Discrete Ginzburg-Landau solitons

Nikolaos K. Efremidis; Demetrios N. Christodoulides

doi:10.1103/PhysRevE.67.026606

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Discrete Ginzburg-Landau solitons

Nikolaos K. Efremidis and Demetrios N. Christodoulides

School of Optics/CREOL, University of Central Florida, Orlando, Florida 32816-2700

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Phys. Rev. E 67, 026606 – Published 12 February, 2003

DOI: https://doi.org/10.1103/PhysRevE.67.026606

Abstract

We demonstrate that discrete solitons are possible in Ginzburg-Landau lattices. As a result of discreteness, we find that this system exhibits a host of features that have no counterpart whatsoever in either the continuous limit or in other conservative discrete models.

References (29)

M.C. Cross and P.C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer, Berlin, 1984).
I.S. Aranson and L. Kramer, Rev. Mod. Phys. 74, 99 (2002).
N.N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams (Chapman and Hall, London, 1997).
P. Manneville, Dissipative Structures and Weak Turbulence (Academic, San Diego, 1990).
L.M. Hocking and K. Stewartson, Proc. R. Soc. London, Ser. A 326, 289 (1972); N.R. Pereira and L. Stenflo, Phys. Fluids 20, 1733 (1977).
K. Nozaki and N. Bekki, J. Phys. Soc. Jpn. 53, 1581 (1984).
W. van Saarloos and P.C. Hohenberg, Physica D 56, 303 (1992).
R. Conte and M. Musette, Physica D 69, 1 (1993); ibid.P. Marcq, H. Chaté, and R. Conte, 73, 305 (1994).
N.N. Akhmediev, V.V. Afanasjev, and J.M. Soto Crespo, Phys. Rev. E 53, 1190 (1996).
L.C. Crasovan, B.A. Malomed, and D. Mihalache, Phys. Lett. A 289, 59 (2001).
J.M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, Phys. Rev. Lett. 85, 2937 (2000).
P.S. Hagan, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 42, 762 (1982).
L.C. Crasovan, B.A. Malomed, and D. Mihalache, Phys. Rev. E 63, 016605 (2001).
H. Willaime, O. Cardoso, and P. Tabeling, Phys. Rev. Lett. 67, 3247 (1991).
S.S. Wang and H.G. Winful, Appl. Phys. Lett. 52, 1774 (1988); ibid.H.G. Winful and S.S. Wang, 53, 1894 (1988).
K. Otsuka, Nonlinear Dynamics in Optical Complex Systems (KTK Scientific Publishers, Tokyo, 1999).
K. Otsuka, Phys. Rev. Lett. 65, 329 (1990); ibid.H.G. Winful and L. Rahman, 65, 1575 (1990).
D.N. Christodoulides and R.I. Joseph, Opt. Lett. 13, 794 (1988).
H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, and J.S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1986).
B.P. Anderson and M.A. Kasevich, Science 282, 1686 (1998).
A. Trombettoni and A. Smerzi, Phys. Rev. Lett. 86, 2353 (2001).
B. Kneer, T. Wong, K. Vogel, W.P. Schleich, and D.F. Walls, Phys. Rev. A 58, 4841 (1998).
F.T. Arecchi, J. Bragard, and L.M. Castellano, in Bose-Einstein Condensates and Atom Lasers, edited by S. Martellucci, A.N. Chester, A. Aspect, and M. Inguscio (Kluwer, New York, 2002).
It is often useful to consider Eq. (1) in the so-called continuous limit at the base (edge) of the Brillouin zone, where it can be approximately written as ${iU}_{z} - i ε^{'} U \pm α U_{xx} + p | U |^{2} U + q | U |^{4} U = 0,$ where $ε^{'} = ε \mp 2 α_{i},$ respectively.
Y.S. Kivshar, Opt. Lett. 18, 1147 (1993).
S. Darmanyan, A. Kobyakov, and F. Lederer, Zh. Éksp. Teor. Fiz. 113, 1253 1998 [Sov. Phys. JETP 86, 682 (1998)]; P.G. Kevrekidis, A.R. Bishop, and K.O. Rasmussen, Phys. Rev. E 63, 036603 (2001).
N.K. Efremidis and D.N. Christodoulides, Phys. Rev. E 65, 056607 (2002).

