We consider a discrete two-dimensional model of a crystal with particles having rotational degrees of freedom. We derive the equations of motion and analyze its continuum analog obtained in the long-wave limit. The continuum equations are shown to be the ones of the micropolar elasticity theory. The conditions when the micropolar elasticity equations can be reduced to the equations of conventional elasticity theory are discussed. We show that the rotational degrees of freedom are responsible for the anomalies in the elastic properties of some of the dielectric crystals.

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We derive the multi-field, micropolar-type continuum theory for the two-dimensional model of crystal having finite-size particles. Continuum theories are usually valid for waves with wavelength much larger than the size of primitive cell of crystal. By comparison of the dispersion relations, it is demonstrated that in contrast to the single-field continuum theory constructed in our previous paper the multi-field generalization is valid not only for long but also for short waves. We show that the multi-field model can be used to describe spatially localized short- and long wavelength distortions. Short-wave external fields of forces and torques can be also naturally taken into account by the multi-field continuum theory.Comment: 14 pages, 4 figures, submitted to International Journal of Solids and Structure

We construct a two-field higher-order gradient micropolar model for Cosserat media on the basis of a square lattice of elements with rotational degrees of freedom. This model includes equations of single-field higher-order gradient micropolar theory, and additional ones, which allow modelling of short wavelength phenomena. We demonstrate an example of short wavelength spatially localized static deformations in a structural system, which could not be obtained in the classical single-field framework, but which are captured by the proposed two-field model.

The three-soliton solutions to the sine-Gordon equation describing the collision of a kink with a breather or with a kink-antikink pair are given and several separatrix three-soliton solutions are extracted from these solutions. The influence of a small perturbation on the threesoliton collisions is studied numerically. As the perturbed system the Frenkel-Kontorova model with a small degree of discreteness is considered. We show that in the three-soliton collisions, in the presence of small perturbation, energy exchange between solitons can take place. The degree of inelasticity of a three-soliton collision is extremely sensitive to the parameters of the collision.