Abstract

We analyze the stability of a recently found exact analytical spatial soliton in binary waveguide arrays—an analog of the relativistic Dirac soliton. We demonstrate that this soliton class is very robust. The soliton dynamics and different scenarios of soliton interactions are systematically investigated.

© 2014 Optical Society of America

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  1. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
    [CrossRef]
  2. T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
    [CrossRef]
  3. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
    [CrossRef]
  4. M. Ghulinyan, C. J. Oton, Z. Gaburro, L. Pavesi, C. Toninelli, and D. S. Wiersma, “Zener tunneling of light waves in an optical superlattice,” Phys. Rev. Lett. 94, 127401 (2005).
    [CrossRef]
  5. H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual observation of Zener tunneling,” Phys. Rev. Lett. 96, 023901 (2006).
    [CrossRef]
  6. S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
    [CrossRef]
  7. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
    [CrossRef]
  8. Tr. X. Tran and F. Biancalana, “Diffractive resonant radiation emitted by spatial solitons in waveguide arrays,” Phys. Rev. Lett. 110, 113903 (2013).
    [CrossRef]
  9. Tr. X. Tran and F. Biancalana, “Mimicking the nonlinear dynamics of optical fibers with waveguide arrays: towards a spatiotemporal supercontinuum generation,” Opt. Express 21, 17539–17546 (2013).
    [CrossRef]
  10. Tr. X. Tran, D. C. Duong, and F. Biancalana, “Supercontinuum generation in both frequency and wave number domains in nonlinear waveguide arrays,” Phys. Rev. A 89, 013826 (2014).
    [CrossRef]
  11. S. Longhi, “Klein tunneling in binary photonic superlattices,” Phys. Rev. B 81, 075102 (2010).
    [CrossRef]
  12. F. Dreisow, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Klein tunneling of light in waveguide superlattices,” Europhys. Lett. 97, 10008 (2012).
    [CrossRef]
  13. S. Longhi, “Photonic analog of Zitterbewegung in binary waveguide arrays,” Opt. Lett. 35, 235–237 (2010).
    [CrossRef]
  14. F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Classical simulation of relativistic Zitterbewegung in photonic lattices,” Phys. Rev. Lett. 105, 143902 (2010).
    [CrossRef]
  15. S. Longhi, “Classical simulation of relativistic quantum mechanics in periodic optical structures,” Appl. Phys. B 104, 453–468 (2011).
    [CrossRef]
  16. J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109, 023602 (2012).
    [CrossRef]
  17. Tr. X. Tran, S. Longhi, and F. Biancalana, “Optical analogue of relativistic Dirac solitons in binary waveguide arrays,” Ann. Phys. 340, 179–187 (2014).
    [CrossRef]
  18. Y. Nogami, F. M. Toyama, and Z. Zhao, “Nonlinear Dirac soliton in an external field,” J. Phys. A 28, 1413–1424 (1995).
    [CrossRef]
  19. A. A. Sukhorukov and Y. S. Kivshar, “Discrete gap solitons in modulated waveguide arrays,” Opt. Lett. 27, 2112–2114 (2002).
    [CrossRef]
  20. A. A. Sukhorukov and Y. S. Kivshar, “Generation and stability of discrete gap solitons,” Opt. Lett. 28, 2345–2347 (2003).
    [CrossRef]
  21. M. Conforti, C. De Angelis, and T. R. Akylas, “Energy localization and transport in binary waveguide arrays,” Phys. Rev. A 83, 043822 (2011).
    [CrossRef]
  22. R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov, and Y. S. Kivshar, “Observation of discrete gap solitons in binary waveguide arrays,” Opt. Lett. 29, 2890–2892 (2004).
    [CrossRef]
  23. W. Heisenberg, “Quantum theory of fields and elementary particles,” Rev. Mod. Phys. 29, 269–278 (1957).
    [CrossRef]
  24. D. C. Ionescu, R. Reinhardt, B. Muller, and W. Greiner, “Nonlinear extensions of the Dirac equation and their implications in QED,” Phys. Rev. A 38, 616–620 (1988).
    [CrossRef]
  25. A. Zecca, “Dirac equation in space-time with torsion,” Int. J. Theor. Phys. 41, 421–428 (2002).
    [CrossRef]
  26. M. J. Esteban and E. Séré, “An overview on linear and nonlinear Dirac equations,” Discrete Contin. Dyn. Syst. 8, 381–397 (2002).
    [CrossRef]
  27. I. Bialynicki-Birula and J. Mycielski, “Nonlinear wave mechanics,” Ann. Phys. 100, 62–93 (1976).
    [CrossRef]
  28. L. H. Haddad and L. D. Carr, “The nonlinear Dirac equation in Bose-Einstein condensates: foundation and symmetries,” Physica D 238, 1413–1421 (2009).
    [CrossRef]
  29. L. H. Haddad and L. D. Carr, “Relativistic linear stability equations for the nonlinear Dirac equation in Bose-Einstein condensates,” Europhys. Lett. 94, 56002 (2011).
    [CrossRef]
  30. N. Kemmer, “Zur theorie der neutron-proton wechselwirkung,” Helv. Phys. Acta 10, 47–67 (1937).
  31. E. Fermi and C. N. Yang, “Are mesons elementary particles?” Phys. Rev. 76, 1739–1743 (1949).
    [CrossRef]
  32. G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2013).
  33. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fiber to Photonic Crystals, 5th ed. (Academic, 2003).
  34. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).
  35. N. N. Rosanov and Tr. X. Tran, “Interaction of dissipative Bragg solitons in active nonlinear fibers,” Chaos 17, 037114 (2007).
    [CrossRef]