[c1] M.C. Cross and P.C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).

[c2] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer, Berlin, 1984).

[c3] I.S. Aranson and L. Kramer, Rev. Mod. Phys. 74, 99 (2002).

[c4] N.N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams (Chapman and Hall, London, 1997).

[c5] P. Manneville, Dissipative Structures and Weak Turbulence (Academic, San Diego, 1990).

[c6] L.M. Hocking and K. Stewartson, Proc. R. Soc. London, Ser. A 326, 289 (1972); N.R. Pereira and L. Stenflo, Phys. Fluids 20, 1733 (1977).

[c7] K. Nozaki and N. Bekki, J. Phys. Soc. Jpn. 53, 1581 (1984).

[c8] W. van Saarloos and P.C. Hohenberg, Physica D 56, 303 (1992).

[c9] R. Conte and M. Musette, Physica D 69, 1 (1993); ibid.P. Marcq, H. Chaté, and R. Conte, 73, 305 (1994).

[c10] N.N. Akhmediev, V.V. Afanasjev, and J.M. Soto Crespo, Phys. Rev. E 53, 1190 (1996).

[c11] L.C. Crasovan, B.A. Malomed, and D. Mihalache, Phys. Lett. A 289, 59 (2001).

[c12] J.M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, Phys. Rev. Lett. 85, 2937 (2000).

[c13] P.S. Hagan, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 42, 762 (1982).

[c14] L.C. Crasovan, B.A. Malomed, and D. Mihalache, Phys. Rev. E 63, 016605 (2001).

[c15] H. Willaime, O. Cardoso, and P. Tabeling, Phys. Rev. Lett. 67, 3247 (1991).

[c16] S.S. Wang and H.G. Winful, Appl. Phys. Lett. 52, 1774 (1988); ibid.H.G. Winful and S.S. Wang, 53, 1894 (1988).

[c17] K. Otsuka, Nonlinear Dynamics in Optical Complex Systems (KTK Scientific Publishers, Tokyo, 1999).

[c18] K. Otsuka, Phys. Rev. Lett. 65, 329 (1990); ibid.H.G. Winful and L. Rahman, 65, 1575 (1990).

[c19] D.N. Christodoulides and R.I. Joseph, Opt. Lett. 13, 794 (1988).

[c20] H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, and J.S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).

[c21] C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1986).

[c22] B.P. Anderson and M.A. Kasevich, Science 282, 1686 (1998).

[c23] A. Trombettoni and A. Smerzi, Phys. Rev. Lett. 86, 2353 (2001).

[c24] B. Kneer, T. Wong, K. Vogel, W.P. Schleich, and D.F. Walls, Phys. Rev. A 58, 4841 (1998).

[c25] F.T. Arecchi, J. Bragard, and L.M. Castellano, in Bose-Einstein Condensates and Atom Lasers, edited by S. Martellucci, A.N. Chester, A. Aspect, and M. Inguscio (Kluwer, New York, 2002).

[c26] It is often useful to consider Eq. (1) in the so-called continuous limit at the base (edge) of the Brillouin zone, where it can be approximately written as ${iU}_{z} - i ε^{'} U \pm α U_{xx} + p | U |^{2} U + q | U |^{4} U = 0,$ where $ε^{'} = ε \mp 2 α_{i},$ respectively.

[c27] Y.S. Kivshar, Opt. Lett. 18, 1147 (1993).

[c28] S. Darmanyan, A. Kobyakov, and F. Lederer, Zh. Éksp. Teor. Fiz. 113, 1253 1998 [Sov. Phys. JETP 86, 682 (1998)]; P.G. Kevrekidis, A.R. Bishop, and K.O. Rasmussen, Phys. Rev. E 63, 036603 (2001).

[c29] N.K. Efremidis and D.N. Christodoulides, Phys. Rev. E 65, 056607 (2002).

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