Abstract
We show that the interplay of discrete diffraction in a waveguide array and nonlinear phase modulation evoked by either a quadratic or cubic on-site nonlinearity may lead to the formation of discrete solitons. Beyond the canonical types of solitons there are novel solutions, peculiar for discrete systems, as twisted and flat-top bright solitons as well as asymmetric dark solitons. Stability issues are discussed in detail.
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References
Fermi, E., Pasta, J., and Ulam, S. (1955) Studies of Nonlinear Problems. Los Alamos Report No. LA1940.
Scott, A. C. and Macneil, L. (1983) Binding Energy versus Nonlinearity for a’ small’ Stationary Soliton. Phys. Lett. A 98, 87–90
Eilbeck J. C, Lomdahl P. S., and Scott A. C. (1985) The Discrete Self-Trapping Equation. Physica D 14, 318–338
Sievers A. J. and Takeno S.(1988) Intrinsic Localized Modes in Anharmonic Crystals. Phys. Rev. Lett. 61, 970–973
Page J. B. (1990) Asymptotic Solutions for Localized Vibrational Modes in Strongly Anharmonic Periodic Systems. Phys. Rev. B 41, 7835–7838
Kivshar Y. S. and Campell D. K.(1993) Peierls-Nabarro Potential Barrier for Higly Localized Nonlinear Modes. Phys. Rev. E 48, 3077–3081
Cai D., Bishop A. R., and Gronblech-Jensen N.(1993) Localized States in Discrete Nonlinear Schrödinger Equation. Phys. Rev. Lett. 70, 591–595
Christodoulides D. N. and Joseph R. I. (1988) Discrete Self-Focusing in Nonlinear Arrays of Coupled Waveguides. Opt. Lett. 13, 794–796
Aceves A., De Angelis C, Peschel T., Muschall R., Lederer F., Trillo S., and Wabnitz S. (1996) Discrete Self-Trapping, Soliton Interactions, and Beam Steering in Nonlinear Waveguide Arrays. Phys. Rev. E 53, 1172–1189
Millar P., Aitchison J. S., Kang J.U., Stegeman G. I., Villeneuve A., Kennedy, G. T., and Sibbett W. (1997) Nonlinear Waveguide Arrays in AlGaAs. J. Opt. Soc. Am. B 14, 3224–3231
Eisenberg H. S., Silberberg Y., Morandotti R., Boyd A. R., and Aitchison J. S. (1998) Discrete Spatial Solitons in Waveguide Arrays. Phys. Rev. Lett. 81, 3383–33386
Morandotti, R., Peschel U., Aitchison J. S., Eisenberg H. S., and Silberberg Y. (1999) Dynamics of Discrete Solitons in Optical Waveguide Arrays. Phys. Rev. Lett. 81, 2726–2729
Pertsch T., Dannberg P., Elflein W., Bräuer A., and Lederer F. (1999) Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays. Phys. Rev. Lett. 83, 4752–4755
Morandotti R., Peschel U., Aitchison J. S., Eisenberg H. S., and Silberberg Y. (1999) Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations. Phys. Rev. Lett. 83, 4756–4759
Krolikowski W. and Kivshar Y. S. (1996) Soliton-Based Optical Switching in Waveguide Arrays. J. Opt. Soc. Am. B 13, 876–887
Bang O. and Miller P. (1996) Exploiting Discretness for Switching in Waveguide Arrays. Opt. Lett. 21, 1105–1107
Peschel T., Muschall R., and Lederer F. (1997) Power-Controlled Beam Steering in Nonequidistant Arrays of Nonlinear Waveguides. Opt. Commun. 136, 16–21.
Lederer F. and Aitchison J. S. (1999) Discrete Solitons in Nonlinear Waveguide Arrays. In: Zakharov V.E., Wabnitz S. (Eds.) Optical Solitons, Theoretical Challenges and Industrial Perspectives. Springer, Berlin Heidelberg, 349–365
Darmanyan S., Kobyakov A., and Lederer F. (1998) Stability of Strongly Localized Excitations in Discrete Media with Cubic Nonlinearity. JETP 86, 682–685
Darmanyan S., Kobyakov A., Lederer F., and Vazquez L. (1998) Discrete Fronts and Quasirectangular Solitons. Phys. Rev. B 59, 5994–5997
Darmanyan S., Kobyakov A., Lederer F. (2000) Asymmetic dark solitons. Phys. Rev. Lett, (submitted)
Darmanyan S., Kobyakov A., Schmidt E., and Lederer F. (1998) Strongly Localized Vectorial Modes in Nonlinear Waveguide Arrays. Phys.Rev.E 57, 3520–3529
Kobyakov A., Darmanyan S., Lederer F., and Schmidt E. (1998) Dark Spatial Solitons in Discrete Cubic Media with Self-and Cross-Phase Modulation. Opt.Quantum Electron. 30, 795–808
Darmanyan S., Kobyakov A., and Lederer F. (1998) Strongly Localized Modes in Discrete Systems with Quadratic Nonlinearity. Phys.Rev.E 57, 2344–2349
Kobyakov A., Darmanyan S., Pertsch T., and Lederer F. (1999) Stable Discrete Domain Walls and Quasi-Rectangular Solitons in Quadratically Nonlinear Waveguide Arrays. J. Opt. Soc. Am B 16, 1737–1742
Peschel T., Peschel U., and Lederer F. Discrete Bright Solitary Waves in Quadratically Nonlinear Media. (1998) Phys.Rev.E 57, 1127–1133
Etrich C, Lederer F., Malomed B., Peschel T., and Peschel U. (2000) Optical Solitons in Media with a Quadratic Nonlinearity. Progress in Optics, Elsevier, Amsterdam.
Kivshar Y. S., Peyrard M. (1992) Modulational Instabilities in Discrete Lattices. Phys. Rev. A 46, 3198–3205
Darmanyan S., Relke I., and Lederer F. (1997) Instability of Continuous Waves and Rotating Solitons in Waveguide Arrays. Phys. Rev. E 56, 7662–7668
Miller P. D. and Bang O. (1998) Macroscopic Dynamics in Quadratic Nonlinear Lattices. Phys.Rev.E 57, 6038–6049
Etrich C, Peschel U., Lederer, F., Malomed, B., and Kivshar Y. (1996) Origin of Persistent Oscillations of Solitary Waves in Nonlinear Quadratic Media. Phys. Rev. E 54, 4321–4324
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Lederer, F., Darmanyan, S., Kobyakov, A. (2001). Discrete Solitons in Nonlinear Waveguide Arrays. In: Abdullaev, F., Bang, O., Sørensen, M.P. (eds) Nonlinearity and Disorder: Theory and Applications. NATO Science Series, vol 45. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0542-5_11
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DOI: https://doi.org/10.1007/978-94-010-0542-5_11
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