2014

Tr. X. Tran, D. C. Duong, and F. Biancalana, “Supercontinuum generation in both frequency and wave number domains in nonlinear waveguide arrays,” Phys. Rev. A 89, 013826 (2014).
[CrossRef]

Tr. X. Tran, S. Longhi, and F. Biancalana, “Optical analogue of relativistic Dirac solitons in binary waveguide arrays,” Ann. Phys. 340, 179–187 (2014).
[CrossRef]

2013

Tr. X. Tran and F. Biancalana, “Diffractive resonant radiation emitted by spatial solitons in waveguide arrays,” Phys. Rev. Lett. 110, 113903 (2013).
[CrossRef]

Tr. X. Tran and F. Biancalana, “Mimicking the nonlinear dynamics of optical fibers with waveguide arrays: towards a spatiotemporal supercontinuum generation,” Opt. Express 21, 17539–17546 (2013).
[CrossRef]

2012

J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109, 023602 (2012).
[CrossRef]

F. Dreisow, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Klein tunneling of light in waveguide superlattices,” Europhys. Lett. 97, 10008 (2012).
[CrossRef]

2011

S. Longhi, “Classical simulation of relativistic quantum mechanics in periodic optical structures,” Appl. Phys. B 104, 453–468 (2011).
[CrossRef]

M. Conforti, C. De Angelis, and T. R. Akylas, “Energy localization and transport in binary waveguide arrays,” Phys. Rev. A 83, 043822 (2011).
[CrossRef]

L. H. Haddad and L. D. Carr, “Relativistic linear stability equations for the nonlinear Dirac equation in Bose-Einstein condensates,” Europhys. Lett. 94, 56002 (2011).
[CrossRef]

2010

S. Longhi, “Photonic analog of Zitterbewegung in binary waveguide arrays,” Opt. Lett. 35, 235–237 (2010).
[CrossRef]

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Classical simulation of relativistic Zitterbewegung in photonic lattices,” Phys. Rev. Lett. 105, 143902 (2010).
[CrossRef]

S. Longhi, “Klein tunneling in binary photonic superlattices,” Phys. Rev. B 81, 075102 (2010).
[CrossRef]

2009

L. H. Haddad and L. D. Carr, “The nonlinear Dirac equation in Bose-Einstein condensates: foundation and symmetries,” Physica D 238, 1413–1421 (2009).
[CrossRef]

