DISCRETE BREATHERS IN A SQUARE LATTICE BASED ON DELOCALIZED MODES

10.25712/ASTU.1811-1416.2023.03.002

Abstract

Interest in nonlinear lattice vibrations has increased in recent decades due to the fact that crystalline materials are subjected to high-amplitude effects in many areas of human activity. One of the effects of nonlinearity in discrete periodic structures is the possibility of the existence of large-amplitude oscillations localized in space, called discrete breathers (DBs) (or intrinsic localized modes). The problem of searching for DBs in nonlinear chains, that is, in one-dimensional crystals, is quite simply solved, since the variety of possible DBs in this case is small. However, for high-dimensional crystal lattices, no general approaches to the search for discrete breathers have been developed so far. This approach appeared due to the work of Chechin, Sakhnenko et al., where the theory of bushes of nonlinear normal modes was developed, which later, as applied to crystals, began to be called delocalized nonlinear vibrational modes (DNVM). Relatively recently, it was noticed that all known DBs can be obtained by superimposing localizing functions on DNVM with a frequency outside the phonon spectrum of the lattice. Since the Chechin and Sakhnenko theory makes it possible to find all possible DNVMs by considering the symmetry of the lattice, it became possible to formulate the problem of finding all possible DBs in a given lattice. This approach has recently been successfully applied to the search for DBs in a two-dimensional triangular lattice. The study and description of discrete breathers in a two-dimensional square crystal lattice obtained using a localized function is the subject of this article. As a result, new types of DBs of a square lattice were obtained, including one-dimensional, that is, localized only in one of two orthogonal directions, and zero-dimensional, that is, localized in two directions.