Discrete Breathers in a Square Lattice Based on Delocalized Modes

Abstract

In recent decades, much interest has been shown in nonlinear lattice vibrations because crystalline materials are subjected to high-amplitude impacts in many fields of human activity. One of the effects of nonlinearity in discrete periodic structures is the possibility of existence of spatially localized high-amplitude vibrations, referred to as discrete breathers (DBs), or intrinsic localized modes. The problem of searching for DBs in nonlinear chains (i.e., one-dimensional crystals) can be solved in a fairly simple way, because the variety of possible DBs is small in this case. However, no general approaches to the search for DBs have been developed for high-dimension crystal lattices. Such an approach was derived based on the works by Chechin, Sakhnenko et al., who developed the theory of bushes of nonlinear normal modes, which (as applied to crystals) were later referred to as delocalized nonlinear vibrational modes (DNVMs). It has recently been noted that all known DBs can be obtained by superimposing localizing functions on DNVMs with a frequency beyond the phonon spectrum of the lattice. Since the Chechin and Sakhnenko theory makes it possible to find all possible DNVMs by considering the lattice symmetry, it has become possible to formulate the problem of determining all possible DBs in a given lattice. This approach has recently been applied with success to the search for DBs in a two-dimensional triangular lattice. The purpose of this study is to analyze and describe DBs in a two-dimensional square lattice obtained using a localizing function. As a result, new types of DBs of a square lattice are obtained, including one-dimensional DBs (i.e., those localized only in one of two orthogonal directions) and zero-dimensional DBs (i.e., those localized in two directions).