2008

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[CrossRef]

2007

N. N. Rosanov and Tr. X. Tran, “Interaction of dissipative Bragg solitons in active nonlinear fibers,” Chaos 17, 037114 (2007).
[CrossRef]

2006

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual observation of Zener tunneling,” Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef]

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef]

2005

M. Ghulinyan, C. J. Oton, Z. Gaburro, L. Pavesi, C. Toninelli, and D. S. Wiersma, “Zener tunneling of light waves in an optical superlattice,” Phys. Rev. Lett. 94, 127401 (2005).
[CrossRef]

2004

2003

2002

A. A. Sukhorukov and Y. S. Kivshar, “Discrete gap solitons in modulated waveguide arrays,” Opt. Lett. 27, 2112–2114 (2002).
[CrossRef]

A. Zecca, “Dirac equation in space-time with torsion,” Int. J. Theor. Phys. 41, 421–428 (2002).
[CrossRef]

M. J. Esteban and E. Séré, “An overview on linear and nonlinear Dirac equations,” Discrete Contin. Dyn. Syst. 8, 381–397 (2002).
[CrossRef]

1999

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[CrossRef]

1995

Y. Nogami, F. M. Toyama, and Z. Zhao, “Nonlinear Dirac soliton in an external field,” J. Phys. A 28, 1413–1424 (1995).
[CrossRef]

1988

D. C. Ionescu, R. Reinhardt, B. Muller, and W. Greiner, “Nonlinear extensions of the Dirac equation and their implications in QED,” Phys. Rev. A 38, 616–620 (1988).
[CrossRef]

1976

I. Bialynicki-Birula and J. Mycielski, “Nonlinear wave mechanics,” Ann. Phys. 100, 62–93 (1976).
[CrossRef]

1972

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

1957

W. Heisenberg, “Quantum theory of fields and elementary particles,” Rev. Mod. Phys. 29, 269–278 (1957).
[CrossRef]

1949

E. Fermi and C. N. Yang, “Are mesons elementary particles?” Phys. Rev. 76, 1739–1743 (1949).
[CrossRef]

1937

N. Kemmer, “Zur theorie der neutron-proton wechselwirkung,” Helv. Phys. Acta 10, 47–67 (1937).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2013).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fiber to Photonic Crystals, 5th ed. (Academic, 2003).

Aitchison, J. S.

R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov, and Y. S. Kivshar, “Observation of discrete gap solitons in binary waveguide arrays,” Opt. Lett. 29, 2890–2892 (2004).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[CrossRef]

Akylas, T. R.

M. Conforti, C. De Angelis, and T. R. Akylas, “Energy localization and transport in binary waveguide arrays,” Phys. Rev. A 83, 043822 (2011).
[CrossRef]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[CrossRef]

Avidan, A.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Bialynicki-Birula, I.

I. Bialynicki-Birula and J. Mycielski, “Nonlinear wave mechanics,” Ann. Phys. 100, 62–93 (1976).
[CrossRef]

Biancalana, F.

Tr. X. Tran, S. Longhi, and F. Biancalana, “Optical analogue of relativistic Dirac solitons in binary waveguide arrays,” Ann. Phys. 340, 179–187 (2014).
[CrossRef]

Tr. X. Tran, D. C. Duong, and F. Biancalana, “Supercontinuum generation in both frequency and wave number domains in nonlinear waveguide arrays,” Phys. Rev. A 89, 013826 (2014).
[CrossRef]

Tr. X. Tran and F. Biancalana, “Mimicking the nonlinear dynamics of optical fibers with waveguide arrays: towards a spatiotemporal supercontinuum generation,” Opt. Express 21, 17539–17546 (2013).
[CrossRef]

Tr. X. Tran and F. Biancalana, “Diffractive resonant radiation emitted by spatial solitons in waveguide arrays,” Phys. Rev. Lett. 110, 113903 (2013).
[CrossRef]

Bräuer, A.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual observation of Zener tunneling,” Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

Carr, L. D.

