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A technique for obtaining an approximate breather solution of the Klein–Gordon equation is presented. A breather solution of the equation describing the propagation of nonlinear waves in a graphene-based superlattice is investigated. [ABSTRACT FROM AUTHOR]

A model of a superlattice consisting of alternating strips of single-layer and bilayer graphene is proposed, whose parameters of the energy spectrum can be controlled by changing the external electric field perpendicular to the surface of the sample. Using the Kronig-Penney model, the dispersion equation is obtained based on the analysis of which the energy spectrum of a graphene superlattice is studied depending on the ratio of the strip widths of single-layer and bilayer graphene. For the considered superlattice, it is shown that there are two types of dispersion surfaces corresponding to two branches in the spectrum of bilayer graphene. In the absence of a transverse electric field, neighboring minibands obtained from the solution of different types of the dispersion equation touch at the edges of the first Brillouin band, and the conduction band and the valence band touch in the center of the first Brillouin band of the superlattice. The results of the analytical solution are compared with the results of modeling by methods of the density functional theory. It is shown that the low-energy approximation used to derive the dispersion equation is valid when considering a superlattice with narrow strips of bilayer graphene and wide strips of singlelayer graphene. Under this condition, the dispersion surfaces are symmetrical with respect to the K-point of the inverse space for the basic material – single-layer graphene. Quantum chemical modeling has shown that the band gap in the superlattice spectrum appears even in the absence of a transverse external field due to a violation of symmetry between states in different layers of bilayer graphene in the superlattice, and has confirmed the dependence of the band gap width on the transverse electric field. [ABSTRACT FROM AUTHOR]

The problem of the spontaneous generation of a transverse electric field with a variation in the pulling field in a two-dimensional superlattice with an additive spectrum is considered. It is shown that if the miniband width is the same in the longitudinal and transverse direction, two equiprobable states of the system corresponding to the appearance of a transverse field of different signs appear upon reaching a certain threshold value by the pulling field, which leads to a knee in the longitudinal current–voltage characteristic. If the miniband widths deviate from equality, one of these states becomes more energetically favorable, but the system can transfer into the second state under a certain pulsed effect, which leads to a jump-like change in the longitudinal current in the superlattice. [ABSTRACT FROM AUTHOR]

Existence of the soliton solution to the generalized sine–Gordon equation (also known as the Kryuchkov–Kukhar' equation) is numerically studied. The equation can be used to describe propagation of electromagnetic waves in a graphene-based superlattice. Calculation errors related to implicit representation of the kink solution to the equation under study are estimated. Variations in the shapes of kinks that move in opposite directions are studied prior to and after collision. The results show that the kink solution is not a soliton. [ABSTRACT FROM AUTHOR]