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    Nonlinear Dynamics of Torsion Lattices

    2018, Vol. 14, no. 2, pp.  179-193

    Author(s): Smirnov V. V., Kovaleva M. A., Manevitch L. I.

    We present an analysis of torsion oscillations in quasi-one-dimensional lattices with periodic potentials of the nearest neighbor interaction. A one-dimensional chain of point dipoles (spins) under an external field and without the latter is the simplest realization of such a system. We obtained dispersion relations for the nonlinear normal modes for a wide range of oscillation amplitudes and wave numbers. The features of the short wavelength part of the spectrum at large-amplitude oscillations are discussed. The problem of localized excitations near the edges of the spectrum is studied by the asymptotic method. We show that the localized oscillations (breathers) appear near the long wavelength edge, while the short wavelength edge of the spectrum contains only dark solitons. The continuum limit of the dynamic equations leads to a generalization of the nonlinear Schrödinger equation and can be considered as a complex representation of the sine-Gordon equation.
    Keywords: essentially nonlinear systems, coupled pendulums, nonlinear normal modes, limiting phase trajectories
    Citation: Smirnov V. V., Kovaleva M. A., Manevitch L. I., Nonlinear Dynamics of Torsion Lattices, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 2, pp.  179-193

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    [1] Stroscio, M. and Dutta, M., Phonons in Nanostructures, Cambridge Univ. Press, Cambridge, 2001, 274 pp.
    [2] Mukherjee, P. K., “Phase Transitions among the Rotator Phases of the Normal Alkanes: A Review”, Phys. Rep., 588 (2015), 1–54  crossref  mathscinet  adsnasa
    [3] Volkenstein, V. M., Configuration Statistics of Polymeric Chains, Akad. Nauk, Moscow, 1959, 466 pp. (Russian)
    [4] Braun, O. M. and Kivshar, Yu. S., The Frenkel – Kontorova Model: Concepts, Methods, and Applications, Springer, Berlin, 2004, XVIII+472 pp.  mathscinet  zmath
    [5] J. Cuevas-Maraver, P. Kevrekidis, F. Williams (eds.), The sine-Gordon Model and Its Applications: From Pendula and Josephson Junctions to Gravity and High-Energy Physics, Springer, Cham, 2014, XIII+263 pp.  mathscinet  zmath
    [6] Takeno, Sh. and Homma, Sh., “A sine-Lattice (sine-Form Discrete sine-Gordon) Equation: One- and Two-Kink Solutions and Physical Models”, J. Phys. Soc. Japan, 55:1 (1986), 65–75  crossref  mathscinet  adsnasa
    [7] Yomosa, S., “Soliton Excitations in Deoxyribonucleic Acid (DNA) Double Helices”, Phys. Rev. A, 27:4 (1983), 2120–2125  crossref  mathscinet  adsnasa
    [8] Manevitch, L. I. and Smirnov, V. V., “Limiting Phase Trajectories and the Origin of Energy Localization in Nonlinear Oscillatory Chains”, Phys. Rev. E, 82:3 (2010), 036602, 9 pp.  crossref  mathscinet  adsnasa
    [9] Smirnov, V. V. and Manevich, L. I., “Limiting Phase Trajectories and Dynamic Transitions in Nonlinear Periodic Systems”, Acoust. Phys., 57:2 (2011), 271–276  crossref  adsnasa
    [10] Sagdeev, R.Ż., Usikov, D. A., and Zaslavsky, G. M., Nonlinear Physics: From the Pendulum to Turbulence and Chaos, Harwood Acad. Publ., Chur, 1990, 675 pp.  mathscinet
    [11] Takeno, Sh. and Peyrard, M., “Nonlinear Rotating Modes: Green’s-Function Solution”, Phys. Rev. E, 55:2 (1997), 1922–1928  crossref  adsnasa
    [12] Dauxois, Th. and Peyrard, M., Physics of Solitons, Cambridge Univ. Press, Cambridge, 2010, 436 pp.  mathscinet  zmath
    [13] Lichtenberg A.,Livi R., Pettini M., Ruffo S., “Dynamics of Oscillator Chains”, The Fermi – Pasta – Ulam Problem: A Status Report, Lect. Notes Phys., 728, ed. G. Gallavotti, Springer, Berlin, 2008, 21–121  crossref  mathscinet  zmath  adsnasa
    [14] Scott, A., Nonlinear Science: Emergence and Dynamics of Coherent Structures, 2nd ed., Oxford Univ. Press, New York, 2003, 504 pp.  mathscinet  zmath
    [15] Kivshar, Yu. S. and Luther-Davies, B., “Dark Optical Solitons: Physics and Applications”, Phys. Rep., 298:2–3 (1998), 81–197  crossref  mathscinet  adsnasa

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