L. H. Haddad and L. D. Carr, “Relativistic linear stability equations for the nonlinear Dirac equation in Bose-Einstein condensates,” Europhys. Lett. 94, 56002 (2011).
[CrossRef]

L. H. Haddad and L. D. Carr, “The nonlinear Dirac equation in Bose-Einstein condensates: foundation and symmetries,” Physica D 238, 1413–1421 (2009).
[CrossRef]

Christodoulides, D. N.

J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109, 023602 (2012).
[CrossRef]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[CrossRef]

R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov, and Y. S. Kivshar, “Observation of discrete gap solitons in binary waveguide arrays,” Opt. Lett. 29, 2890–2892 (2004).
[CrossRef]

Cianci, E.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef]

Conforti, M.

M. Conforti, C. De Angelis, and T. R. Akylas, “Energy localization and transport in binary waveguide arrays,” Phys. Rev. A 83, 043822 (2011).
[CrossRef]

Dannberg, P.

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

De Angelis, C.

M. Conforti, C. De Angelis, and T. R. Akylas, “Energy localization and transport in binary waveguide arrays,” Phys. Rev. A 83, 043822 (2011).
[CrossRef]

Dreisow, F.

J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109, 023602 (2012).
[CrossRef]

F. Dreisow, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Klein tunneling of light in waveguide superlattices,” Europhys. Lett. 97, 10008 (2012).
[CrossRef]

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Classical simulation of relativistic Zitterbewegung in photonic lattices,” Phys. Rev. Lett. 105, 143902 (2010).
[CrossRef]

Duong, D. C.

Tr. X. Tran, D. C. Duong, and F. Biancalana, “Supercontinuum generation in both frequency and wave number domains in nonlinear waveguide arrays,” Phys. Rev. A 89, 013826 (2014).
[CrossRef]

Efremidis, N. K.

J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109, 023602 (2012).
[CrossRef]

Eisenberg, H. S.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[CrossRef]

Elflein, W.

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

Esteban, M. J.

M. J. Esteban and E. Séré, “An overview on linear and nonlinear Dirac equations,” Discrete Contin. Dyn. Syst. 8, 381–397 (2002).
[CrossRef]

Fermi, E.

E. Fermi and C. N. Yang, “Are mesons elementary particles?” Phys. Rev. 76, 1739–1743 (1949).
[CrossRef]

Foglietti, V.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef]

Gaburro, Z.

M. Ghulinyan, C. J. Oton, Z. Gaburro, L. Pavesi, C. Toninelli, and D. S. Wiersma, “Zener tunneling of light waves in an optical superlattice,” Phys. Rev. Lett. 94, 127401 (2005).
[CrossRef]

Ghulinyan, M.

M. Ghulinyan, C. J. Oton, Z. Gaburro, L. Pavesi, C. Toninelli, and D. S. Wiersma, “Zener tunneling of light waves in an optical superlattice,” Phys. Rev. Lett. 94, 127401 (2005).
[CrossRef]

Greiner, W.

D. C. Ionescu, R. Reinhardt, B. Muller, and W. Greiner, “Nonlinear extensions of the Dirac equation and their implications in QED,” Phys. Rev. A 38, 616–620 (1988).
[CrossRef]

Haddad, L. H.

L. H. Haddad and L. D. Carr, “Relativistic linear stability equations for the nonlinear Dirac equation in Bose-Einstein condensates,” Europhys. Lett. 94, 56002 (2011).
[CrossRef]

L. H. Haddad and L. D. Carr, “The nonlinear Dirac equation in Bose-Einstein condensates: foundation and symmetries,” Physica D 238, 1413–1421 (2009).
[CrossRef]

Heinrich, M.

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Classical simulation of relativistic Zitterbewegung in photonic lattices,” Phys. Rev. Lett. 105, 143902 (2010).
[CrossRef]

Heisenberg, W.

