Summary
In waveguide arrays the interplay of discrete diffraction and nonlinear phase modulation evoked by either quadratic or cubic on-site nonlinearities may lead to the formation of discrete solitons. Domains of existence as well as peculiarities of this particular type of spatial solitons will be identified. It will turn out that in addition to the canonical types of solitons there are novel solutions, peculiar for discrete systems, as twisted and flat-top bright solitons. Stability issues are addressed in detail.
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References
E. Fermi, J. Pasta, S. Ulam. Studies of Nonlinear Problems (Los Alamos National Laboratory, Los Alamos, NM, 1955) Report No. LA1940.
A.C. Scott, L. Macneil, Binding energy versus nonlinearity for a “small” stationary soliton, Phys. Lett. A 98, 87–90 (1983).
J.C. Eilbeck, P.S. Lomdahl, A.C. Scott, The discrete self-trapping equation, Physica D 14. 318–338 (1985).
A.J. Sievers, S. Takeno, Intrinsic localized modes in anharmonic crystals, Phys. Rev. Lett. 61. 970–973 (1988).
J.B. Page, Asymptotic solutions for localized vibrational modes in strongly anharrnonic periodic systems, Phys. Rev. B 41, 7835–7838 (1990).
Y.S. Kivshar, D.K. Campell, Peierls-Nabarro potential barrier for higly localized nonlinear modes, Phys. Rev. E 48, 3077–3081 (1993)
D. Cai, A.R. Bishop, N. Gronblech-Jensen, Localized states in discrete nonlinear Schrödinger equation, Phys. Rev. Lett. 70, 591–595 (1993).
D.N. Christodoulides, R.I. Joseph Discrete self-focusing in nonlinear arrays of coupled waveguides, Opt. Lett. 13, 794–796 (1988).
A. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, S. Wabnitz, Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays, Phys. Rev. E 53, 1172–1189 (1996).
P. Millar, J.S. Aitchison, J.U. Kang, G.I. Stegeman, A. Villeneuve, G.T. Kennedy, W. Sibbett, Nonlinear waveguide arrays in AlGaAs, J. Opt. Soc. Arrr. B 14, 3224–3231 (1997).
H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, J.S. Aitchison, Discrete spatial solitons in waveguide arrays, Phys. Rev. Lett. 81, 3383–3386 (1998).
R. Morandotti, U. Peschel, J.S. Aitchison, H.S. Eisenberg, Y. Silberberg, Dynamics of discrete solitons in optical waveguide arrays, Phys. Rev. Lett. 81, 2726–2729 (1999).
T. Pertsch, P. Dannberg, W. Elflein, A. Brïi,uer, F. Lederer, Optical Bloch oscillations in temperature tuned waveguide arrays, Phys. Rev. Lett. 83, 4752–4755 (1999).
R. Morandotti, U. Peschel, J.S. Aitchison, H.S. Eisenberg, Y. Silberberg, Experimental observation of linear and nonlinear optical Bloch oscillations, Phys. Rev. Lett. 83, 4756–4759 (1999).
W. Krolikowski, Y.S. Kivshar, Soliton-based optical switching in waveguide arrays, J. Opt. Soc. Am. B 13, 876–887 (1996).
O. Bang, P. Miller, Exploiting discretness for switching in waveguide arrays, Opt. Lett. 21, 1105–1107 (1996).
T. Peschel, R. Muschall, F. Lederer, Power-controlled beam steering in nonequidistant arrays of nonlinear waveguides, Opt. Commun. 136, 16–21 (1997).
F. Lederer, J.S. Aitchison, Discrete solitons in nonlinear waveguide arrays, in Optical Solitons, Theoretical Challenges and Industrial Perspectives, ed. by V.E. Zakharov, S. Wabnitz ( Springer, Berlin, Heidelberg, 1999 ) pp. 349–365.
S. Darmanyan, A. Kobyakov, F. Lederer, Stability of strongly localized excitations in discrete media with cubic nonlinearity, JETP 86, 682–685 (1998).
S. Darmanyan, A. Kobyakov, F. Lederer, L. Vazquez, Discrete fronts and quasirectangular solitons, Phys. Rev. B 59, 5994–5997 (1998).
S. Darmanyan, A. Kobyakov, E. Schmidt, F. Lederer, Strongly localized vectorial modes in nonlinear waveguide arrays, Phys. Rev. E 57, 3520–3529 (1998).
A. Kobyakov, S. Darmanyan, F. Lederer, E. Schmidt, Dark spatial solitons in discrete cubic media with self-and cross-phase modulation, Opt. Quantum Electron. 30, 795–808 (1998).
S. Darmanyan, A. Kobyakov, F. Lederer, Strongly localized modes in discrete systems with quadratic nonlinearity, Phys. Rev. E 57, 2344–2349 (1998).
A. Kobyakov, S. Darmanyan, T. Pertsch, F. Lederer, Stable discrete domain walls and quasi-rectangular solitons in quadratically nonlinear waveguide arrays. J. Opt. Soc. Am B 16, 1737–1742 (1999).
T. Peschel, U. Peschel, F. Lederer, Discrete bright solitary waves in quadratically nonlinear media, Phys. Rev. E 57. 1127–1133 (1998).
C. Etrich, F. Lederer, B. Malomed, T. Peschel, U. Peschel, Optical Solitons in Media with a Quadratic Nonlinearity, Progress in Optics ( Elsevier, Amsterdam, 2000 ).
Y.S. Kivshar, M. Peyrard, Modulational instabilities in discrete lattices, Phys. Rev. A 46, 3198–3205 (1992).
S. Darmanyan, I. Relke, F. Lederer, Instability of continuous waves and rotating solitons in waveguide arrays, Phys. Rev. E 56, 7662–7668 (1997).
P.D. Miller, O. Bang, Macroscopic dynamics in quadratic nonlinear lattices, Phys. Rev. E 57, 6038–6049 (1998).
C. Etrich, U. Peschel, F. Lederer, B. Malorned, Y. Kivshar, Origin of persistent oscillations of solitary waves in nonlinear quadratic media, Phys. Rev. E 54, 4321–4324 (1996).
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Lederer, F., Darmanyan, S., Kobyakov, A. (2001). Discrete Solitons. In: Trillo, S., Torruellas, W. (eds) Spatial Solitons. Springer Series in Optical Sciences, vol 82. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44582-1_10
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DOI: https://doi.org/10.1007/978-3-540-44582-1_10
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