Floquet spectrum of charge carriers in a 2D crystal with initially displaced Dirac points has been derived. The phase and amplitude dependences of the energy gap induced by elliptically polarized and bichromatic high-frequency fields has been investigated. In contrast to graphene the linearly polarized electric field has been shown to be able to transform the initially semi-metallic state of Dirac crystal into the Floquet-insulator state. The conditions for such a transition are indicated, one of which is the mismatch between the orientation of the field polarization line and the direction of the crystallographic axes.

Multi-mode dynamics with Zitterbewegung of an electron in 2D Dirac crystal placed in the field of monochromatic radiation is studied. For calculations a model Hamiltonian taking into account two independent Dirac points has been used. Calculations have shown that the spectrum of electron oscillations contains a series of new (compared to the usual Zitterbewegung) frequencies. The latter, in the case of a high radiation frequency, are a combination of the Zitterbewegung frequency and frequencies that are multiples of the field frequency. In the case when the field frequency is comparable to the Zitterbewegung frequency, the spectrum of electron oscillations is determined by the field amplitude. The character of this dependence has been shown to be changed by changing of the direction of radiation polarization. The possibility of the appearance of a constant component of the electron velocity in the field of monochromatic radiation is also discussed.

Dispersion relation for plasma waves in graphene bilayer has been investigated. Influence of the bias voltage on the dispersion curve for plasmon in bigraphene has been studied within random phase approximation. The possibility of controlling of energy and group velocity for plasmon by changing of bias voltage has been shown. The dependence of plasmon energy on the bias voltage has been predicted to have the nonmonotonous character. Effect of the temperature on the plasmon dispersion has been analyzed.

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The possibility of the propagation of a solitary electromagnetic wave in bigraphene, between the layers of which a potential difference is applied, is studied. The condition is found under which the formation of such a wave is possible. The dependence of the solitary-wave shape on the potential difference between graphene layers is studied. As possible solitary-wave identification, the electric charge carried away by the wave in the direction of its propagation is calculated.