W. Heisenberg, “Quantum theory of fields and elementary particles,” Rev. Mod. Phys. 29, 269–278 (1957).
[CrossRef]

Ionescu, D. C.

D. C. Ionescu, R. Reinhardt, B. Muller, and W. Greiner, “Nonlinear extensions of the Dirac equation and their implications in QED,” Phys. Rev. A 38, 616–620 (1988).
[CrossRef]

Keil, R.

J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109, 023602 (2012).
[CrossRef]

F. Dreisow, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Klein tunneling of light in waveguide superlattices,” Europhys. Lett. 97, 10008 (2012).
[CrossRef]

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Classical simulation of relativistic Zitterbewegung in photonic lattices,” Phys. Rev. Lett. 105, 143902 (2010).
[CrossRef]

Kemmer, N.

N. Kemmer, “Zur theorie der neutron-proton wechselwirkung,” Helv. Phys. Acta 10, 47–67 (1937).

Kivshar, Y. S.

Lahini, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Laporta, P.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef]

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[CrossRef]

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual observation of Zener tunneling,” Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

Lobino, M.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef]

Longhi, S.

Tr. X. Tran, S. Longhi, and F. Biancalana, “Optical analogue of relativistic Dirac solitons in binary waveguide arrays,” Ann. Phys. 340, 179–187 (2014).
[CrossRef]

F. Dreisow, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Klein tunneling of light in waveguide superlattices,” Europhys. Lett. 97, 10008 (2012).
[CrossRef]

S. Longhi, “Classical simulation of relativistic quantum mechanics in periodic optical structures,” Appl. Phys. B 104, 453–468 (2011).
[CrossRef]

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Classical simulation of relativistic Zitterbewegung in photonic lattices,” Phys. Rev. Lett. 105, 143902 (2010).
[CrossRef]

S. Longhi, “Photonic analog of Zitterbewegung in binary waveguide arrays,” Opt. Lett. 35, 235–237 (2010).
[CrossRef]

S. Longhi, “Klein tunneling in binary photonic superlattices,” Phys. Rev. B 81, 075102 (2010).
[CrossRef]

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef]

Mandelik, D.

Marangoni, M.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef]

Michaelis, D.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual observation of Zener tunneling,” Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef]

Morandotti, R.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov, and Y. S. Kivshar, “Observation of discrete gap solitons in binary waveguide arrays,” Opt. Lett. 29, 2890–2892 (2004).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[CrossRef]

Muller, B.

D. C. Ionescu, R. Reinhardt, B. Muller, and W. Greiner, “Nonlinear extensions of the Dirac equation and their implications in QED,” Phys. Rev. A 38, 616–620 (1988).
[CrossRef]

Mycielski, J.

I. Bialynicki-Birula and J. Mycielski, “Nonlinear wave mechanics,” Ann. Phys. 100, 62–93 (1976).
[CrossRef]

Nogami, Y.

Y. Nogami, F. M. Toyama, and Z. Zhao, “Nonlinear Dirac soliton in an external field,” J. Phys. A 28, 1413–1424 (1995).
[CrossRef]

Nolte, S.

J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109, 023602 (2012).
[CrossRef]

F. Dreisow, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Klein tunneling of light in waveguide superlattices,” Europhys. Lett. 97, 10008 (2012).
[CrossRef]

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Classical simulation of relativistic Zitterbewegung in photonic lattices,” Phys. Rev. Lett. 105, 143902 (2010).
[CrossRef]

Oton, C. J.

M. Ghulinyan, C. J. Oton, Z. Gaburro, L. Pavesi, C. Toninelli, and D. S. Wiersma, “Zener tunneling of light waves in an optical superlattice,” Phys. Rev. Lett. 94, 127401 (2005).
[CrossRef]

Pavesi, L.

