Nonlinear Dynamics of Torsion Lattices

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.