp ( ( Δ n ) 2 2 η 2 ) cos Δ ϕ + P ( 8 d y + P γ π η ) q cos ( π Δ n )
V 0 = 2 P d ( 1 y ) P 2 γ 4 π η
Δ ϕ ( Δ n ) = c Δ n 4 ,
F = d 2 d t 2 Δ n ( t )
V = Δ n i Δ n f F ( Δ n ) d ( Δ n ) .
c η eq 4 = π 2 .
ψ n = A e ( n n 1 ( t ) ) 2 η ( t ) 2 + i k 1 ( t ) ( n n 1 ( t ) ) + i β ( t ) ( n n 1 ( t ) ) 2 + i ϕ 1 ( t ) + A e ( n n 2 ( t ) ) 2 η ( t ) 2 + i k 2 ( t ) ( n n 2 ( t ) ) + i β ( t ) ( n n 2 ( t ) ) 2 + i ϕ 2 ( t ) .
n 0 ( t ) = n 1 ( t ) + n 2 ( t ) , Δ n ( t ) = n 1 ( t ) n 2 ( t ) , k 0 ( t ) = k 1 ( t ) + k 2 ( t ) , Δ k ( t ) = k 1 ( t ) k 2 ( t ) , ϕ 0 ( t ) = ϕ 1 ( t ) + ϕ 2 ( t ) , Δ ϕ ( t ) = ϕ 1 ( t ) ϕ 2 ( t ) ,
L = P 4 [ ( k 0 n ˙ 0 + Δ k Δ n ˙ η 2 β ˙ 2 ϕ ˙ 0 ) ] V .
V = V s s + V s w
V s s = V 0 [ k 0 , Δ k , β , η ] + P 4 π η [ P γ + 8 π d η exp ( 1 2 η 2 ) ( 1 cosh ( Δ n η 2 ) ) ] exp ( ( Δ n ) 2 2 η 2 ) cos Δ ϕ
V s w = P [ 8 d exp ( 1 2 η 2 ) + P γ π η ] exp ( π 2 η 2 2 ) cos ( π Δ n ) cos ( π n 0 ) ,
V 0 = 2 P d [ 1 exp ( 1 2 η 2 ) cos ( k 0 2 ) cos ( Δ k 2 ) ] P 2 γ 4 π η ,
n ˙ 0 = 2 d exp ( π 2 η 2 2 ) cos ( Δ k 2 ) sin ( k 0 2 ) ,
Δ n ˙ = 2 d exp ( η 2 β 2 2 ) sin ( Δ k 2 ) cos ( k 0 2 ) ,
k ˙ 0 = 4 P d V s w d n 0 ,
Δ k ˙ = 4 P ( d V s s d ( Δ n ) + d V s w d ( Δ n ) ) .
Δ n ¨ = 4 d P d V d ( Δ n ) .
Δ n ¨ = c sin ( π Δ n ) ,
Δ n ( t ) = 2 am ( 1 2 c 1 2 c ( x + c 2 ) 2 , 4 c 2 c c 1 ) ,
n ¨ 0 = 4 d P d V s w d ( Δ n ) .

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