M. Ghulinyan, C. J. Oton, Z. Gaburro, L. Pavesi, C. Toninelli, and D. S. Wiersma, “Zener tunneling of light waves in an optical superlattice,” Phys. Rev. Lett. 94, 127401 (2005).
[CrossRef]

Pertsch, T.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual observation of Zener tunneling,” Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

Peschel, U.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual observation of Zener tunneling,” Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[CrossRef]

Pozzi, F.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Ramponi, R.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef]

Reinhardt, R.

D. C. Ionescu, R. Reinhardt, B. Muller, and W. Greiner, “Nonlinear extensions of the Dirac equation and their implications in QED,” Phys. Rev. A 38, 616–620 (1988).
[CrossRef]

Rosanov, N. N.

N. N. Rosanov and Tr. X. Tran, “Interaction of dissipative Bragg solitons in active nonlinear fibers,” Chaos 17, 037114 (2007).
[CrossRef]

Segev, M.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[CrossRef]

Séré, E.

M. J. Esteban and E. Séré, “An overview on linear and nonlinear Dirac equations,” Discrete Contin. Dyn. Syst. 8, 381–397 (2002).
[CrossRef]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[CrossRef]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov, and Y. S. Kivshar, “Observation of discrete gap solitons in binary waveguide arrays,” Opt. Lett. 29, 2890–2892 (2004).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[CrossRef]

Sorel, M.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov, and Y. S. Kivshar, “Observation of discrete gap solitons in binary waveguide arrays,” Opt. Lett. 29, 2890–2892 (2004).
[CrossRef]

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[CrossRef]

Streppel, U.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual observation of Zener tunneling,” Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef]

Sukhorukov, A. A.

Szameit, A.

J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109, 023602 (2012).
[CrossRef]

F. Dreisow, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Klein tunneling of light in waveguide superlattices,” Europhys. Lett. 97, 10008 (2012).
[CrossRef]

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Classical simulation of relativistic Zitterbewegung in photonic lattices,” Phys. Rev. Lett. 105, 143902 (2010).
[CrossRef]

Toninelli, C.

M. Ghulinyan, C. J. Oton, Z. Gaburro, L. Pavesi, C. Toninelli, and D. S. Wiersma, “Zener tunneling of light waves in an optical superlattice,” Phys. Rev. Lett. 94, 127401 (2005).
[CrossRef]

Toyama, F. M.

Y. Nogami, F. M. Toyama, and Z. Zhao, “Nonlinear Dirac soliton in an external field,” J. Phys. A 28, 1413–1424 (1995).
[CrossRef]

Tran, Tr. X.

Tr. X. Tran, S. Longhi, and F. Biancalana, “Optical analogue of relativistic Dirac solitons in binary waveguide arrays,” Ann. Phys. 340, 179–187 (2014).
[CrossRef]

Tr. X. Tran, D. C. Duong, and F. Biancalana, “Supercontinuum generation in both frequency and wave number domains in nonlinear waveguide arrays,” Phys. Rev. A 89, 013826 (2014).
[CrossRef]

Tr. X. Tran and F. Biancalana, “Diffractive resonant radiation emitted by spatial solitons in waveguide arrays,” Phys. Rev. Lett. 110, 113903 (2013).
[CrossRef]

Tr. X. Tran and F. Biancalana, “Mimicking the nonlinear dynamics of optical fibers with waveguide arrays: towards a spatiotemporal supercontinuum generation,” Opt. Express 21, 17539–17546 (2013).
[CrossRef]

N. N. Rosanov and Tr. X. Tran, “Interaction of dissipative Bragg solitons in active nonlinear fibers,” Chaos 17, 037114 (2007).
[CrossRef]

Trompeter, H.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual observation of Zener tunneling,” Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef]

Tünnermann, A.

F. Dreisow, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Klein tunneling of light in waveguide superlattices,” Europhys. Lett. 97, 10008 (2012).
[CrossRef]

J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109, 023602 (2012).
[CrossRef]

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Classical simulation of relativistic Zitterbewegung in photonic lattices,” Phys. Rev. Lett. 105, 143902 (2010).
[CrossRef]

Wiersma, D. S.

M. Ghulinyan, C. J. Oton, Z. Gaburro, L. Pavesi, C. Toninelli, and D. S. Wiersma, “Zener tunneling of light waves in an optical superlattice,” Phys. Rev. Lett. 94, 127401 (2005).
[CrossRef]

Yang, C. N.

E. Fermi and C. N. Yang, “Are mesons elementary particles?” Phys. Rev. 76, 1739–1743 (1949).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Zecca, A.

A. Zecca, “Dirac equation in space-time with torsion,” Int. J. Theor. Phys. 41, 421–428 (2002).
[CrossRef]

Zeuner, J. M.

J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109, 023602 (2012).
[CrossRef]

Zhao, Z.

Y. Nogami, F. M. Toyama, and Z. Zhao, “Nonlinear Dirac soliton in an external field,” J. Phys. A 28, 1413–1424 (1995).
[CrossRef]

Ann. Phys.

Tr. X. Tran, S. Longhi, and F. Biancalana, “Optical analogue of relativistic Dirac solitons in binary waveguide arrays,” Ann. Phys. 340, 179–187 (2014).
[CrossRef]

I. Bialynicki-Birula and J. Mycielski, “Nonlinear wave mechanics,” Ann. Phys. 100, 62–93 (1976).
[CrossRef]

Appl. Phys. B

S. Longhi, “Classical simulation of relativistic quantum mechanics in periodic optical structures,” Appl. Phys. B 104, 453–468 (2011).
[CrossRef]

Chaos

N. N. Rosanov and Tr. X. Tran, “Interaction of dissipative Bragg solitons in active nonlinear fibers,” Chaos 17, 037114 (2007).
[CrossRef]

Discrete Contin. Dyn. Syst.

M. J. Esteban and E. Séré, “An overview on linear and nonlinear Dirac equations,” Discrete Contin. Dyn. Syst. 8, 381–397 (2002).
[CrossRef]

Europhys. Lett.

L. H. Haddad and L. D. Carr, “Relativistic linear stability equations for the nonlinear Dirac equation in Bose-Einstein condensates,” Europhys. Lett. 94, 56002 (2011).
[CrossRef]

F. Dreisow, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Klein tunneling of light in waveguide superlattices,” Europhys. Lett. 97, 10008 (2012).
[CrossRef]

Helv. Phys. Acta

N. Kemmer, “Zur theorie der neutron-proton wechselwirkung,” Helv. Phys. Acta 10, 47–67 (1937).

Int. J. Theor. Phys.

A. Zecca, “Dirac equation in space-time with torsion,” Int. J. Theor. Phys. 41, 421–428 (2002).
[CrossRef]

J. Phys. A

Y. Nogami, F. M. Toyama, and Z. Zhao, “Nonlinear Dirac soliton in an external field,” J. Phys. A 28, 1413–1424 (1995).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rep.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[CrossRef]

Phys. Rev.

E. Fermi and C. N. Yang, “Are mesons elementary particles?” Phys. Rev. 76, 1739–1743 (1949).
[CrossRef]

Phys. Rev. A

D. C. Ionescu, R. Reinhardt, B. Muller, and W. Greiner, “Nonlinear extensions of the Dirac equation and their implications in QED,” Phys. Rev. A 38, 616–620 (1988).
[CrossRef]

M. Conforti, C. De Angelis, and T. R. Akylas, “Energy localization and transport in binary waveguide arrays,” Phys. Rev. A 83, 043822 (2011).
[CrossRef]

Tr. X. Tran, D. C. Duong, and F. Biancalana, “Supercontinuum generation in both frequency and wave number domains in nonlinear waveguide arrays,” Phys. Rev. A 89, 013826 (2014).
[CrossRef]

Phys. Rev. B

S. Longhi, “Klein tunneling in binary photonic superlattices,” Phys. Rev. B 81, 075102 (2010).
[CrossRef]

Phys. Rev. Lett.

F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Classical simulation of relativistic Zitterbewegung in photonic lattices,” Phys. Rev. Lett. 105, 143902 (2010).
[CrossRef]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[CrossRef]

M. Ghulinyan, C. J. Oton, Z. Gaburro, L. Pavesi, C. Toninelli, and D. S. Wiersma, “Zener tunneling of light waves in an optical superlattice,” Phys. Rev. Lett. 94, 127401 (2005).
[CrossRef]

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual observation of Zener tunneling,” Phys. Rev. Lett. 96, 023901 (2006).
[CrossRef]

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96, 243901 (2006).
[CrossRef]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Tr. X. Tran and F. Biancalana, “Diffractive resonant radiation emitted by spatial solitons in waveguide arrays,” Phys. Rev. Lett. 110, 113903 (2013).
[CrossRef]

J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109, 023602 (2012).
[CrossRef]

Physica D

L. H. Haddad and L. D. Carr, “The nonlinear Dirac equation in Bose-Einstein condensates: foundation and symmetries,” Physica D 238, 1413–1421 (2009).
[CrossRef]

Rev. Mod. Phys.

W. Heisenberg, “Quantum theory of fields and elementary particles,” Rev. Mod. Phys. 29, 269–278 (1957).
[CrossRef]

Sov. Phys. JETP

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Other

G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2013).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fiber to Photonic Crystals, 5th ed. (Academic, 2003).

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Figures (4)

Fig. 1.
Fig. 1.

Formation of the Dirac solitons during the propagation of beams, which are initially different from the exact Dirac solitons. (a), (b) Propagation of beams with parameter r=1.15 and 0.85, respectively. (c), (d) The input and output beam profiles with parameter r=1.15 and 0.85, respectively. Parameters: κ=1; γ=1; σ=1.2; and beam width parameter n0=5.

Fig. 2.
Fig. 2.

Interaction of two Dirac solitons. (a)–(c) Interaction of two Dirac solitons that are initially identical with initial phase difference θ=0,π/2, and π, respectively. (d) Interaction of one Dirac soliton with another in-phase Dirac quasi-soliton with r=1.1. Parameters: Δn=22; the other ones are the same as in Fig. 1.

Fig. 3.
Fig. 3.

Interaction of three Dirac solitons. (a) Solitons are initially identical. (b) Initial peak amplitude decreases monotonically from the first to third quasi-solitons: r1=1.05; r2=1.0; r3=0.95. (c) Peak amplitude of the central soliton is initially smaller than that of two quasi-solitons on wings: r1=1.05; r2=1.0; r3=1.05. Solitons are in-phase in (a)–(c). (d) The first two solitons have the same amplitude at the beginning with beam width parameter n0=5, but the width of the third soliton is reduced by a factor of 0.9 as compared to that of the first two solitons. The second and third solitons are in-phase, but out-of-phase with the first soliton. Solitons on wings are initially equidistant from the central soliton in (a)–(c), but have different initial center-to-center distances in (d). Parameters: κ=1; γ=1; σ=1.2; beam width parameter n0=5 for quasi-solitons in (a)–(c).

Fig. 4.
Fig. 4.

(a), (b) Interaction of four and five Dirac solitons, respectively. (c) The input and output soliton profiles taken from (a) with the initial parameter r=1.1, 1.05, 1.0, 0.95 from the first to fourth quasi-solitons. (d) The input and output soliton profiles taken from (b) with the initial parameter r=1.15, 1.1, 1.05, 1.0, 0.95 from the first to fifth quasi-solitons. All quasi-solitons are initially in-phase, spacing between two adjacent quasi-solitons is 44 waveguides, the other parameters are the same as in Fig. 1.

Equations (2)

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

idan(z)dz=κ[an+1(z)+an1(z)]+(1)nσanγ|an(z)|2an(z),
[a2n(z)a2n1(z)]=[i2n2κn0σγsech(2nn0)eiz(2κ2n02σσ)i2n2κ2n02σσγsech(2n1n0)tanh(2n1n0)eiz(2κ2n02σσ)].

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